INDEX

S. No Topic Page No
Week 1
1 Introduction Fuzzy Sets, Logic and Systems & Applications 1
2 Introduction Fuzzy Sets, Logic and Systems & Applications 17
3 Fuzzy Sets and Fuzzy Logic Toolbox in MATLAB - I 38
4 Fuzzy Sets and Fuzzy Logic Toolbox in MATLAB - II 48
5 Membership Functions 62
Week 2
6 Membership Functions-II 83
7 Nomenclatures used in Fuzzy Set Theory-I 103
8 Nomenclatures used in Fuzzy Set Theory-II 117
9 Nomenclatures used in Fuzzy Set Theory-III 129
10 Set Theoretic Operations on Fuzzy Sets-I 145
Week 3
11 Set Theoretic Operations on Fuzzy Sets-II 160
12 Properties of Fuzzy Sets-I 171
13 Properties of Fuzzy Sets-II 187
14 Properties of Fuzzy Sets-III 215
15 Properties of Fuzzy Sets-IV 234
Week 4
16 Properties of Fuzzy Sets-V 261
17 Distance between Fuzzy Sets-I 301
18 Distance between Fuzzy Sets-II 312
19 Distance between Fuzzy Sets-III 327
20 Arithmetic Operations on Fuzzy Numbers-I 353
Week 5
21 Arithmetic Operations on Fuzzy Numbers-II 367
22 Arithmetic Operations on Fuzzy Numbers-III 387
23 Complement of Fuzzy Sets 408
24 T-norm Operators 426
25 S-norm Operators 443
 Week 6
26 Parameterized T-Norm Operators 459
27 Parameterized S-Norm Operators 477
28 Fuzzy Relation-I 493
29 Fuzzy Relation-II 511
30 Operations on Crisp and Fuzzy Relations 526
Week 7
31 Projection of Fuzzy Relation Set 547
32 Cylindrical Extension of Fuzzy Set 558
33 Properties of Fuzzy Relation-I 571
34 Properties of Fuzzy Relation-II 589
35 Extension Principle 615
Week 8
36 Composition of Fuzzy Relations 631
37 Properties of Composition of Fuzzy Relations 642
38 Fuzzy Tolerance and Equivalence Relations-I 663
39 Fuzzy Tolerance and Equivalence Relations-II 674
40 Fuzzy Tolerance and Equivalence Relations-III 683
Week 9
41 Linguistic Hedges 693
42 Linguistic Hedges and Negation/ Complement and Connectives 705
43 Concentration and Dilation & Composite Linguistic Term and 717
44 Dilation and Composite Linguistic Term and Some Examples 728
45 Some Examples on Composite Linguistic Terms 743
Week 10
46 Contrast Intensification of Fuzzy Sets 752
47 Orthogonality of Fuzzy Sets 763
48 Fuzzy Rules and Fuzzy Reasoning-I 776
49 Fuzzy Rules and Fuzzy Reasoning-II 790
50 Fuzzy Inference System 801
Week 11
51 Mamdani Fuzzy Model-I 824
52 Mamdani Fuzzy Model-II 836
53 Mamdani Fuzzy Model-III 852
54 Example on Mamdani Fuzzy Model for Single Antecedent with Three Rules 871
55 Example on Mamdani Fuzzy Model for Two Antecedents with Four Rules 886
Week 12
56 Larsen Fuzzy Model-I 897
57 Larsen Fuzzy Model-II 911
58 Larsen Fuzzy Model-III 924
59 Tsukamoto Fuzzy Model 942
60 TSK Fuzzy Model 968
 Fuzzy Sets, Logic and Systems & Applications
Prof. Nishchal K. Verma
Department of Electrical Engineering
Indian Institute of Technology, Kanpur

Lecture - 01
Introduction: Fuzzy Sets, Logic and Systems & Applications

So, welcome to the first lecture of the course on Fuzzy Sets Logic and Systems and
Applications. This lecture is based on the Introduction of Fuzzy Logic and then the little bit
of introduction to artificial intelligence and I will try here to relate fuzzy logic with Artificial
Intelligence. So, before I move to that part, I would like to tell you that fuzzy logic is a multi
valued logic and of course, as I have mentioned already that there is a linkage of fuzzy
system with artificial intelligence.

So, or in other words I would like to tell you that the fuzzy systems is one of the very key
agents of artificial intelligence. So, when we talk of artificial intelligence, or machine
intelligence in other words let me briefly define what is artificial intelligence.

(Refer Slide Time: 01:41)

So, artificial intelligence is nothing but it is a discipline which involves all sort of
mechanisms, algorithms that deal with mimicking the activities of our brain.

1
(Refer Slide Time: 01:56)

I would like to go to brief history of artificial intelligence, since we are going finally, to study
fuzzy systems and as I mentioned that there is a linkage of fuzzy systems with the artificial
intelligence. So, it is necessary to have our brief history of artificial intelligence. So, artificial
intelligence starts with the Allen’s Universal Turing Machine it is here and it was the time in
1936 - 37. So, around that it was proposed and with this the beginning of artificial
intelligence is seen here and then in 1942 - 43 Warren McCulloch and Walter Pitts created a
computational model for neural networks and that was also called threshold logic.

So, this was the time when artificial neural network started. So, here it was basically a
preposition of the ANN model and it was based on the biological neuron and then in 1950 a
Turing Test was proposed, in 1955 the formal name artificial intelligence has come up and
this name was given by John McCarthy. As it’s written here that in 1955 the John McCarthy
founding father of artificial intelligence has coined the word artificial intelligence.

In 1957 a perceptron model was introduced. So, a perceptron model is nothing, but it is again
artificial neuron model, the difference here is just the activation function. So, perceptron
model is ANN model with activation function as binary linear.

2
(Refer Slide Time: 04:18)

And then in 1960’s the genetic algorithm was proposed, in 1965 it is very important to note
here that in 1965 fuzzy logic was proposed by professor Lotfi A Zadeh. And this is a time
when the deep learning term which is very very relevant, very very popular term being used
nowadays, it was coined in 1965 by Evancho and Lapa. So, I would say here that this year
there were two main concepts were proposed, first concept was fuzzy logic and then the deep
learning. And these two have a very high correlation.

In 1970’s evolutionary computing was proposed. So, various algorithms of evolutionary

computing were proposed and then 1980’s witnessed neural computing swarm intelligence
and then 1990’s hybrid models of these like neuro fuzzy systems, neuro fuzzy genetic, fuzzy
genetic like that the models were proposed. So, in nutshell I would say the fuzzy neuro
genetic all these were used together to give a better model performance and these were
proposed when studied.

3
(Refer Slide Time: 06:01)

Beyond 90’s the research areas based on all of these agents were helpful in giving rise to
various models, various systems. For example, if systems, evolutionary computing, data
mining, simulated annealing, particle swarm algorithm, deep neural networks, deep fuzzy
networks etcetera.

(Refer Slide Time: 06:33)

So, the artificial intelligence which started which was seen right from, I would say the birth
of artificial intelligence is seen around 1936 – 37, and it you know with the advent of all
these agents for example, fuzzy systems, neural network, genetic algorithm and many more

4
which I am I will be describing in due course of time. So, we can call these are the agents of
artificial intelligence, like fuzzy systems, artificial neural systems, evolutionary systems.

And in evolutionary algorithms we mainly cover genetic algorithms differential evolution and
then in evolutionary systems we have meta heuristic and swarm intelligence. Under these we
have ant colony optimization, Bees Algorithm, Bat Algorithm, Cuckoo Algorithm, Harmony
search, Firefly Algorithm, Artificial Immune Systems, Particle Swarm Optimization.

(Refer Slide Time: 07:54)

And then as agents of artificial intelligence again we have probabilistic systems, for example,
Bayesian networks, Gaussian mixture models, hidden Marco models which is not mentioned
here and then we have as agent of AI, we have chaos theory simulated annealing, rough set
theory, support vector machines and there are many more agents which I have not been
mentioned here.

5
(Refer Slide Time: 08:25)

So, fuzzy system is also one of the key agents of computational intelligence. So,
computational intelligence is an equivalent name of artificial intelligence, these two names go
hand in hand and I would like to just briefly tell you the definition of a computational
intelligence.

(Refer Slide Time: 08:53)

So, Computational Intelligence basically a set of nature inspired computational

methodologies and approaches to address complex real world problems to which
conventional mathematical or traditional modelling can be useless.

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For a few reasons, like the processes might be too complex, the processes for which we are
developing the model that could be too complex for mathematical reasoning. It might contain
some uncertainties during the process or the process might simply be stochastic in nature. So,
the major constituents of computational intelligence are fuzzy systems, neural networks
evolutionary algorithms and other hybrid intelligent systems.

So, we can clearly see here fuzzy system which is also a key agents of computational
intelligence. So, what I mean here is that that artificial intelligence and computational
intelligence although these two go hand in hand are being interchangeably being used, the
fuzzy system is a key component of these two.

(Refer Slide Time: 10:46)

Let us now just look at the artificial intelligence, Machine Learning and Deep Learning. So,
at this stage let us now understand as to how the artificial intelligence which we have just
discussed and then machine learning and deep learning how are these three terms related.

So, we see here that the artificial intelligence is a very broad term; what do I mean by broad
term here is that the artificial intelligence is a bigger set and machine learning is the is
actually part of artificial intelligence because the agents of artificial intelligence like a fuzzy
systems, neural network, artificial neural network, genetic algorithm and all other which we
have already mentioned. So, they help us in managing the machine learning process or
machine learning activities.

7
And then comes the deep learning which is again you see is a very smaller set than the
machine learning. So, it means that machine learning is a bigger set and deep learning is
smaller set and deep learning is contained in the machine learning and artificial intelligence.

It means that the deep learning is part of artificial intelligence, deep learning is part of
machine learning and deep learning here would mean that it’s a part of machine learning and
in machine learning when there is an intense learning process or repetitive learning
hierarchically, so, this is termed as deep learning. In nutshell I would say the deep learning is
also part of artificial intelligence as machine learning.

(Refer Slide Time: 12:57)

So, let me just briefly describe to how artificial intelligence the theory getting developed
through various the research experts of from various fields for example, the statistics,
mathematics, engineering, natural sciences, computer science. So, the artificial intelligence is
an interdisciplinary area and experts from these areas, but not limited to these areas, they are
contributing to the theory development of artificial intelligence.

And if we see here that the artificial intelligence is used by various fields for example,
computer vision and we see here the natural language processing, information, retrieval
information filtering, predictive analysis, decision analysis, robotics, but not limited to these,
again these are the few fields that are mentioned and there are many more where artificial
intelligence is contributing.

8
And then if we see they are separate applications again, I would say these applications are the
applications that are mentioned here are only a few applications, but there are so many
applications in respective fields which are being practiced which are being carried out by the
with the help of artificial intelligence.

(Refer Slide Time: 14:53)

So, now coming to the fuzzy systems theory which is based on fuzzy logic, professor Lotfi A
Zadeh which is who is also known as the father of fuzzy systems theory. So, he proposed the
idea of fuzzy logic in 1965. Fuzzy systems theory differs from conventional computing
because the conventional computing is based on bivalent logic or the Boolean logic whereas,
fuzzy logic is based on the multi valued logic. So, we can also say that the conventional
computing is based on one of the cases of or we can also say that the conventional computing
that we have done. So, far is one of the cases of fuzzy logic or the mathematics based on the
fuzzy logic.

Fuzzy systems theory involves soft or partial truth or partial false; soft because the truth if it
is soft it means that it is true, but not 100 percent true or somewhere in between 0 to 100
percent or false in between 0 to 10 percent. So, if it is 100 percent true the truth is hard or the
false is 100 percent it means the false is hard, but if there is a truth or the false which is not
100 percent or somewhere in between 0 and 100 percent it is termed as soft.

Fuzzy systems theory also deals with the uncertainties due to ambiguity, imprecision and
vagueness. So, these are the uncertainties which are very special kinds of uncertainties

9
because these uncertainties cannot be dealt by any other artificial intelligent agents, so far
what we have done mainly about uncertainties due to randomness and which is which can be
dealt or which are dealt by probability theory because the uncertainty here is due to a
randomness, but this is different from the uncertainties due to randomness. So, that is why
fuzzy system theory is very well suited for tackling these uncertainties.

And let me make it very clear here that these uncertainties cannot be dealt by the probability
theory very well. So, another thing is that fuzzy system theory is a multidisciplinary area,
multidisciplinary area here would mean that the concepts of fuzzy systems theory can be very
well utilized by many disciplines.

(Refer Slide Time: 18:22)

For example, the engineering science, humanities and so on and so forth. As I already
mentioned that fuzzy systems theory is based on fuzzy logic and fuzzy logic is multivalent
logic.

I already explained difference between bivalent logic which is nothing, but the Boolean logic
and the multivalent logic. So, let us now understand in bivalent logic truth is bivalent means
every proposition is either true or false with no degree of truth allowed. Means that truth is
truth and the false is hard. In multivalent logic as I already explained this also truth is a
matter of degree or I would say here as I mentioned just before the slide the truth or false is
soft.

10
A Multivalent logic can take in multivalent logic the values of truth or false they can take any
value in between 0 and 1. So, that is why if we talk of the degree, so degree can be can be
infinite in number, the number of values that can be assigned can be infinite. So, fuzzy logic
deals with partial which is a matter of degree.

So, partial information, imprecise information fuzzy logic deals with the granular information
granular here would mean that if we have linguistic information the fuzzy logic can deal with
this kind of information and manage to understand, manage to quantify from the linguistic
information and then fuzzy logic can also help in perception based information. In other
worse words perception based information can be quantified by fuzzy logic.

(Refer Slide Time: 20:22)

So, let me go through little historical background of fuzzy logic, classical logic of Aristotle
was proposed in 400 BC, 400 Before Christ. So, it is now very clear here that the bivalent
logic or the Boolean logic was proposed by Aristotle, in 400 BC, the law of bivalence which
is in use for more than 2000 years. So, it means every proposition is either true or false, it
means here true is hard and false is also hard it means 100 percent true or a 100 percent false.
So, there is no intermediate value of true or false.

So, another logic here is was proposed by Jan Lukasiewics who proposed three valued logic
in 1900 AD, this logic is a three valued logic it means that we have true, false and possible.
And then Lotfi A Zadeh proposed a fuzzy logic in 1965 which is bivalued logic and this is
again this is very popularly known as fuzzy logic.

11
(Refer Slide Time: 21:45)

So fuzzy logic is much more general as I already mentioned, than the traditional logic or
conventional logic these systems based on the traditional logic system. So, I can say here the
traditional logical systems. So, this statement goes like this the fuzzy logic is much more
general than traditional logical system, fuzzy logic provides a foundation for the development
of new tools for natural language processing like computing with words. So, this is a very
important area where fuzzy logic is very very helpful.

In other words if fuzzy logic has the ability to deal with to understand the linguistic
information and to quantify it in such a way that linguistic information is properly understood
and processed and a suitable output is created.

12
(Refer Slide Time: 22:51)

The Aristotle came us with the binary logic which has been the principle foundation of
conventional mathematics. Boolean logic states a glass can be full or not full. So, if we have a
case for example, that a glass is either half way filled. So, by using the Boolean logic we
cannot manage to define the half full glass of water or anything.

So, this disapproves the Aristotle’s low of bivalent logic or in other words we can say that the
Boolean logic is not sufficient to manage to take up this kind of situation. This concept of
certain degree or multivalence is the fundamental concept is stated by Lotfi A Zadeh this
helps us in defining such situations very well.

13
(Refer Slide Time: 22:55)

So, basis on which the fuzzy logic was proposed is here as the complexity of a system
increases it becomes more difficult and eventually impossible to make a precise statement
about its behavior, eventually arriving at a point of complexity where the fuzzy logic method
born humans is the only way to get at the problems. So, this statement was made by professor
Lotfi A Zadeh.

(Refer Slide Time: 24:24)

A professor Lotfi A Zadeh claimed that many sets in the world surrounded by us are defined
by a non distinct boundary. So, we will have few examples later and then we will see that the

14
claim of professor Lotfi A Zadeh is true. Let us also understand and let us also know that why
should we use fuzzy logic or fuzzy systems theory.

(Refer Slide Time: 24:56)

So, we use fuzzy logic when we have systems with uncertainties due to imprecision. I mean
the systems which suffers from the uncertainty is due to imprecision, vagueness ambiguity,
randomness, partial truth and approximation. Fuzzy logic can be very very helpful in
managing with the black box model or gray box model of a system, many times we do not
have the idea of the exact physical laws of the system, defining the defining the system and
are many times the system which is which we are studying are trying to model that is not
accessible to us.

So, we do not have all the physical laws which are governing the system in order to model
the system mathematically. So, we cannot get the exact mathematical equations. So, when
these situations occur we take the help of black box modeling where fuzzy system is also one
of the tools, one of the agents, one of the methods we can manage we can we can model the
system using black box modeling approach or the gray box modeling approach.

So, when we say gray box gray box means that a part of the system part of the system’s
mathematical equations are known or either known or can be known and that is how you
know the gray box kind of system can be can also use the fuzzy logic or fuzzy systems theory
in order to get the final model.

15
(Refer Slide Time: 27:16)

We should also know that when should we do not use the fuzzy logic like whenever we
already have the physics of the model known; obviously, we can have the mathematical
equations known, it means we have the model available mathematical model available. And
when we have mathematical model available then of course, this kind of model is also termed
as white box means the mathematical all the mathematical equations governing the model is
known. So, we do not require any such agents like fuzzy systems theory or fuzzy based
theory to go for this black box modeling approach.

So, in this case when the model is completely known then fuzzy logic is not needed and then
here when we have a system which is a linear system, then also we do not need to use fuzzy
systems theory for understanding the behavior or the getting the model and then systems with
moderate nonlinearities we can use simple models and fuzzy logic is not needed. Also the
systems with moderate complexities, so unless we have a very high complexity we should not
use fuzzy systems theory for studying such models. So, with this now I would like to stop
here and in the next lecture I will discuss some real time applications of fuzzy system.

16
 Fuzzy Sets, Logic and Systems & Applications
Prof. Nishchal K. Verma
Department of Electrical Engineering
Indian Institute of Technology, Kanpur

Lecture - 02
Introduction: Real Life Applications of Fuzzy Systems

Welcome to the lecture number 2 of a Fuzzy Sets, Logic and Systems and Applications.

(Refer Slide Time: 00:37)

So, today we will be discussing the real-life applications of fuzzy systems. So, first and
foremost application that I would like to discuss here is the Hitachi subway which is in
Sendai, Japan and in 1988 this turned into the fuzzy system. So, this means that the controller
that was already there was converted into a fuzzy controller and this was perhaps the most
visible application of fuzzy logic that time and this was in Sendai, Japan.

17
(Refer Slide Time: 01:18)

And the second application that I would like to mention here is the washing machines, very
popularly we see even nowadays also these are fuzzy logic based washing machines. So, in
these machines we see lots of linguistic terms like, if we have to select the water supply we
see cold, hot; for water levels we select high, medium, low, extra low, and then similarly for
the functions like soak, wash, rinse, spin and then for course we see digital, blanket, speedy,
wool.

So, like that we see lots of linguistic terms here. So, here in this washing machine the fuzzy
logic controller is sitting and this controller is taking inputs from the users in terms of a
linguistic variable. So, linguistic variable here as I mentioned here are like cold, hot, then
high, medium, low, extra low and like that for function soak, wash, rinse, spin.

So, instead of the you know the crisp input like the numbers here the inputs that consumer or
the user feeds here are these values and these control, the controller which is sitting in this
machine takes these as the input and accordingly produces the control output and which in
turn you know helps in managing the performance of the machine.

18
(Refer Slide Time: 03:20)

Then comes the fuzzy auto controller in cars, like a very recently Nissan patented a fuzzy
automatic transmission that saves fuel by 12 to 17 percent. So, here also the controller in this
car is a fuzzy controller.

(Refer Slide Time: 03:41)

Then coming to the next controller which is the fuzzy control of a cement kiln. So, here also
the controller which is sitting is fuzzy control and takes the inputs in terms of the linguistic
variables like, if the oxygen percentage is high and the temperature is low, then increase air
flow. So, what does it mean? It means that the inputs that are coming in the fuzzy controller

19
is in terms of the linguistic values, like high, the oxygen percentage is high and then the
temperature is low.

So, if these two are existing then the output should be in the region, the air flow has to be
increased so like that and similarly we can have multiple rule basis fuzzy rule basis and based
on that this fuzzy controller works. So, what I am trying to say here is also we have the fuzzy
controller which is acting based on the input, inputs that are fuzzy inputs. Fuzzy inputs means
the linguistic variables or values and then the output is also output out of this fuzzy controller
is a fuzzy and this is again used for further decision making.

(Refer Slide Time: 05:15)

Another application which you see here is the elevator monitoring and control. So, here also
the controller that is sitting here for this elevator monitoring is fuzzy controller which takes
inputs as the waiting time short, priority is high. So, like that we have so many inputs,
linguistic inputs coming in and then based on the inputs the decisions the controller give
controller produces the output in terms of, again either the linguistic output or the you know
the crisp output. So, based on that further decision is made.

20
(Refer Slide Time: 06:08)

So, then another application which is which we see here is the fuzzy controller-based copying
machine. So, we see here the copying machine and here also the fuzzy controller in many of
the machines fuzzy controllers are being used and here the drum voltage is adjusted based on
the picture density, humidity and temperature.

And these the variation of these parameters are basically course parameters, I mean the
linguistic terms like humidity, it can be either low, medium, high and so on and based on that
the controller makes the decision.

(Refer Slide Time: 07:01)

21
Here we see another application which is fuzzy based palmtop computer. So, here this
palmtop computer recognizes the handwritten kanji characters.

(Refer Slide Time: 07:15)

Then we come to another application of fuzzy logic here is used in golf diagnostic system.
So, what is done here is that this fuzzy logic helps in selecting the golf club based on golfer’s
swing. So, the golfer’s swing and physique. So, based on these two factors the golf club is
selected and then another point here a very important point here is to be noted here is that it
also determines the shaft flex profile for a golfer based on these parameters.

So, here also we see that based on the linguistic terms the fuzzy controller decides the what I
mean a particular golf club based on the parameters which are fed which are in the linguistic
terms.

22
(Refer Slide Time: 08:28)

Then we see a fuzzy logic application here in celerity in the courts. So, what we mean by,
celerity in the court is like fuzzy logic is helping various courts in managing the decision very
quickly in or in accelerating the decision making process.

So, a model case complexity of criminal justice systems. So, basically if we use fuzzy logic
the complexity is very well dealt by the fuzzy logic and you know with linguistic terms which
otherwise it is very difficult to be understood, fuzzy logic is helping this system the court
system to manage the complexity which is present in the you know this justice system and
then in the decision making in selection of courthouse building and similarly lots of other in
decision making this fuzzy logic is helping us very well.

23
(Refer Slide Time: 09:47)

Very interesting application we can see here is the fuzzy logic full in image processing. So, in
image processing fuzzy logic helps in contrast enhancement, the edge detection,
classification, segmentation, filtering. So, here we see you know some of the some of the
applications that is done by the fuzzy by the use of fuzzy logic or fuzzy logic-based system.

So, the fuzzy logic is very very helpful in image processing and if we are further interested in
few more application related application we should try going to or referring to the IEEE
transactions and fuzzy systems and we may see so many other similar applications.

(Refer Slide Time: 10:55)

24
So, here also we see one application which is fuzzy logic-based application and this is a
condition-based monitoring of machine. So, what do we do here is that various parameters for
example, the vibration, temperature, voltages and few other parameters are you know picked
up from the machine and these parameters are used for diagnosing the condition of the
machine.

So, the fuzzy based fault classifications are very helpful in recognizing the the status of the
machine, whether the machine is healthy or faulty and if the machine is healthy its fine, but if
the machine is faulty then what kind of fault in the machine is present. So, like that fuzzy
system or fuzzy logic based system especially the classifiers, the feature selectors, feature
extractors all these are helping us in managing the fault recognition process very well.

And another thing here is that fuzzy based algorithms are very very helpful for estimating the
remaining life prediction means, remaining useful life of a particular machine.

(Refer Slide Time: 12:43)

Another application that we here we see here is the fuzzy logic in aerospace. So, we see here
that the altitude control of a spacecraft is managed by the fuzzy based controllers and then
fuzzy based controllers are also helpful in managing the satellite altitude control flow and
mixture regulation and like that we use fuzzy logic based controllers in similar aerospace
applications.

25
(Refer Slide Time: 13:22)

So, these are very helpful in aerospace. And next application here is the fuzzy logic in
psychology. So, in psychology also the fuzzy logic based techniques, fuzzy logic based
algorithm approaches are very very useful in analyzing the human behavior and criminal
investigation and with this through you know the prevention is also prevention of the criminal
attitudes are also done.

(Refer Slide Time: 14:03)

26
So, next application is the fuzzy logic based air conditioning system. So, here also in air
conditioning many of the air conditioners nowadays coming with fuzzy logic based fuzzy
logic controller and these controllers are taking inputs in linguistic terms again.

And these input inputs are in terms of like linguistic values like room temperature control and
this room temperature inputs will be like the temperature, low temperature, high temperature,
medium temperature and like that. And then humidity control if this has the fuzzy controller
based humidity control then we have the low humidity, high humidity, medium humidity or
like that.

So, all these linguistic values are selected and based on this the controller takes the
appropriate decision and this decision values are fed to the respective systems to manage to
give the appropriate performance.

(Refer Slide Time: 15:28)

And another application here is the fuzzy logic-based recipes recommendation. So, here also
we see that the we have certain linguistic values that is a given as the input for decision
making and this linguistic values could be based on the person mood, person’s mood, healthy
eating, balanced meal, appetite, spare time.

And all and all these will be affecting the decision and here the decision maker, the decision
you know the system is fuzzy and this decision system is taking the inputs in terms of the
linguistic values.

27
(Refer Slide Time: 16:24)

So, fuzzy logic can also be very very helpful in automatic gear selection. For example, based
on the road conditions and driving style and so many other features could also be added in
order to make the decision in terms of the gear selection and other things. So, this can also be
these are these are also possible and being used in some of the cars.

So, road conditions here would mean like if the road condition is very good or bad, very bad
or like that the linguistic values if we select and based on that the driving style also if this is
also you know this is also given as the input like good style, bad style or whatever. So, based
on these inputs the gears selector prompts are it helps in selecting the gear, appropriately to
give the better performance of the car.

28
(Refer Slide Time: 17:36)

And fuzzy logic in another application here is very helpful for diagnosis of coronary artery
disease. So, if we look at the features here and based on these features this diagnosis this
diagnostic system which is based on fuzzy logic takes the decision. And based on the age,
like age could be young, old, very old or similar values linguistic values and then the gender
and this gender could be male value or the female value.

And then the cholesterol, obesity, smoking and all these are normally used as input of the
fuzzy logic based diagnostic system and based on these inputs the fuzzy logic based
diagnostic system gives us the appropriate output and this output helps us in the diagnosing
the condition of the health of the heart.

29
(Refer Slide Time: 18:53)

Similarly, we have another application which is in agriculture. So, fuzzy logic is being
utilized, fuzzy logic based systems are utilized in agriculture in soil, moisture, water, weather,
another environmental conditions based you know decisions are made.

And as I already mentioned that based on these the linguistic values appropriately you know
like soil moisture is low, high like that very high or like that we can have the inputs, similarly
for water for weather for you know the environmental conditions. So, all these are fed as the
inputs and based on these inputs the appropriately decisions are taken by the fuzzy based
decision system and this enhances the overall performance of the agriculture.

30
(Refer Slide Time: 20:02)

So, if we look at real life applications we see that we have nowadays so many areas where
fuzzy logic-based systems are being used and only a few of these areas I have covered. But
there are so many applications existing. So, if you are interested you may go ahead and
explore other areas also where the fuzzy logic-based systems are being used. Fuzzy system is
a universal approximator. So, what does it mean here is that see the fuzzy system can
approximate any function.

So, when we say any function means if the function is linear we do not need fuzzy system,
but if the function is the non-linear highly non-linear very complex function then the fuzzy
logic can approximate it means it actually finds the f by using the fuzzy logic.

31
(Refer Slide Time: 21:23)

So, let me also discuss here the fuzzy logic versus probability. So, many of us often get
confused by the fuzzy logic and probability because both of these operates on the values in
between a 0 and 1. So, that is the major confusion normally students face. So, both operate as
I mentioned, both operate over the same numeric range; same numeric range and at first
instance both have similar values in between 0 and 1.

What does it mean here is that the fuzzy logic when we see takes the values in between 0 and
1 for its belongingness and similarly the probability values are also in between 0 and 1. So,
that is how we often get confused. However there is a clear distinction between the two. So,
let us understand the distinction between the two. So, the semantic difference here is the
significant as the first is based on degree of randomness. So, when we talk off the probability,
probability is based on the degree of randomness whereas, the fuzzy system fuzzy logic is
based on the degree of belongingness.

32
(Refer Slide Time: 23:19)

So, let us now take some of the examples and by these examples we will be able to
understand the distinction, clear distinction in between these two. So, fuzzy logic basically
has the fuzzy boundaries, fuzzy logic deals with the linguistic values and if we have the, you
know the groups and these groups, they do not have the clear-cut boundary the sharp
boundary.

So, like if we see, if we see here in this picture the height of people. So, if we see that we
have three categories here in terms of the height. So, three category categories of people are
present. So, tall, medium, short. So, it is very difficult to draw a line in between the tall
group, medium group, short group.

So, if we talk off the tallest, he is the tallest, but since there this in when we talk of tall. So,
tall is the group of person who has the in terms of height who are tall. So, this person can
belong to that group, this person can also belong to that group, this person can also belong to
that group. When we talk of medium so, we do not know whether I should put this person,
this lady in the medium group or the tall group. So, that is why we can always say that this
lady can be in can be partially present in the tall group and can be present in the medium
group. Similarly, for other group also this you know distinction this partial belongingness can
be there.

33
(Refer Slide Time: 25:31)

If you take another example, similarly for weight of the people. So, similar clear boundaries
cannot be drawn like heavy weight, middle weight, light weight, fly weight. So, it is very
difficult to clearly sharply draw a boundary and based on that we cannot separate these
people in terms of these groups.

(Refer Slide Time: 26:02)

And when we talk of probability so, probability you see if we toss a coin let us say 10 number
of times, 10 times if we toss coin. So, few times let us say here as you see the 6 times we get
the heads and 4 times we get tails right, so, for probability you see the probability is 1 over 6,

34
probability of getting the head and then probability of getting the tail is 1 by 4 and here the
probability is 1 by 6.

So, if we see these values which are coming here because of the randomness these probability
values are coming in between 0 and 1. But if we talk of fuzzy, so there if a person belongs, let
us say this person belongs to a particular group completely like a tall group completely it
means this person is belonging to that group with 100 percent belongingness.

And if this person may belong to that group with a less than 100 percent say, 95 percent, so
the degree of belongingness here could be 1 and here that degree of belongingness could be
0.95. So, here also the degrees of all the degree of belongingness, all the degrees, will be
coming in between 0 and 1. So, that is why we all we often gets confused get confused with
these values and we confuse with the cases whether this is fuzzy logic case or probability
case.

So, we need to be very very careful while dealing with some of the problems. So, we look we
should first check whether the process is based on the randomness or process is based on the
belongingness.

(Refer Slide Time: 28:31)

So, now then let us come to the block diagram of the fuzzy logic in control and decision
making. So, we have a fuzzy model here. So, input of this fuzzy model is this fuzzy model is

35
basically from the real-world measurements or assessments of system conditions like
temperature, market data and all.

So, here the input to this model can be either crisp or fuzzy, if this is crisp we need to fuzzify
it in order to feed this data into the fuzzy model or if it is already fuzzy that we can straight
away feed this data as input to this fuzzy model. And then the output of this fuzzy model is
generated, this could be fuzzy or this could be the crisp if this is fuzzy value as the output of
this model then we defuzzify.

Then we defuzzify before we use. If this output is already crisp then we can straight away use
this for further processes as input. So, this way this fuzzy model is or this fuzzy model can be
regarded as the decision maker.

(Refer Slide Time: 30:12)

Let me briefly tell you here about a fuzzy system. So, what is a fuzzy system is system which
contains these four blocks. So, first block is the fuzzifier and then the second block is the
fuzzy inference engine, third block is fuzzy rule base, fourth block is defuzzifier. So, any
typical fuzzy system will have these four blocks.

So, if we come across any system, any fuzzy system we have must check whether all these
four blocks are there or not. If any of these blocks are missing it means the system which we
are dealing is not a fuzzy system. So fuzzifier here is fuzzifying the input data which if which

36
is a crisp, if it is crisp fuzzifier fuzzifies the input data and feeds it to the fuzzy inference
engine.

And fuzzy inference engine takes the help of fuzzy rule base and based on that the suitable
output is generated and if this output here is fuzzy a defuzzifier is used for generating the
crisp output. And here this can also be possible that this output is a crisp output then we do
not need a defuzzifier, we can straight away use this output as the input to the other system.

(Refer Slide Time: 32:30)

So, I would stop here for this lecture and in the next lecture we will discuss fuzzy sets, the
representations and fuzzy logic toolbox in MATLAB.

Thank you very much.

37
 Fuzzy Sets, Logic and Systems & Applications
Prof. Nishchal K. Verma
Department of Electrical Engineering
Indian Institute of Technology, Kanpur

Lecture – 03
Fuzzy Sets and Fuzzy Logic Toolbox in MATLAB

Welcome to lecture 3 of the course on Fuzzy Sets, Logic and Systems and Applications. So,
in today’s lecture we will discuss Fuzzy Sets and logic Fuzzy Logic Toolbox in MATLAB.
So, before we discuss fuzzy sets let me introduce classical sets just before going to fuzzy sets,
because we need to first understand what is a set, I mean conventional set. And, then we will
from the classical sets we will transition to the fuzzy sets and that this is needed for better
understanding. So, what is a classical set? Classical set normally is a collection of objects
from the universe of discourse.

(Refer Slide Time: 01:24)

So, if we take here classical set whose universe of discourse is X . So, let X be the universe of
discourse and x ∈ X. So, x, x is an element that belongs to the universe of discourse. Then a
classical set A can be defined as A is equal to the collection of all the x’s in the set and the
condition here is that the x should meet certain criteria or conditions. And, as I mentioned
that this x must be belonging to the universe of discourse.

So, any element which is not belonging to the universe of discourse should not be part of this
set. So, here we all know now that conventional set A is the collection of all the elements and

38
these elements are from, are drawn from the universe of discourse X . So, this kind of
collection is termed as a set.

(Refer Slide Time: 02:54)

If we take an example here, like if we write a set of positive integers less than 20 and more
than 15, so, and the universe of discourse of course, that is X is set of all positive integers that
is I +¿ ¿ . So, if we are writing a classical set A, so this classical set A will be containing all the
elements that satisfies this criteria. So, if we write set A; set A will be like this. So, A is equal
to the set the collection of all the elements that is x which is satisfying this criteria.

What is this criteria? A={x∨15< x<20 , x ∈ X } So, now, we can write this set as the A which
is the name of the set, the classical set as collection of all the elements in between 15 and 20.
So, in between 15 and 20 are coming out to be 16, 17, 18, 19 the part of this set and if we
look at the set here all the elements here, then we find that these all the elements are drawn
from the positive integers, the set of positive integers I +¿ ¿ .

So, this way we can say that A is a classical set. Now, here a very important point that we
should note is that’s the element that are present in the classical set are completely present.
What do I mean by completely means 16 is present with 100 percent, 17 is present with 100
percent, 18 is present with 100 percent, 19 is present with 100 percent. So, 16, 17, 18 are
completely present in the system. So, the classical set, in classical set if we have the element
that present in the classical set they all are present with 100 percent.

39
Now, if we look at the above classical set, it is clear that any of the elements A are either
completely belongs to A, means they are present with the with 100 percent or completely
does not belong to A. Means, the elements that are not present in the set, means they are not
present in the set with 100 percent, means their percentage is 0. So, any element which is not
present in the set, it indicates here, it means here that the element is present with 0 percent,
means element is not present.

(Refer Slide Time: 06:34)

So, for the same classical set A let us let me make this thing more clear, in other words we
make say that the membership value. So, a new term is coming over here is membership
value. So, when I was mentioning 100 percent, it means 100 percent is the membership value;
means here is that if we have classical set or conventional set, so, this classical set or
conventional set, the element that are present they are present with 100 percent membership
value.

So, the membership value here, the membership value of 16 is 100 percent; although it is not
written, membership value of 17 is 100 percent, membership value of 18 is 100 percent,
membership value of 19 is 100 percent. But, all these membership value are not written in the
classical set, because it is understood that they are completely present. So now, if we look at
this set again, what we see here the elements that do not belong to the classical set A have
their corresponding membership value zero.

40
Of course, because the conventional set is based on the Boolean logic. So, any element which
is present will have 100 percent presence or element or any element which is not present will
have 100 percent absence. So, if we have any element present in the classical set, we will say
that it is true and if the element is not present with 100 percent means the 0 percent it is
termed as false. So, there is no possible case in between any of the element.

So, this point is very important to be noted because in classical set either an element in the set
is present or it’s not present. But, if we talk of fuzzy logic, since fuzzy logic is based on the
multi-valued logic; so, the element can be present in the set in fuzzy set with varied
membership value. What does this, what does this mean here is, that any element may be
present with the membership value more than 0 and less than 1; the elements with this value
may be present, but not completely present. Any element which is present with 100 percent
membership value is completely present.

So, as I just mentioned that multivalued logic here, with the multivalued logic in fuzzy
system; so, here not only the true or false, the, so every element in a fuzzy logic A will be
assigned its membership value. So, let me make it very clear that any element, any element
which has the membership value more than 0 will be the part of a fuzzy set. And, also if the
element is not completely present means if the element has the membership value 0, the
element will not be part of the fuzzy set.

(Refer Slide Time: 10:48)

41
So, let us now move to fuzzy set from the classical set and in classical set we do not write any
membership value as I mentioned. Because, it is assumed that any element which is present
in the classical set it is, it is assumed, it is understood that the element that are in the
conventional set, or classical set, or traditional set which is based on the Boolean logic, the
elements are with 100 percent membership value.

So, that is why it is not written, but if we talk of fuzzy set here unlike the conventional,
traditional fuzzy logic, fuzzy logic, in fuzzy logic the elements that are present in the set they
these will have the membership values in between 0, in between 0 plus and 1. So, the
elements, these elements will be with the respective membership values. This is needed
because, otherwise we may not be knowing with what value, with what degree, with what
membership value these elements are present in the sets.

So, here like in the classical set that we have had just before we had 16, so, 16 was 100
percent present, 17 was 100 percent present, 18 was 100 percent present, 19 was 100 percent
present. And of course, all these elements were drawn from the universe of discourse. So,
here also we will have a universe of discourse and these elements must be from the universe
of discourse, some universe of discourse say X. Now, in classical sets these all were present
with 100 percent means completely present, but let us assume a case where this 16 is not
completely present, I would say partially present. And, if we say partially present means it is,
it has some value, it has some membership value less than 1.

So, if it has some membership value less than 1, say 0.6. So, this 0.6 is the membership value
of 16. So, this 16 and 0.6 these two are very important for together to be included in the in the
set. And, since this is not 100 percent present; so, we need to know with what degree it is
present. So, that is why the degree along with that element is needed and that’s why if we see
in a fuzzy set every element is with some degree. So, every element is paired with some
degree and that is how a fuzzy set is formed.

So, if we see here this fuzzy set A, now this is not a crisp set, it is a fuzzy set because the
element 16, 17, 18, 19 they all are having some degree associated with these element. So, 16
is with 0.6, 17 is with 0.9, 18 is with 0.2, 19 is with 1. So, if we see here that fuzzy set is the
ordered pair of all the elements and this pair is nothing, but the first element of the this pair is
the element and the second element of this pair is the membership value. So, 16, 0.6 means
that this 16 is present with 0.6 membership value, similarly 17 is present with 0.9

42
membership value, 18 is present with 0.2 membership value, 19 is present with 1.0
membership value means 100 percent. So, this 19 is completely present, 18 is partially
present, 17 is partially present and 16 is also partially present. So, this way fuzzy set is a set
which has all the elements with its membership values. So, we can also say the same thing as
it is written over here; a fuzzy set A can be written as a set of ordered pairs of the element
and its belongingness, and this belongingness is also called as membership. So, this way
fuzzy set can be written and this is a transition from the classical set.

In classical set we saw that 16 was 100 percent, 17 was 100 percent, 18 was 100 percent, 19
was 100 percent. So, that is why there was no need to include any membership value along
with these elements. So, that is why it was not needed, but here since the since fuzzy set
fuzzy logic fuzzy system is based on fuzzy logic which is of course, a multivalued logic and
because of that the no matter whether a element an element is present with 100 percent or not,
all the element from the universe of discourse must be included in the fuzzy set. So, that is
why any element which is present, even if it is not even if it is not present with 100 percent
all the element has been included here.

(Refer Slide Time: 17:23)

So, this is how we transition from the classical set to a fuzzy set. So, we clearly see there is a
need of writing, there is a need of including here the membership value with the elements
with element, without that it is difficult for us to write a fuzzy set. So, that is why we include
a membership value and the membership value here is represented as μ A , membership value

43
of any element is represented by μ A ( x). So, if we have any element x and the corresponding
membership value will be μ A and thisμ( x ) and this A is nothing, but the A signifies the
particular the name of the set. So, A is basically a particular set.

So, a fuzzy set in the fuzzy set representation we can write a fuzzy set either in discrete form
or in continuous form. So, in discrete form if we write a fuzzy set, this is the way we write
like, you see here that same fuzzy set; that we already discussed could be written as if we
have let’s say are already had fuzzy set A which was like this 16 with ok. I have I am taking
some other member membership values like let’s say 0.2 and then 17 with 0.1 and 18, 0.5 say
19, 1 like this. So, if we have this as a fuzzy set, this is a crude representation of the fuzzy set.

So, the same fuzzy set can be represented as you see first the membership value and then
corresponding membership corresponding element and then with plus sign we add, but,
please understand that there is there will not be any addition of this ah this values. So, 0.1, 17
and then 0.5, 18 plus 1, 19. So, the same fuzzy set which you see here can be written as
A={0.2/16 +0.1/ 17+0.5/ 18+1.0/ 19 }. So, we see that first we write the membership value
and then with oblique sign with line, a slanted line we write here the corresponding element.

So, if we have any discrete; so, let me first make it clear that a fuzzy set can be the discrete or
a fuzzy set can be continuous. So, if we are writing a discrete fuzzy set, we write the discrete
fuzzy set this way as I just mentioned. And, then we see here the, you know the
representation wise here the same thing can be written like this, like form in which we write a
fuzzy set. So, if we have a fuzzy set A fuzzy set A should be written as like

μ A ( x 1) μ A ( x 2)
A= + …. .
x1 x2

, if we have a x 1 , x 2 , x 3 all these the corresponding generic variable value values drawn from
the universe of discourse.

So, this plus sign is very important here because this plus sign this should be noted, it should
be noted that this plus sign this plus sign is not here for addition. So, please understand that
although we have plus signs here, all the elements are separated by plus sign, but this plus
sign is not here for addition. So, please do not add these values with this plus sign, if you add
these values together this will be, this will not be a fuzzy set. So, this will become this will be
wrong thing. So, please do not add this, these plus signs you leave as it is.

44
So, this is and please understand this is another way of writing the same thing, you can use

μA ( xi)
summation also. Like if we do not want to write it like this, ∑ ={( x i , μ i ( xi ) )∨ xi ∈ X }.
X xi
So, this way the discrete fuzzy set is written, then if we want to write, if we want to represent
a continuous fuzzy set. So, a continuous fuzzy set of course, this will be from the infinite
universe of discourse.

And, this fuzzy set will be written as you see a fuzzy set A is written as A is equal to we use
here instead of this summation sign we use the integration sign.

So, please understand here that this if we have continuous fuzzy set, we use the integration

sign. A=∫ μ A ( x i )/ x i
X

And, then with this slanted line we separate and then we write the corresponding generic
variable value.

Please understand this μ A ( xi ) will be continuous function, because if we are writing fuzzy set
in a continuous format of course ( x i ) will be a continuous function. So, this is this will be a
continuous, this corresponding the values here the generic variable values will be ( x i ). So,
this way we write continuous fuzzy set in a continuous, in a continuous format. And, as it is
written here that summation and integration signs indicate the collection of all the element in
the universe of discourse.

So, along with their associated membership values. So, please do not use this summation and
integration, so, summation for addition and do not integrate the function that is coming so,
but you just leave it leave as it is.

45
(Refer Slide Time: 26:16)

So, μ A ( x) as I mentioned μ A ( x) is nothing, but a membership value. So, μ A ( x) is termed as

the membership value for element x in fuzzy set A and this gives a single value for every
element contained in fuzzy set. So, membership value, values for every x can be found from
membership function. So, if we have a membership function let us μ A ( xi ), it should be ( x i )
here, it should ( x i ). And, then if it is a corresponding if this is the membership value,
membership function and if we want to have a corresponding membership value. If we have a
generic variable let us say ( x i ), ith value i of the in the universe of discourse, ithelement some
element ( x i ). So, corresponding that ( x i )we will have μ A ( xi ) and then this is a this will be a
single value. So, and one more thing that to be noted here is that this membership values will
be in between 0 and 1. So, the membership value which is written here, that membership
values lie between the interval 0 and 1 for a normal fuzzy sets. So, when we say a normal
fuzzy set, it means they it means there may be a fuzzy sets which are not normal.

So, we ah normally call those sets as subnormal fuzzy sets. So, so fuzzy set which has at least
one membership value equal to 1 is a normal fuzzy set. But, any fuzzy set which does not
have any of the values as equal to any of the membership values equal to 1, it is called
subnormal fuzzy sets. So, in other words you can say the subnormal fuzzy sets are those sets
whose which does not have any membership value up to 1. So, with this I would like to stop
here.

46
(Refer Slide Time: 29:01)

And, in the next lecture we will be discussing few examples on fuzzy sets and fuzzy logic
toolbox in MATLAB will also be discussed.

47
 Fuzzy Sets, Logic and Systems & Applications
Prof. Nishchal K. Verma
Department of Electrical Engineering
Indian Institute of Technology, Kanpur

Lecture - 04
Fuzzy Sets and Fuzzy Logic Toolbox in MATLAB

So, welcome to lecture number 4 of the course on Fuzzy Sets, Logic and Systems and
Applications. So, in today’s lecture we will be discussing Fuzzy Sets and Fuzzy Logic
Toolbox in MATLAB and this will be in continuation to my previous lecture.

(Refer Slide Time: 00:36)

So, let us take some examples and then whatever we have understood so far let us try to
understand even further. We have a set of cities and this set is here for this example taken as
universe of discourse. So, capital X is the universe of discourse. It means that whatever
elements that will be taking or including in our fuzzy set will be taken from this fuzzy set.

So, these are the places the cities which we may choose and then if we want to write a fuzzy
set like this that a fuzzy set A that we would like to write. So, a fuzzy set that we would like
to write is based on the desirable places to live in.

So if we write this in form of a fuzzy set we will write it like this. So, A if A is a fuzzy set
and this will be equal to the collection of all the elements. So, if I take Delhi. So, Delhi is the
element which is drawn from the universe of discourse and then since if since this is a fuzzy

48
set I must have the belongingness or the membership value. So, we will have a pair of city
and then corresponding membership value.

So, corresponding membership value can be any value in between 0 plus and 1. Why I am
saying 0 plus 1? Because this value must be some value which is more than 0. This value can
be 0 plus epsilon. So, if it is 0 we are not we will not be including the corresponding element.
So, this value this membership value must be more than 0.

So you see if we are writing like this. So, this membership value will indicate as if it is the
desirable place to live in means the it is my liking to live in a particular city. So, Delhi has 0.9
membership value. It means that the I am liking this comparatively you know if we have
other places to choose.

So, Delhi with 0.9 means that we are writing we are liking this 90 percent or something like
that. So, if one is a 100 one is taken as a 100 percent and then if I take Agra, so I may choose
Agra with 0.6 membership value. It means I like Agra less than Delhi and then if I choose
Mumbai then I like Mumbai more than Agra and less than Delhi that is with 0.7.

If I take Kolkata, Kolkata is 0.7. So, it means I like Kolkata same as Mumbai and if I take
Kanpur. So, Kanpur membership value is 0.1. So, I like Kanpur you know less than all the
cities as discuss like Delhi, Agra, Mumbai, Kolkata, and then if I talk off IIT Kanpur, I like
100 percent. So, that is why you know Kanpur IIT Kanpur has the membership value 1, so, 1
means 100 percent here.

So, as I mentioned when I when we write fuzzy set. So, fuzzy set we will have the pair of the
elements and the corresponding membership value. So, if you see here in this fuzzy set you
see here this fuzzy set includes the city and then corresponding membership value. So, this
way this is a this can be called as a fuzzy set.

So the same fuzzy set see this is the crude this is the form which I mentioned above before.
So, the same fuzzy set A can be written as

A=0.9/ Delhi+0.6 / Agra+0.7 / Mumbai+0.7/ Kolkata+0.1/ Kanpur+1.0/(IIT Kanpur )

So, we see that the same fuzzy set will have first membership value and then we separated by
the slanted line and then we write the corresponding elements. So, like that all the elements in
the fuzzy sets are included.

49
(Refer Slide Time: 06:38)

Now, if we take another example here to make it more clear. So, fuzzy set A is defined as
below right A in summation format. So, here A is given like this. If this is a fuzzy set that is
given to you and please understand that these x 1 , x 2 , x 3 , x 4 , x5 , x 6 , x 7 , x 8 all these elements
must be drawn from the universe of discourse x.

So I am writing here x i’s for all these and this is the capital X which is my universe of
discourse. Where i is the first i is equal to 1 to 8. So, i if I change this i is equal to 1 this will
become x 1. So, x 1 and then similarly up to x 8 so, all these elements will be drawn from the
universe of discourse capital X .

So, how we will be writing in this summation this a fuzzy set in the summation format. We
have already seen that how fuzzy set can be written in this summation format. So, let us know
quickly write and understand. So, you see here this x in summation format can be written as;
can be written as in summation format what do we do we first write the membership value.
See the membership value here which is coming from here for the first element. So,
membership value and then we separated by a slanted line we can also call this as oblique.

So, 0.6/ x 1 and then 0.2/ x2 you see here this is the membership value here which is coming
here and then this is the corresponding you see here corresponding element. So, this way we
first write the, we first write the membership value and then the corresponding elements
separated by a slanted line and we also use plus in between just to show the collection of
these elements in the fuzzy set.

50
So, this way we write the same fuzzy set in the summation format. Now let us take another
example here if we are interested in writing a fuzzy set in continuous form whose
membership function is given as μ( x ) is equal to π x 2. So π x 2 is a continuous function. So,
and of course, needless to mention that this x should be coming from, all x's should be
coming from the universe of discourse.

So, if we are interested in writing fuzzy a continuous fuzzy set. So, we will be writing it like
this. So, A is the continuous fuzzy set and as I mentioned that when we are writing a
continuous fuzzy set we use integration sign of integration. So, here you see the sign of
integration and then just below this we write the universe of discourse and which is x and
then we write the continuous membership function you see the continuous membership
function is given as π x 2.

A=∫ μ ( x ) / x =∫ π x 2 / x
X X

So, we write this here and then we write the corresponding elements and this elements is
coming from again the universe of discourse which is a generic variable normally we call
generic variable, so we write the generic variable x here and now if you replace this μ( x ) by
π x 2. So, in place of μ( x ) we write π x 2 and then we separate by a slanted line the generic
variable and then this becomes our, the fuzzy set in continuous format ok.

So, this is how we write of continuous fuzzy set in this format. So, similarly if we have the
other continuous functions for μ( x ) we can replace we can write and that will be our
continuous fuzzy set, okay. So, and one more thing I would like to tell here is that the fuzzy
membership functions can take any shape, but the highest value at any instant can never be
more than 1, this is because you know these fuzzy membership functions they provide the
membership values and the membership value cannot be more than 100 percent.

So, that is why the 100 percent is actually for is represented as 1. So, that is why any
membership function can never go more than the value of 1.

51
(Refer Slide Time: 12:54)

So, this way we understand that how do we write a fuzzy set and we also understood at this
point that why it is necessary to include the membership values along with its elements in a
fuzzy set. I would like to repeat here that since in fuzzy set all the elements can be either
present are partially present. So, all the elements are included that is why the membership
values are necessary to be included along with the elements and these membership values
from where do we get these membership values these membership values functions.

So, we will discuss this in more detail in coming slides coming lectures and but these
membership values are also known as the belonging belongingness our degree you know are
these other names. So, membership functions are very important and these membership
functions provide us the membership values and there are certain criteria’s. Which we
normally follow while we choose the membership functions.

So at this point I would just like to tell you that the nature of the variables are very important
when we choose the membership functions because these membership functions as I
mentioned they are providing us the belongingness they are providing us the membership
values. So, we need to be very intelligently choosing these membership functions in order to
have the proper fuzzy set. In order to form right fuzzy set.

So, nature of the variable is very important and then resolutions are the level of details to be
included and then the nature of applications design and optimization, suitability, concepts to
represent the variables. So, all of these the factors which we normally take into account and

52
based on these we try to choose the membership functions. So, this will be discussed in detail
in coming lectures, but at this point I would just like to like you to know that these points are
these factors are important for taking choosing the membership functions.

(Refer Slide Time: 16:06)

Now, since in fuzzy logic fuzzy in this course we will be doing some exercise we will be
doing certain tasks by using MATLAB. So, I would like to briefly introduce the fuzzy logic
tool box in MATLAB and with this I would like you to get yourself acquainted that a fuzzy
logic toolbox is available in the MATLAB those of you who have not done any prior exercise
in MATLAB or do not know anything in mat lab you can start from here.

And this is the this is basically I thought that this will be very helpful while going through the
fuzzy logic and the corresponding exercises you can implement using MATLAB, but
MATLAB is not the only platform you can use other platforms like R ,you can use you know
Java, C whatever you are interested with, but here I will be going through the MATLAB
platform.

So, fuzzy logic toolbox is there in the MATLAB and if we see if we have a MATLAB in
MATLAB the fuzzy inference system is already there and then we have fuzzy logic toolbox,
we have fuzzy simulink and then here if you want we can write the MATLAB files. So, in
MATLAB like other tool boxes if fuzzy logic tool boxes also there.

53
(Refer Slide Time: 18:24)

So, I would like to very briefly tell you that if you are interested you can go to this MATLAB
fuzzy logic toolbox and this is the snapshot which you see here of that of the fuzzy logic
toolbox page.

(Refer Slide Time: 18:47)

And if we go to that we have these pages available like getting started. So if you do not have
any prior idea of a MATLAB either with a reference to the fuzzy logic toolbox or even
otherwise you should go to a fuzzy logic toolbox and then go to getting started. So, this

54
explains the basic theory about fuzzy and guides towards the beginning of the toolbox with
basic functions and then fuzzy inference system modeling.

So, this explains the building tools for Mamdanis and Sugenos type 1, type 2 fuzzy inference
systems. This also has the fuzzy inference systems tuning. So, it explains the commands and
functions required for tuning of the membership functions and rules of the fuzzy inference
systems, this also has data clustering.

So, in data clustering we have the tools and commands that are used for finding the clusters in
the data being provided as input output through fuzzy c mean clustering and then the fuzzy
logic simulink is there where we have graphical user interface and we can you know connect
these the blocks and then, so, same thing can we can implement using this and then we have
the deployment, so using this feature given in the MATLAB one can deploy a fuzzy inference
system by generating code in either simulink or MATLAB.

(Refer Slide Time: 20:43)

So, here are some functions some inbuilt functions that are provided in the MATLAB. So,
you may like to go through. So, like a mamfis which is a nothing, but the inbuilt function for
Mamdani fuzzy inference system and then sugfis is the sugeno fuzzy inference system. So,
like that we have so many functions you can go through all these are listed in these slides.

So, I will just read these functions like genfis and then genfisoptions, mamfistype 2
sugfistype 2. So, type 2 means the there are the fuzzy system is actually divided into two

55
types. So, first there is a first is a type 1 and then type 2. Now type 3 is also coming up, but
whatever we will be dealing here with we will be with respect to type 1 initially and then
fistree convert to sugeno, convert to type 1, convert to type 2.

(Refer Slide Time: 22:01)

So, all these functions are given and then here we have few more functions that you can go
through.

(Refer Slide Time: 22:06)

56
(Refer Slide Time: 22:08)

All these functions all these functions are and these functions will be helping you to directly
use, give the input and get the output.

(Refer Slide Time: 22:19)

57
(Refer Slide Time: 22:21)

So, these are the functions that are included in the MATLAB.

(Refer Slide Time: 22:24)

58
(Refer Slide Time: 22:25)

So, please go through these functions and uh try to use these functions and get yourself
acquainted. So, further for those of you who do not have any idea of MATLAB before? So,
when you open the MATLAB you basically you get to see this slide and in this slide you get
to see this page and in this page you will have the four boxes.

(Refer Slide Time: 23:14)

If you see here the first box is the editor and then the current folder.

59
(Refer Slide Time: 23:17)

And then the work command window.

(Refer Slide Time: 23:20)

And then the workspace. So, let me quickly tell you what are all these the editor window
where you can basically write your program in m file MATLAB m is designated as a
MATLAB. So, here you can write your program and edit the MATLAB subscripts and then
the current folder will tell you as to the paths for showing the current operating folder then
the command window basically will show you the running simulation. This is the, this is
where you get to see all your results.

60
Work space window will show you the space for storage of variable data and that’s how we
can start working on the MATLAB and if we go through all these functions that were
mentioned we will certainly have a fair idea of MATLAB with respect to fuzzy systems.

So, I would suggest you to go through all these inbuilt function first, but at the same time I
would like to tell you that you try to build your, you try to write your own MATLAB code in
a MATLAB editable file and avoid using these slowly avoid using these functions and try to
learn further in writing the better programs with respect to the fuzzy logic.

(Refer Slide Time: 25:11)

In the next lecture I will be covering the fuzzy membership functions of different types.

61
 Fuzzy Sets, Logic and Systems & Applications
Prof. Nishchal K. Verma
Department of Electrical Engineering
Indian Institute of Technology, Kanpur

Lecture - 05
Membership Functions

Welcome to lecture number 5 of Fuzzy Sets Logic and Systems and Applications. So, today
in this lecture we will discuss various types of Fuzzy Membership Functions.

(Refer Slide Time: 00:34)

As we know we have seen in the previous lectures that we can define a fuzzy A={¿∨x ∈ X .
So, you see here x is the generic variable this is generic variable and this is along with μ A ( x).
So, this μ A ( x) is nothing but the fuzzy membership function.

And this fuzzy membership function is responsible for giving the membership values to the
corresponding value of the generic variable. So, we see here this μ A ( x) is very important in
the sense that this provides us the membership values. So, fuzzy membership function is very
important in the sense that this function, this membership function assigns the corresponding
membership values which are more than 0 up to 1.

So, since fuzzy membership value can be more than 0 up to 1, any value in between that and
that is why the membership value, fuzzy members function can be in between 0 to 1 or I
would say the fuzzy membership function can go up to at the most up to 1.

62
(Refer Slide Time: 03:19)

So, there are some commonly implied membership functions in fuzzy systems theory and
these membership functions are also called a standard membership functions which are listed
as the triangular membership function, trapezoidal membership function, Gaussian
membership function, generalized bell-shaped membership functions, sigmoidal membership
function, left-right membership function, pi membership function, open left membership
function, open right membership function and S-shaped membership functions. So, the details
of these membership functions will be described given in the coming slides.

63
(Refer Slide Time: 04:09)

So, if we take up first the triangular membership function, so this fuzzy membership function
we defined by this syntax.

0 x ≤a

{ x−a
triangle ( x ; a ,b , c )= b−a
c−x
c−b
a≤ x≤b

b≤ x ≤c
0c ≤ x

So, this is how we define triangular membership function. We write a triangle membership
function. And here we have an alternate expression for triangular membership function

x−a c−x
{ [
triangle ( x ; a ,b , c )=max ⁡ min ,
b−a c−b]}
,0

So, this called min-max function for triangle triangular membership fuzzy function.

64
(Refer Slide Time: 06:48)

So, either of these we can use to generate a triangular fuzzy membership function. So, here is
here you see we have a MATLAB code for generating triangular fuzzy membership function.
So, you we see here the code, MATLAB code which generates a triangular membership
fuzzy function whose the it generates the triangular shaped fuzzy membership function whose
vertices are a, b, c. So, a here is equal to 5, b is 7 and c is equal to 9.

So, we have basically if we make a triangular members fuzzy function it will be like this. If
we have let us say a here, so a ,b ,c and here the highest membership value that it can attain is
1. So, you see here that if this is a let us say and this is c, so it starts from a goes up to c and
this is the generic variable x’s and here is the membership value. So, the highest membership
value here that it attains is 1, ok.

So, the vertex the touches at 1, ok. So, a is equal to 5, b is equal to 7, c is equal to 9. So, if we
use this MATLAB program we will be getting this kind of shape whose vertices will be at 5,
7 and 9 like this. So, this is 5, this is 7, this is 9. So, if we use this program MATLAB
program we will be able to generate a shape, a triangular membership function like this.

We see here the values of the vertices that are given. So, 5, 7, 9 are given, if we change these
values we will get accordingly the triangular membership function whose vertices we assign
here. So, this needs to be noted.

65
(Refer Slide Time: 09:34)

Now, we see here the plotted shape that we used in MATLAB. So, if we run the, this program
in the MATLAB we will be able to get this shape plotted. So, this is the fuzzy membership
function which is triangular membership function whose vertex whose vertices are, whose
vertices are at 5 and then 7 and then at 9. So, as I mentioned before that if we change if we
assign the values of vertices a different values of vertices we can different get the different
kinds of triangular membership fuzzy functions.

(Refer Slide Time: 10:28)

66
So, as we see here if we assign a is equal to 1, b is equal to 3 and c is equal to 7 it means one
of the vertex is, one of the vertices is 1 and then the other one is at 3 the and then the last
vertex is 7 at 7. So, we see this kind of shape gets generated. So, this is how a triangular
membership function whose vertices are 1 ,3 ,7 gets generated.

So, by using this program, this MATLAB program you can generate any kind of triangular
membership functions. So, the input that you need to give to this function is only the vertex,
the, and the vertices and accordingly you can get the ship membership functions generated,
ok.

(Refer Slide Time: 11:38)

So, let us know using say MATLAB program let us give two vertices at 0 ,0, and the other
the third one 7. So, let’s see what is happening. So, if we give two vertices at 0 ,0, means two
vertices are at 0, so we see here these two vertices of the triangle are membership function
they are at 0 here and then the third one is at 7.

So, we see this kind of shape gets generated. So, this is also a triangular membership function
whose two vertices are at 0 and the third vertex is at 7. So, this kind of membership is also
called left side open, left open triangular membership function, all right.

67
(Refer Slide Time: 12:42)

So, then we have another kind of a membership, fuzzy membership function. So, this is a
trapezoidal membership function and this can be defined by this function, you see here we
can write simply

0 x ≤a

{
x−a
a≤ x ≤b
b−a
trapezoid ( x ; a , b , c ,d )= 1b ≤ x ≤ c
d −x
c ≤ x≤ d
d−c
0d ≤ x

And same as the triangular membership function here we this also has an alternate expression
for generating trapezoidal membership function.

x−a d−x
{ [
trapezoid ( x ; a , b , c ,d )=max min
b−a
, 1,
d −c ]}
,0

So, this is how, so, either of these can be used for generating a trapezoidal membership
function.

68
(Refer Slide Time: 15:06)

Now, here we have a member a MATLAB code for generating or plotting the trapezoidal
membership function. So, in this example MATLAB code we have a trapezoidal membership
function with a is equal to 3, b is equal to 4 and c is equal to 8 ,d is equal to 10, means we
have assigned the values a ,b ,c , d as 3 , 5 ,8 ,10 respectively. And if we run this code we
generate this kind of function.

(Refer Slide Time: 15:41)

So, you see we have 4 vertices and these 4 vertices are like this, that first vertex is at 3 and
then the second vertex is at 5 and the third vertex is at 8 and a 4th vertex is at 10. So, if we

69
change the values of a ,b ,c means we can change we can assign accordingly the place of
these vertices and similarly, we can change the here if we assign the values of these vertex
these vertices we get different shapes of the trapezoidal membership function.

(Refer Slide Time: 16:20)

Very interesting thing here is to note is that if we make the two vertices common, so means v
and c here they are at 5 , 5 if we assign a ,b ,c value; a ,b ,c , d values like that we get at
triangular membership function. So, we can use a trapezoidal membership function formula
to you know generate a triangular membership function also. So, what we do here is that we
simply have b and c common point, means here in this case b is at 5 and c is also 5 at 5.

So, with this we are generating a triangular membership function instead of trapezoidal
membership function. So, by using this MATLAB program for trapezoidal membership
function we can generate a different kinds of trapezoidal membership functions by changing
vertices suitably.

70
(Refer Slide Time: 17:51)

So, then here we have another kind of plot generated by the trapezoidal membership function.
So, if we keep a is equal to 0 ,b is equal to 0, we see that a and b come at the same point, 0.
So, here the two vertices a and b are here at this point and then c and c is equal to 3, d is
equal to 5, so we have two more vertices at different points. So, here we can make a left open
membership function also using this formula, this trapezoidal membership functions formula.

(Refer Slide Time: 18:38)

71
Now, comes the Gaussian membership functions. So, this is a very interesting membership
function, fuzzy membership function. So, this can be defined by a Gaussian and then x, so
this is actually the syntax so, this is the way we write a Gaussian membership function. So,

2
−1 x−c

gaussian ( x ; c , σ ) =e 2 ( )
σ

So, this way Gaussian membership function gets generated.

(Refer Slide Time: 20:07)

So, here is an example by using a MATLAB code the Gaussian membership function has
been generated and if you like to use this MATLAB code you can use for generating a
membership Gaussian membership function. So, gaussmf is the function which is being used
here. It is already in built MATLAB function, so here this is sigma and here this value is the
mean or the center. So, if we substitute these values in this syntax the gaussmf MATLAB
function, so we see this plot generated.

72
(Refer Slide Time: 21:07)

So, you see here the center of the Gaussian and the mean of the Gaussian comes at c, so c is
equal to 2 and then we have the standard deviation 0.5 which is used to generate this plot. So,
this is called a fuzzy Gaussian membership function.

(Refer Slide Time: 21:44)

Now, if we change these values we will get different types of or different you know Gaussian
function membership functions, so if we change here see here if we take c is equal to 5 and σ
is equal to 0.5, so we get this Gaussian membership function generated. See the center is
changed here. So, now as the center is at c is equal to 5 and the standard division remains 0.5.

73
(Refer Slide Time: 21:55)

Now, if we change the standard deviation, we see the shape of this Gaussian membership
function changing. So, this gets this the later one has the more spread. So, by substituting
sigma is equal to 1, we are spreading the Gaussian membership function. Similarly, if we
increase the sigma we will get more spread like this.

So, if we see that the center remains the same, center is at 5, but we see 3 Gaussian
membership functions with different σ ’s. So, sigma is increasing. So, with the increase of
sigma the spread is increasing. So, if we increased the sigma the spread is increasing. So, that
is how if you would like to plot a fuzzy Gaussian membership function we can very easily
plot Gaussian membership function by using this MATLAB code, ok.

74
(Refer Slide Time: 23:21)

Then comes the generalized bell-shaped membership function. So, the formula for
generalized bell-shaped membership function is here. So, this the syntax of this bell bell-
shaped membership function is written as

1
bell ( x ; a , b , c )= 2b

| |.
1+
x−c
a

So, the a here defines the width of the membership function, so I would say a controls the
width of the membership function, that means, the larger value creates a wider membership
function. So, larger value of a creates a wider membership function. And b defines the shape
of the curve on either side of the central plateau; that means, the larger value of b creates a
steeper transition; c defines the center of the membership function means the either we say
center or we say mean.

75
(Refer Slide Time: 24:50)

So, here also we have a MATLAB code for generating generalized bell-shaped membership
function. So, as we saw the formula for generalized bell-shaped membership function, if we
substitute the values of a ,b and c we can generate the bell-shaped membership function like
this.

(Refer Slide Time: 25:19)

So, we can use the same MATLAB code for generating this generalized bell-shaped
membership function. We have to substitute the values of a ,b and c and then we will get this
bell-shaped membership function generated.

76
(Refer Slide Time: 25:42)

So, here if we change the values of a ,b ,c accordingly we get the different generalized bell-
shaped membership functions generated.

(Refer Slide Time: 25:52)

Here also we see that as we change these values we get different kinds of membership,
generalized bell-shaped membership functions generated.

77
(Refer Slide Time: 26:05)

Here also we see by changing the, these values of a ,b ,c ; a , b , c here also we get these values
they generalized membership functions generated.

(Refer Slide Time: 26:21)

Now, we come to another kind of a membership function which is sigmoidal membership

function. So, sigmoidal membership function is given by

1
Sig ( x ; a ,c ) = −a( x−c )
1+e

78
So, here also a controls the width of the transition area and this a here is control the, a
controls the width of the transition area and c is nothing but again the center of the plot, the
center of the transition area, center of the transition area, this is center of the transition area.

(Refer Slide Time: 27:24)

So, like other membership functions we have the MATLAB code for sigmoidal membership
function also. So, this MATLAB code is given for the, for generating sigmoidal membership
function. If we use this MATLAB code and we feed the value of a and c, suitably we can get
the sigmoidal membership function generated.

So, like in previous MATLAB codes here also we have a sigmf function which is nothing but
MATLAB function, MATLAB function for generating sigmoidal membership function and x
is nothing but the generic variable, and this 4 and 5 are the values of a and c. So, this way if
we suitably input these values we can get you know suitable sigmoidal membership function
generated.

79
(Refer Slide Time: 28:42)

So, for a=4 and c=5 you see this kind of sigmoidal function will be generated. This is for
a=4 and c=5.

(Refer Slide Time: 29:00)

Now, if we choose another set of a and c we see that we get different type of a different
sigmoidal curve here, different sigmoidal plot here. So, this plot is for a=2 and c=5 and this
plot is for a=7, c=5.

80
(Refer Slide Time: 29:30)

Similarly, we can have other values like this and we can get other sigmoidal membership
functions created.

(Refer Slide Time: 29:41)

So, far we have discussed the following membership functions in this lecture. So, these
membership functions are triangular membership function, trapezoidal membership function,
Gaussian membership functions, generalized bell-shaped membership function, sigmoidal
membership functions. And remaining membership functions, remaining 5 membership

81
functions out of 10 commonly used membership functions we will be discussing in the next
lecture.

Thank you.

82
 Fuzzy Sets, Logic and Systems and Applications
Prof. Nishchal K. Verma
Department of Electrical Engineering
Indian Institute of Technology, Kanpur

Lecture - 06
Membership Functions

So, welcome to lecture number 6 of Fuzzy Sets Logic and Systems and Applications. So, in
this lecture we will be discussing fuzzy Membership Functions and these membership
functions are basically single dimensional membership functions. So, in the last lecture, while
discussing the fuzzy membership functions, we had already seen these commonly employed
membership functions.

(Refer Slide Time: 00:45)

And the you can see the names of these membership functions. So, out of these 10
membership functions we have already covered 5 membership functions. So, remaining 5
membership functions that is from left-right membership function, we will cover in this
lecture.

83
(Refer Slide Time: 01:23)

So, let us begin with this membership function; that is left-right membership function. Left-
right membership function is defined here by this function. So, you see here, the left-right
membership function can simply be written as or defined as

c−x

{
LR ( x ; c ,α , β )=
FL(
F (
R
α
x−c
β )
) x ≤c

x≤ c

So, if we see here F L and F R and these are just the functions. So, F Lfunction and F R function
you see here F L function and then F R functions and these F L, F R functions are monotonically
decreasing functions, so this has to be noted. And these are defined on strict interval here
with F L; when we have F L of 0 and this will be equal to 1. And if we have F R also, so F R of
0 will also be equal to 1.

So, we can also write like this as it is written. So, F L of 0 is equal to F R of 0 equal to 0. So,
what does this mean here is that, both the functions when we have taken of 0 this will be
equal to 1. And F L when x → ∞ means when x is approaching infinite this F L will be equal to
1 and so not only F L the F R also.

84
So, both the F L and F R x → ∞ is going to give you 0. Here these parameter c, alpha, beta
these three functions basically, these are the parameters which determine this shape of the left
right membership function.

So, c here defines the point, c this c defines the point where the value of membership function
is 1. And alpha, this alpha controls the width of the left region of left region for x less than or
equal to c; beta here, this beta in this left right function controls the width of the right region
for x more than or equal to c, so this way we define a left right membership function.

(Refer Slide Time: 05:00)

So, let us take an example here to understand this better. So, if we write a code in MATLAB
for implementing left right membership function, with α=40, β=10 and c=70. So, these
three parameters if we substitute in this function and based on this function left-right function
we will be able to plot left-right membership function by using this MATLAB code.

So, this MATLAB code you can simply write or you can copy and you can take this two
MATLAB. And you can run and if you run this MATLAB code you will see that with these
parameters you are going to get this shape.

85
(Refer Slide Time: 06:04)

So, when you have α=40, β=10, c=70, this kind of shape you are going to get. So, please
note that the c=70 is the point here at here at which the membership function has its value 1.
So, in other words we can say at c=70 the function has reached at the highest level and that
is 1.

And let me go back to the previous slide here and we see here that as I have already spoken
about F R here and F L; so, F R and F L both are the functions. So, in this case apart from these
parameters α, β and c we also take we also choose a suitable monotonically decreasing
functions. So, here in this case F R and F L has been have been chosen as

3 3

F R ( u ) =e−¿ u ∨¿ ¿ and F L ( u )=e−¿ u ∨¿ ¿

So, if we have substituted this function these parameters. And we have taken the function
included in this we’ll be able to get the plot here as, you see after getting these MATLAB
code executed.

86
(Refer Slide Time: 07:51)

So, let us take another example here to understand this better. Here in this example we have
three cases for left-right membership function plot. So, the first case here you see this first
case, in which we have α=30, β=12 and c=50. So, if we take this value α=30, you see it
means α=30, β=12 and c=50.

So, c=50; means this is the point 50 is a point where this particular plot will have it’s, this
particular plot will have its highest value. So, if we see here the plot for these values and let
me make it clear here that here also we have used the same monotonically decreasing
function.

So, but we are only varying these alpha the values of alpha, beta and c. So, these are the three
cases the first case is where α=30, β=12, c=50. And the second case we have α=30 and
β=12 and c=6 0. And the third case here is α=30, β=12 and c=7 0.

So, we can clearly see here that, what we are wearing here is the c only. So, in all the three
cases we are changing only the value of c; it means we are shifting the point where we have
the membership functions value 1. So, let me go back to the function definition. Here, you
see here it is clearly written that c defines the point where the value of membership function
is 1.

So, in this example, in this example this, this c is changing in one case we have 50, in the
other case we have 60 and then the third one we have 70. So, we can clearly see that for the

87
first membership function, which is shown in blue. So, we see that here the membership
value is 1; we see here. And the second case we have its membership value 1. So, this is the
other membership function the second membership function the second case. And the third
case here we have the membership value 1. So, this way we see that, the c is the parameter
which is changing in all the three cases rest two parameters are not changing at all.

So, as we clearly know that the α is responsible for change in the left half of the region about
c. And β is responsible for the change of the shape the right half of the c. So, we can clearly
see once again I would like to tell you that these two values are same alpha, beta are same in
all the three cases only the c is changing.

(Refer Slide Time: 12:08)

So, let us take another example, here where you see we have again three cases where we are
only changing alpha. So, values are alpha are changing see here see in one case we have 60,
the second case we have 50 and the third case we have 40. And other two parameters in all
the cases are same like beta, c are same in all the three cases. So, if we see here c, since c the
value of c remain same in all the three cases.

So, let us see where the c is c=70; so, you see 70 is this point this point and around this I
mean this side is left side of this c and this side is right side of this c. So, if you see here as I
already mentioned that, when we vary when we change alpha. So, we see clearly that left side
of the curve is changing. So, this is responsible for the change in the shape of the left side on
the left side.

88
(Refer Slide Time: 13:26)

And if we change in the other example we see here that, c remain same in all the three
examples and α remains same see here 40. So, in all three cases alpha, c remains the same
only β is changing. So, in one case we have β as 5 and the second case we have β 12 and the
third case we have β 20.

So, beta is changing as 5, 12, 20. So, we see that c is here c is the point c is basically I can say
it can be regarded as a center around which you know we have the left side on the left side
change β is responsible right side change the α is responsible.

So, c here is at 70 and if we see very clearly by changing the values of β, we see that this
side is the shape of curve is changing. The plot we can clearly see that the blue is for beta=5
and a red is for β=12 and yellow is for β=20. So, this way we can control the shape of the
left right membership function by changing alpha beta and c suitably.

89
(Refer Slide Time: 15:10)

Now we can go to the other membership function, which is a called pi shaped membership
function. So, pi membership function there are two types of pi membership functions; there
are two functions; basically, first function here is represented by π 1. So, π 1 function is having
only two parameters.

1
π1 ( x ; a , b)= 2
x−a
( )
1+
b

So, this pi membership function has only a and b; that means, two parameters and these two
parameters actually control the shape of π membership function. So, and then, here we have
another four variabled π membership function.

So, but this four variabled π membership function has four parameters. So, this pi
membership function is represented by,

lw

{
x <lp
lp+lw−x
π 2 ( x ;lw , lp ,rp ,rw ) = 1 lp ≤ x ≤ rp
rw
x >rp
x−rp+rw

So, this way we have two kinds of pi membership function. And here in the second case so,
first case is represented by first type of pi membership function is represented by π 1 which is

90
clearly mentioned here. And the other kind of pi other type of pi membership function here is
represented by π 2. So, π 1 is only two variable membership function and this function
basically we can say this function has membership value at point a membership value 0.5 at a
minus b; which is mentioned over here and a plus b respectively.

So, unlike the function pi, function decreases towards 0 asymptotically, as we move away
from point a, so this is here you can see represented by π 1. Now, in the second case which is
represented by second type of pi membership function which is represented by π 2; this is
characterized by four parameters lw, lp, rp, rw.

Parameter lw and rw define the feet of the membership function and lp rp define its shoulders
respectively. So, this is very interesting membership function which was used in the
controller fuzzy controller, which was very popular fuzzy controller used for the sendai
subway train control system.

(Refer Slide Time: 19:34)

So, here we have the MATLAB code which you can use for plotting pi shaped membership
function. And this has been given with an example a case where we have lw, lp, rp, rw values
given as 2, 5, 8, 10 respectively.

So, we can clearly see here that this pi shaped membership function is a second type
membership function which requires four parameters. So, we see here the MATLAB code.

91
So, if we can if we write this code in the MATLAB and run this code in the MATLAB we get
this kind of plot.

(Refer Slide Time: 20:30)

So, we see here the shape of this plot is pi type. So, we see here as I mentioned the lw, lp, rp,
rw values. And I had already explained these parameters as this lw and r w defined the feet.
So, lw defined the left feet, r w defined the right feet, whereas l p defined the defined its left
shoulder and r p defines its right shoulder.

(Refer Slide Time: 21:15)

92
So, now let us understand this pi shape membership function better. And for in this example
we have taken four cases to better understand these parameters which are controlling the
shape of the pi membership function.

So, we have taken first case here where lw=0.5, l p=5, r p=7, r w =10. And I, I already
mentioned that l w is controlling left feet lp is controlling left shoulder, r p is controlling right
shoulder, r w is controlling right feet. So, we have taken four cases here. So, in first second
third fourth we see the only l w is varying.

So, this is this has done intentionally to make you understand that if we vary if we change l w
only what is happening to the pi shaped membership function. And here in all the three cases
only lw is changing rest three parameters are remaining the same. So, here if we see as I
mentioned that lw is controlling the left feet. So, if we see here the left feet is changing by
changing by varying the values of lw; rest of the rest other values all three values are the
constant the lp is constant rp is sorry the lp lp rp rw remains the same.

So, that is why only feet is changing and that is and only left feet is changing rest other things
remain the same for this pi membership function. So, if you see here in all the three cases
right feet remains the same right feet is this. So, right feet remains the same and then shoulder
here the left shoulder remains the same, right shoulder remains the same. So, we clearly see
that only by changing the lw only left feet is changing. So, this has to be noted here.

(Refer Slide Time: 24:18)

93
And then similarly if we take another example, where we see that left feet remains the same
in all the four cases only lp is varied. So, we are changing the values of lp only and we clearly
see that left since the left feet values are same in all the four cases so, left feet remains the
same. And what is interesting here to see that by changing lp the values of lp in all four cases,
you see the left shoulder is changing this is clearly visible here that the shape of the left
shoulder is changing.

(Refer Slide Time: 25:12)

Similarly, if we take another example where the out of all the four parameters of pi shaped
membership function, if we change only rp that is right shoulder and the rest other parameters
remain the same. If you see here, lw is not changing in all the cases it remains at 2. So, that is
why it is not changing it remains the same in all the four cases. r w is 10 and this is also
remains at the at 10. So, this is also not changing.

So, it means that left feet is not changing right feet is also not changing. And here if we look
at the shoulders like left shoulder see left lp here in all the four cases remain the same and it
remains at 4. So, this is this also remains the same in all the four cases. Whereas, since we are
changing rp; so, we see here the rp is changing means the shoulder width is changing in all
the four cases.

So, first case we have rp is equal to 6. So, at 6 you see here the shoulder width is shoulder is
coming at 6 and then again if you are changing rp which is here is at 7 and then it is changing
at 8. In the third case and then again it is changing to 9 at in the fourth case.

94
So, this way we understand as to how all these four parameters are changing the left feet,
right feet, left shoulder, right shoulder of a pi shaped membership function.

(Refer Slide Time: 27:18)

So, yes so here we are changing the right feet. And if we change the right feet here we see
clearly that the feet is changing So, in all the four cases in this example only rw is changing,
rest three parameters lw, lp, rp remains the same. So, here also we clearly see that the feet the
right feet is changing as per rw values.

(Refer Slide Time: 27:57)

95
So, then let us discuss the open right membership function. So, in the open right membership
function, we see that this kind of membership function is defined by

0 x<α

{
OpenR ( x ; α , β ) = x−α α ≤ x ≤ β
β−α
1 x> β

So, when we try to plot the membership function or when we try to draw a membership
function based on this function, we will be going to get this kind of shape. So, this point on
the x axis we will have as alpha and this point here we will have as beta. And we can clearly
see that this function is open in the right side. So, this is important to be noted.

(Refer Slide Time: 29:44)

And similarly for this also we have a MATLAB code for your help. So, you can clearly copy
this MATLAB code or you can rewrite this MATLAB code in the MATLAB. And you can
run this MATLAB code with α=3 and β=6 you will be going you will be getting this kind
of shape.

96
(Refer Slide Time: 30:10)

(Refer Slide Time: 30:12)

Similarly, since we saw here the open right membership function here where open right
membership function because the right side of this membership function remains open. So,
when we say right side of this membership function remains open means limit x tending to
infinite function the open right function will always be equal to 1. So, when this function
finally, is this function is going to attained the highest value which is 1.

So, likewise we have the open left membership function. And this membership function can
be defined by

97
 1 x <α

{
OpenL ( x ; α , β )= β−
β−α
x
α ≤ x≤ β
0 x>β

(Refer Slide Time: 31:43)

So, similarly here also we have a MATLAB code that will help you to plot open left
membership function by substituting different values of alpha and beta. So, in this example
we have chosen the value of α as 5 and value of βas 7.

(Refer Slide Time: 32:12)

98
So, by substituting this these values of α and β, we see here open left membership function.
And we can clearly see that left side of this membership function is open in the sense that we
have the values the final values or we have the left side if we if we take this value as alpha,
this value as beta, this is α=5 and the β=7.

So, left of 5 left of α=5 the membership function, or the membership values or I would say
the membership function will have membership values equal to 1. Now, we come to another
membership function which is called S shaped membership function. So, S shaped
membership function is defined by two parameters and these parameters are a and b.

(Refer Slide Time: 33:09)

So, this is defined by

0 x ≤a

{
x−a 2 a+b
f ( x ; a , b)=
( )
2
b−a
a≤ x ≤
2
2
x−b a+b
( )
1−2
b−a 2
≤ x≤b
1 x ≤b

And let me also tell you that this parameter a defines the feet of the membership function and
b defines its shoulder.

99
(Refer Slide Time: 34:58)

So, like other membership functions here also for S shaped membership function we have a
MATLAB code ready for you. And if you would like to take this and execute in the
MATLAB with a=4, b=8.

(Refer Slide Time: 35:23)

We clearly see that this kind of membership function the S shaped membership function we
are going to get. So, as I already mentioned that a, b these two parameters are characterizing s
shaped membership function a controls the feet of the membership function and b controls
the shoulder of the membership function.

100
(Refer Slide Time: 35:47)

So, here we have few more cases. So, in this example we have two cases; the first case we
have a=4, b=8 and a and the second case we have a=4 and b=9. So, we can clearly see
that in this example the feet of the curve is not changing feet of the curve is remain remaining
at the 4.

So, here a=4 we have taken and you see the shoulder is changing here the shoulder is one
the first shoulder is at b=8 and the second the shoulder for the second plot is at 9. So, by
changing the suitable values of a and b we can keep changing the feet and the shoulder.

(Refer Slide Time: 36:56)

101
So, here as I mentioned that if we change the values of the feet we suitably we can change the
places of the feet. And here in these two cases we have b constant. So, b is not changing so,
that is why the shoulder remains at the same point. So, shoulder remains in both the cases
same. So, this way the S shaped membership function can be plotted, And let me also tell you
here it’s very important to mention that s shaped membership function fuzzy membership
function, can also be used as the right open or left open type of membership functions.

So, this membership function since the left right side of this membership function remains at
finally, at one all the time, so this qualifies to become the right open type of fuzzy
membership function.

(Refer Slide Time: 38:15)

So, with this I would like to stop here. And in the next lecture we will discuss the
nomenclatures used in the fuzzy sets.

Thank you very much.

102
 Fuzzy Sets, Logic and Systems and Applications
Prof. Nishchal K. Verma
Department of Electrical Engineering
Indian Institute of Technology, Kanpur

Lecture - 07
Nomenclatures used in Fuzzy Set Theory

Welcome to lecture number 7 of Fuzzy Sets, Logic and Systems and Application. So, in this
lecture, Nomenclatures used in Fuzzy Set Theory will be covered.

(Refer Slide Time: 00:34)

So, here are few nomenclatures that are listed. So, we have here 15 nomenclatures that are
commonly used in fuzzy set theory. And in this lecture, we will be discussing support, core,
cross over points, height, normality, sub normal fuzzy set.

103
(Refer Slide Time: 01:06)

So, let’s now discuss the nomenclatures used in fuzzy set theory one by one. So, the first one
is support. So, what is a support? So, the support of a fuzzy set A is the set of all the points
small x in universe of discourse capital X for any associated membership function such that
μ( x ).

So, since we are talking of a fuzzy set A. So, that is why we are saying here the μ A . So, this A
represents the fuzzy set; the name of the fuzzy set, so μ A ( x). So, μ A ( x)>0 . So, support of A
is written here.

Support ( A ) ={x∨μ A ( x ) >0 }

So, it is very clear that; if I have a fuzzy set let us say here, so this fuzzy set has here for the
value of x 1, the mu x 1 is 0. See here the μ( x ¿¿ 1)=0 ¿. And then at x 2, μ( x ¿¿ 2)=0 ¿. So, and
I will be writing A here and A here, because we have taken fuzzy set A; and this μ
corresponds to the fuzzy set A, so that is why μ A has been written. So, now, this two points x 1
and x 2 will be excluded from all the values of x here, which has the membership values more
than 1. So, this point will not be x 1 and x 2 point will not be covered in the support of a fuzzy
set.

So, if we have a continuous fuzzy set here as it is shown here, so we will have all those
points; because since this is a continuous point, we will have as per as the number of

104
elements are concerned, we may have the infinite number of points. So, that is why it is
shown by a line here. So, we can clearly represent this line as the support of a fuzzy set A and
this fuzzy set is a continuous fuzzy set.

(Refer Slide Time: 05:01)

So, let us now take an example here where the fuzzy set A is a discrete fuzzy set. So, if you
take a fuzzy set here as A is equal to this

A={( 5 ,0.1 ) , ( 6 ,0.2 ) , ( 7 ,0.3 ) , ( 9 ,0.9 ) , ( 10 , 1 ) , ( 11, 0.5 ) }

, we have a set fuzzy set which includes all these points; where the first element here is the
generic variable value and then here the second element is the second the element of this pair
the value and then the corresponding membership value.

Similarly, we have 1, 2, 3, 4, 5, 6. So, in this discrete fuzzy set we have six elements. And so,
we look at this fuzzy set. It is very clear that all these generic variable values 5 , 6 ,7 , 9 ,10 , 11,
they have their corresponding membership values all greater than 0. And as per the definition
if we are interested in finding the support of a fuzzy set, especially here; this fuzzy set is a
discrete fuzzy set.

So, we can write the support of a fuzzy set A for this case will be the collection of all the
points for which the corresponding membership values ¿ 0. And we can clearly see here, all
the points all the 6 points here; first, second, third, fourth, fifth, sixth. All this points, all the

105
generic variables have their corresponding membership values ¿ 1. So, it means that, the
support of A will have a set of all those generic variable values.

So, it means if we would like to write the support of fuzzy set; specially the discrete fuzzy
set, then we will have here for this case, for this discrete fuzzy set will have a set of points 5,
6, 7, 9, 10, 11. So, this way we will write the support of a fuzzy set like this

Support ( A ) ={5 ,6 ,7 ,9 , 10 ,11},

(Refer Slide Time: 07:48)

Now, let us discuss course the core of a fuzzy set. So, the core of a fuzzy set is the set of all
the points in the universe of discourse such that μ A ( x ) ¿ 1.

So, this can be represented as if you would like to write the core of a fuzzy set and this will
be equal to the collection of all the points or the set of all the points for which the
corresponding membership values are 1. We can very clearly understand the core of a fuzzy
set like this. If we take a fuzzy set A, this is fuzzy set here and we would like to find the core
of this fuzzy set, we’ll take the collection of all the, we’ll take the set of all the points here for
which corresponding membership values are 1.

So, we can clearly see here for this membership function which is a trapezoidal membership
function and of course, this is a continuous fuzzy set. So, we’ll we can say that this line x 1 till
x 2, we have for each and every point which is coming in between we’ll have the membership

106
their corresponding membership values equal to 1. So, this line will represent the core of
fuzzy set; here for this fuzzy set.

(Refer Slide Time: 09:49)

So, let us now take an example to better understand the core of a fuzzy set, especially for the
discrete fuzzy set here. So, let us take this example fuzzy set A here is given by A see here
and this fuzzy set has I, II, III, IV, V, VI points. And this points basically are the fuzzy
points; fuzzy point means it has two values. This is a pair, the first value as we have seen in
the previous case also previous example also. The first element here is the generic variable
value and then the second element here the pair the second element of this pair is the
membership value, corresponding membership value.

So, if we look at this fuzzy set, we have six elements, six points and if we see we find there
are two points. There are two generic variable values for which the corresponding
membership values are 1. So, one is here; the first one is here and then the second one is here.
So, as per the definition, core of a fuzzy set is collection of all the points, all the generic
variable values for which the corresponding membership values are 1.

So, that is why here if we apply that will find 8 and 11 and we can write here as a set. We can
we can write core of a fizzy set, core of a discrete fuzzy set is collection of the generic
variable values which are 8 and 11. So, this is very simple very clear. So, for discrete fuzzy
set we if we are interested in finding core of a fuzzy set we can very easily collect those

107
values those generic variable values for which the membership; the corresponding
membership value is 1.

Rest generic variable values we just leave aside. So, we only take the generic variable values
here and this is very important here to be noted that when we are writing either support of
fuzzy set core of a fuzzy set, we only write the generic variable the collect the collection of
generic variable values. We do not write the corresponding membership values also along
with the generic variable values.

So, we only write the values of x and that is how we get the support of a fuzzy set and core of
a fuzzy set.

(Refer Slide Time: 13:18)

So, now let us move to another nomenclature that is used in fuzzy systems theory; the
crossover points. So, the crossover points of a fuzzy set A is the points x in universe of
discourse capital X at which μ A ( x ) ¿ 0.5. It can be represented as the crossover of A (Refer
time: 14:00) and this crossover of A will give us a set of all the values of x and as we know
this x is nothing, but the generic variable values. And, these values will be only those values
for which μ A ( x ) =¿ 0.5; means the half.

So, if we take an example here to understand the crossover of a fuzzy set A here. So, if we
have a fuzzy set A. This is a fuzzy set A, is A fuzzy set and we try to find the point
corresponding to which a membership value is 0.5.

108
So, x 1 is the point here corresponding to which the membership value is half or 0.5, so x 1. For
this fuzzy set, we have two values x 1 and x 2 for which we see here the membership value is
0.5.

(Refer Slide Time: 15:29)

Similarly, here if we would like to understand better, we can refer to this example for
crossover points.

So, if we have a fuzzy set A like this. Here we have a fuzzy set and in this fuzzy set we have
one, two, three, four, five, six, seven, eight, nine, ten, eleven elements or eleven pairs. So, if
you are interested in finding out crossover points for this fuzzy sets, we only look for the
generic variable values for which the corresponding membership values are 0.5. So, in this
fuzzy set this is a discrete fuzzy set obviously.

So, we see here in this fuzzy set 7 is the generic variable value for which the membership
value is 0.5. So, this is the first generic variable value that is we have found and then let’s
now look at some other value for which we have the membership value of membership value
half. So, we see here 14 is another point for which the membership value 0.5. So, this way we
find two points; 7 and 14 and their corresponding values are their corresponding membership
values are 0.5.

So, that is how we write here the crossover of a fuzzy set.

Crossover ( A ) ={7 , 14 }

109
So, this way we get crossover of a discrete fuzzy set.

(Refer Slide Time: 17:40)

Now, here is representation of core support and crossover points of a fuzzy set. So, if we take
a fuzzy set, let us say some fuzzy set which is trapezoidal fuzzy set here. If you are interested
in core of this fuzzy set, we can clearly see here that as per the definition; core of the fuzzy is
represented by the line CD here. So, if we look at this point here.

So, right from C we have all the generic variable values corresponding to which the
membership values are 1. So, that is how, and this goes still C till D. So, this way CD will
represent the core of a fuzzy set. And this fuzzy set here is the trapezoidal fuzzy set which is
continuous fuzzy set and if you are interested in finding support of the fuzzy set.

So, support of a fuzzy set is as per definition, it its collection it gives us the collection of all
the generic variable values for which their respective membership values are more than 0. So,
this will start right from 0 plus. Right from 0 plus means; the this will start from A. A is a
point where we have the membership value, the A is the point let us say if it is x A .

So, this x A will have it is corresponding membership value 0, but x 1 +ϵ will have it’s, it’s
corresponding membership value more than 0. So, just after A, all the points and just before
F, all the points of the generic variable will be included in the support of a fuzzy set. So, this
way if you are interested in finding the fuzzy set, we can say the support of the fuzzy set is
represented by the this line the red line. And please a please look at the ends of the A and F, I

110
mean this line. So, we see here the hollow circle. So, we see a hollow circles at both ends; it
means that the generic variable value for which the mu x is 0 is not, these are not included at
both the ends.

So, this way we clearly understand what is a support of a fuzzy set and then when we are
interested in finding the crossover of this fuzzy set of course, we will find the points on the
fuzzy sets. The generic variable values on this in this fuzzy set for which the membership
values are 0.5.

So, please see here clearly that we have B point and E point for which the corresponding
membership values are 0.5. So, that is how the crossover points are B and E. So, this way we
here make the clear distinction among the core support and a crossover points of a fuzzy set.

(Refer Slide Time: 21:57)

Now, let us discuss the height of a fuzzy set. So, how can we find height of a fuzzy set? So,
height of a fuzzy set first of all is defined by hgt ( A ) and this

hgt ( A )=1hgt ( B ) =0.40hgt ( C ) =0.75

So, A is represented by the blue line and B is represented by the red line red color and C is
represented by the violet color.

111
So, if we talk of A; so we clearly see that the highest value of membership here goes till 1.
So, if we take all the membership values corresponding all the generic variable values, we
will find the max of the membership values of this fuzzy set A. So, I can write it like this
like.

If I have μ A ( x) here and if I take this by the middle bracket like this and if I take max of
these, I am going to get 1. Similarly, we talk of height of the fuzzy set B here, for B fuzzy set
the height of the B fuzzy set will be 0.4 because if we will take all the corresponding
membership values and take the max of this will be getting 0.4.

(Refer Slide Time: 24:11)

Discrete fuzzy set can be understood by this example.

So, if we take a fuzzy set A here. So, this is fuzzy set A. So, if we look at all the values of the
membership here; membership values. So, we see that these are the values of these are
corresponding membership values and if we collect these values and take max of this, we are
going to get max of the max of μ( x ), μ A ( x) and this will be equal to 1 for the fuzzy set A. So,
this way the height of the fuzzy set here will be this; we can write as hgt ( A ). So, hgt ( A )=1.
Similarly, what will be the height of fuzzy set B? So, by looking at the fuzzy set here we can
clearly see the highest value of membership that it has it contains is 0.9 in this membership,
in this fuzzy set.

112
So, the height of fuzzy set B will be 0.9. Similarly, for B for C fuzzy set, the height of C
height of C fuzzy set will be 0.7 in this example; because the highest value of the
membership, the maximum value of membership values or I can say if we take max of all the
membership values that it contains will be 0.7.

So, this way we clearly find the height of the fuzzy set.

(Refer Slide Time: 26:18)

Now, let us look at the let us now discuss the normality of a fuzzy set is nothing but it is
defined as a fuzzy set A is normal if it is core is non empty. So, this can be defined in other
words as we can find we can always find a point x in universe of discourse capital X such
that μ A ( x ) =1.

So, what does this mean here is that if we have a fuzzy set A let us say and if this fuzzy set A
has at least one point, one generic variable value for which or it’s corresponding membership
value is 1. So, at least we should have a membership value is equal to 1. So, in other words
we can understand this as that if the height of the of a fuzzy set can go up to 1, the fuzzy set
will be a normal fuzzy set.

So, we can see here in this diagram, we have a fuzzy set a here this is the fuzzy set. And if
you see here x 2 is the point, x 2 is the generic variable value for which we have the
membership the member ship value 1. So, if we have such case where we are getting a point
at least a point like this we can say the fuzzy set is a normal fuzzy set.

113
So, there may be case when we may not be able to find any such point for which the
corresponding membership value is 1. So, this kind of membership, this kind of fuzzy set is
referred to as a sub normal fuzzy set. So, here corresponding to this fuzzy set A, the core of A
is x 2and since the core of A is non empty of course, because we are getting at least a point x 2
for this fuzzy set.

So, we can say the core of A is non empty. So, this fuzzy set is a normal fuzzy set. There may
be a case where core of a fuzzy set empty. So, such a fuzzy set is referred to as sub normal
fuzzy set.

(Refer Slide Time: 29:40)

So, let us now take these examples here. A, B of discrete fuzzy sets A , B ,C and we see here
that if we take a fuzzy set A and we clearly see that we have in this fuzzy set; we get a point,
we get a fuzzy point where we get the μ corresponding μ the corresponding membership
value 1.

So, it means that we have at least one point present one fuzzy point present one point present
for which we have the corresponding membership value 1. And please understand when we
say at least it means if we have multiple such points present then also this fuzzy set will be a
fuzzy set; a normal fuzzy set. So, at least means at least one such point is present for which
the corresponding membership value is 1. So, A is a normal fuzzy set, A is a normal fuzzy set
this is a normal fuzzy set. Now what about B fuzzy set? So, let us now check all the points
here for fuzzy set B is a B fuzzy set.

114
So, in B fuzzy set we if we see here in the fuzzy set, all the points. So, we do not find any
such a point such a membership such a generic variable value for which we get the
membership value 1. So, we can say this fuzzy set is not normal or we can say the B fuzzy set
is subnormal. Similarly, now let us check a fuzzy set C.

So, just by going through the all these points here which are in the in this set, fuzzy set. So,
we see that we get here this point, so the generic variable value 13 has it is corresponding
membership value 1. And, so at least we have one such a point present, such generic variable
value present for which we have mu of the membership value is 1.

So, that is why we can say C is normal fuzzy set. So, this way we can clearly check all the
normality of fuzzy set or normality in a fuzzy we can check.

(Refer Slide Time: 33:08)

Now, we have a diagram here and this figure has the fuzzy sets A , B ,C. So, let us now
clearly check whether A is a normal fuzzy set or a sub normal fuzzy set. So, A, B or C all
these are normal fuzzy sets are subnormal fuzzy sets. So, if we see A here, A reaches up to A
is the height of fuzzy set A has or is 1. So, we can say here the height of a fuzzy set A is 1.
So, it means fuzzy set A reaches up to 1. So, it means it is membership value the highest
membership value, the maximum membership maximum of all the membership values is 1.

So, that is why we can say the core of this fuzzy set is non empty and if the core of this
membership core of this fuzzy set is non empty; it means that this fuzzy set is a normal fuzzy

115
set. Whereas, if we see here will not be getting any point any generic variable value for which
we get μ ( x )=1. So, this will be a sub normal fuzzy set. Similarly, here C also will be the sub
normal fuzzy set.

So, only A is a normal fuzzy set, right. So, because the core of A which is written here. So,
core of A; this is nonempty non empty. And if we talk of C, so since core of C is empty here
and core of B is also an empty set. So, that is why this B and C both the sets are subnormal
fuzzy sets.

(Refer Slide Time: 35:45)

At this point I would like to stop for this lecture and in the next lecture we will be discussing
the remaining nomenclature used in the fuzzy set theory. So, it means we will be you know
moving ahead to discuss remaining nomenclature.

Thank you.

116
 Fuzzy Sets, Logic and Systems and Applications
Prof. Nishchal K. Verma
Department of Electrical Engineering
Indian Institute of Technology, Kanpur

Lecture – 08
Nomenclatures used in Fuzzy Set Theory

So, welcome to the lecture number 8 of Fuzzy Sets, Logic and Systems and Applications. In
this nomenclature, some of the terms that we have already discussed remaining terms,
remaining nomenclatures that we will be discussing in this lecture.

(Refer Slide Time: 00:36)

And, as I mentioned; the support, core, crossover points, height, normality, subnormal fuzzy
sets, all these terms in the nomenclatures have already been covered. And in today’s lecture,
we will discuss the remaining terms in the nomenclatures used in the fuzzy set theory.

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(Refer Slide Time: 00:54)

Now, by now we understand what is a normal fuzzy set. So, normal fuzzy set is a fuzzy set,
which whose height is 1. So, if we have fuzzy sets whose height is not 1, let us say and we
would like to increase its height up to 1 or in other words if you say that; we would like to
normalize a fuzzy set A.

So, let us see how we can do that. So, if we have a fuzzy set A and A is a subnormal fuzzy
set. So, if we have here; if we have a fuzzy set A which is subnormal, this fuzzy set is
subnormal. And, if we would like to normalize this fuzzy set; it means, we would like to
increase the height of this fuzzy set up to 1. So, here we can better understand this
normalization by taking a examples and one of the examples is here.

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(Refer Slide Time: 02:13)

So, if we take fuzzy set here A, which is subnormal fuzzy set. We see here, this is subnormal
fuzzy set and so, when we say subnormal means, none of the elements here present in this
fuzzy set has its corresponding membership value 1. So, we can clearly see here that this
fuzzy set is this A fuzzy set is a subnormal fuzzy set. Now, we would like to normalize this
fuzzy set. So, we first find the height of this fuzzy set and height of this fuzzy set is 0.8
because, the maximum of all the membership values present in this fuzzy set is coming out to
be 0.8, which is here.

So, we have the height of a fuzzy set now and then simply we divide this fuzzy set. So, when
we say we divide this fuzzy set, it means we divide the corresponding membership values.
So, this has to be noted here that like the first element in the fuzzy set A here and the
corresponding first element here in the fuzzy set A dash, which is a normalized fuzzy set. So,
the membership, only the membership value is normalized. So, we need to understand that
the generic variable value which is 5 here remains as it is. So, 5 remains as it is, but the
corresponding value is normalized corresponding membership value is normalized.

So, how is it normalized here is. So earlier, the 5 has its corresponding membership value 0.1.
Now, if we divide it by 0.8, this becomes 0.125. So, this way this the corresponding
membership value is changed. And similarly, 6 will have 0.5 and then 7 will have 0.625. And
similarly, all other points will have its corresponding membership values normalized. So,

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clearly here, if we see here 8 will have in A ' 8 will have its membership value 1, which
earlier was 0.8 in the A membership A fuzzy set, which was a subnormal fuzzy set.

So, now we see that the A dash is the normal fuzzy set; normal fuzzy set. And normalization
process we understood very clearly. So, when we say here, A divided by height of A it means
that, we are dividing the respective membership values, the member the generic variable
values are never normalized.

(Refer Slide Time: 05:57)

Now, coming to fuzzy singleton. So, what is a fuzzy singleton in fuzzy systems theory is a
fuzzy set whose support is a single point and of course, in the universe of discourse capital X ,
and so support is a single point and then, there is another condition the single point should
have its membership value equal to 1.

So, if this condition is satisfied, we call the corresponding fuzzy set, the fuzzy set for which
this condition is satisfied is a fuzzy singleton. In other words, a fuzzy singleton is having core
with only one element in its set, what does it mean? It means that, if I have a fuzzy set and if
we try to find the, if we are interested to know whether this fuzzy set is a fuzzy singleton or
not, then we will quickly try to find the support or I would say a core. So, if this core is a
single value, then the core is the single value and the support is also a single value. So, it
means what? It means that, a fuzzy set which has a single element. So, the support is a single
value and the core is also a single value, but if the support is a single value it should have its
mu equal to 1.

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So this way here, if we see that, if we have a fuzzy set A which will be a fuzzy singleton
when the corresponding x here, so if we take a support here, support here will be only one
value which is x 1 and this at this x 1 will have its corresponding membership value 1. So,
support so you see here, the support of A is nothing but x 1 and of course, the support has a
single element and the core is also a single element. So, in both the cases we have x 1 only.
So, this x 1 this kind of fuzzy set is called a fuzzy singleton.

(Refer Slide Time: 08:46)

Now, let us understand the α−cut of a Fuzzy Set in a fuzzy systems theory. So, alpha cut or
alpha level set of a fuzzy set is a crisp set that is defined as

A α ={x∨μ A ( x ) ≥ α }

Alpha can be any value in between 0 to 1, it can be 0 also. So, let us now try to understand,
what is alpha cut of a fuzzy set?

So, if we have a fuzzy set A here, this there is a fuzzy set A, and if we are interested to find
the α−cut of this fuzzy set. So, if we choose to have my α like this, this is nothing but the
membership value alpha. So, the corresponding to this, we try to find the generic variable
values. So, which is here in this case we have a and then here if we extend this we will have
another point here, which is another point of the generic variable b. So, we will get two
generic variable values which are here a and b.

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So, a and b will correspond to the values, the generic variable values at which we have
membership value α. So, if we take α we have to see whether the membership value is alpha
or not. But, this α as I mentioned can be any value in from 0 up to 1. And please understand
that, when we talk of α−cut, so we are not interested in only quality, but we are interested in,
we will have collection all the generic variable values for which, the their corresponding
membership values are more than r =α.

So, in this case all the points in between even in including a and b will be included. So, that
is why, we are showing this all these α−cut . So, all these values the collection of all the
values in between right from a to b, and please understand that, a has been showed by solid
dot and b also has been shown by a solid dot. So, it means what? It has this line will include
all those points, all the points for which we have the membership values corresponding
membership values either α or more than α.

(Refer Slide Time: 12:10)

So, this we this way we understand α−cut. Now, if you are, we have another nomenclature,
which is another nomenclature that we use here, is a strong alpha. So, if we are interested in
finding out a strong alpha cut. So, everything remains same except the quality goes away. So,
if we are interested in a strong alpha cut, so a strong α−cut is represented by A α' . So, A α' is
equal to collection of all the generic variable values for which we have the corresponding
membership values more than α. So, in this case here, if we take the same example as we
took in the α−cut case. So, instead of this solid dot here we will have the hollow dot.

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What does this mean, that we are excluding the quality condition means this the generic
variable value a and b will not be in included because, at this point here α is exactly equal to
the membership value is exactly equal to α. So, we will take only those values of the generic
variable for which, the membership value is more than α. So, more than α will we will be
getting by this line. So, because of that the terminals of a and b will be hollow it means that,
this points will not be included; the start of a will not be included because, here the alpha is
the membership value is α and here at b also the membership value is exactly equal to α.

(Refer Slide Time: 14:34)

So, let us now take an example here, to better understand alpha cut and strong α−cut of a
fuzzy set. So, here is an example. So, if we take a fuzzy set A for universe of discourse X
and if we are interested in finding out a alpha cut and this alpha let us say is 0.4 here, and it
could be any value in between as I mentioned that right from 0 to 1. So, let’s now try to find
the α−cut of this fuzzy set A and the alpha the value of alpha is equal to 0.4. So, we can
represent as I mentioned this alpha cut by A subscript 0.4. So, let’s now try to see here in the
fuzzy set A.

So, we see here this point, the fuzzy set, so 6 we can we will have 6 and then 7 and then 8, 6,
7, 8 and then 9 and then we will have 11, 12, 13, 14, 15. So, all are all these are the generic
variable values for which, the corresponding membership values are either 0.4 or more than
0.4. So, this way we can write A 0.4. Now, if we change the value of alpha to 0.6. So, my A

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0.6 cut of A will be collection of 8, 9,11,12,13. So, this way we will be able to get the alpha
cut of a fuzzy set.

Now, when we are interested in making this alpha cut a strong means strong alpha cut. So, for
alpha is equal to 0.6. We will have collection of all those points for which, their
corresponding membership value is more than or greater than 0.6. So, we will not be taking
those values for which alpha is equal to 0.6 we will be taking only those values only those
generic variable values for which alpha for which the membership value is more than 0.6. So,
we will exclude in the previous case we have included those values for which the
corresponding values of the membership was equal to 0.6 here in this case we will include.

So, this way A strong α−cut a strong α=¿0.6 cut will be collection of A 0.6 ' 8 , 9 ,11 , 12. So,

this way we will be able to find out the α−cut and a A α' of a fuzzy set.

(Refer Slide Time: 18:25)

Let me go through alpha cut and a strong alpha cut little bit again. So, alpha cut of a fuzzy set
that we had already seen is defined by A α ={x∨μ A ( x ) ≥ α. So, what does exactly this mean is
that we have to collect all the generic variable values of a fuzzy set for which, the respective
or the corresponding membership values are either equal to alpha or more than alpha and this
is called alpha cut of a fuzzy set.

So, let us now understand α−cut more clearly with this diagram. So here, we have a fuzzy
set A and if we are interested in finding the α−cut of this fuzzy set that is A, so we will first

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look for the value alpha and alpha will always be value which is, which can be either 0 also,
but normally more than 0 and it can go up to 1. So, let’s say our alpha is some value here
which is in between 0 and 1 and then we will look for the corresponding generic variable
value which is a here. So, this will be the generic variable value for which, the membership
value is equal to α.

So, this will be the start and then we will take up all the values of x that is the generic
variable. So, all the values of x for which the α is either the membership value is either equal
to α or more than α, so for this fuzzy set A we clearly see that, we get another point b here
which also has the corresponding membership value α. So, in between and including a and b,
we get all the values of the generic variable for which we have the respective or the
corresponding membership values either greater than alpha or equal to alpha. So, that makes
the sense here as to include all the values of the generic variable right from a up to b.

So, since this is a continuous fuzzy set and therefore we can have infinite number of such
values of generic variable. So, that is why this has been shown by a line and if we look at the
terminal of this a b the line ab the terminals are the solid dots. So, solid dot would indicate
here that the a is also included point a is also included and b is also included in the α−cut of
the fuzzy set. So, this has to be no noted here that, α−cut is a set of all the values of generic
variable and this will not contain the generic the membership their corresponding
membership values

So, only the α−cut will contain only α−cut is the set which will contain only the generic
variable values.

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(Refer Slide Time: 23:48)

Now, if we talk of a strong α−cut of a fuzzy set. So, here we make a little change and that is
the equality, that is the equality is not there. So, this means that if we are interested in finding
a strong α−cut of a fuzzy set. First of all, we define strong α−cut of a fuzzy set by A α' . So,
dash is the difference here as compared to the α−cut. So, if we talk of strong α−cut the this
will be represented by the a, the a of A α' and this will be equal to the collection of all the
points of the generic variable values for which their respective membership values are greater
than α.

So, this is to be noted that here the membership values the respective membership values are
greater than alpha, here equality sign is not there. So, this is the difference in between the
α−cut strong α−cut . So, if we take a fuzzy set here in this example we have fuzzy set A and
we are interested in finding out strong α−cut of a fuzzy set. So, please understand that since
the equality sign is not there only the membership values which are greater than α are
included. So, that is why as we have already discussed in the α−cut, the line a and b, a and b
here will be the strong α−cut . But, the terminals of this line a and b means the start of this
line at a and end of this line b will be represented by a hollow circle.

It means the point a is not included and b is not included. So, it is not included because
exactly at the point a the generic variable value point a, we have the membership value the
corresponding membership value α. So, that is why it is not included because equality sign is
not there. So, we are only considering those values of generic variables for which their

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respective membership values are greater than α. So, now I think this is very clear difference
in between the α−cut and the strong α−cut of a fuzzy set.

(Refer Slide Time: 27:18)

Here also, we see the clear difference in between α−cut and strong α−cut . So, we see how
is it defined the quality sign is here see the equality and greater than, but in alpha cut here we
see only the greater than sign is there. So, this way we understand this very clearly. Now,
there is another way of understanding this like if we are interested in a one, a one means the
one cut of a fuzzy set A, but this has to be a normal fuzzy set because α is equal to 1 has been
taken here. So, if we take an example here and we have a fuzzy set let us say A this is a fuzzy
set A and if we are interested in alpha cut, this will be the alpha cut here this can be small a
this can be b. So, α−cut is this line a and b.

And strong alpha cut as I have already explained this a here and b here will be the strong
alpha cut. So, we also have if we are interested in finding A 1. So, A 1 means the one cut of the
fuzzy set. So, one cut of the fuzzy set A. So, this will have all those points this will have the
collection of all those points for which, the, their corresponding membership values are equal
to 1. So, obviously, the core is also the same. So, we can say that a core of a is equal to α 1 is
one cut of a fuzzy set A. Similarly, the support of a can also be defined in terms of the strict
α−cut. So, if we take the value of α here as 0.

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(Refer Slide Time: 30:00)

Alright so, now this is exactly the same thing was explained and through diagram you can
understand it little bit more clearly.

(Refer Slide Time: 30:14)

So, at this point we would like to stop for this lecture and the in the next lecture, we will be
discussing the remaining nomenclatures used in the fuzzy set theory. So, it means we will be
you know moving ahead to discuss remaining nomenclatures.

Thank you.

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 Fuzzy Sets, Logic and Systems and Applications
Prof. Nishchal K. Verma
Department of Electrical Engineering
Indian Institute of Technology, Kanpur

Lecture – 09
Nomenclatures used in Fuzzy Sets Theory

So, welcome to lecture number 9 of Fuzzy Sets, Logic and Systems and Applications. And in
the Nomenclatures used in Fuzzy Set Theory we have already covered so many terms in the
nomenclatures that are used in fuzzy set theory in previous lectures. So, these terms that were
used that were discussed so far in previous lectures are support, core, crossover points,
height, normality, subnormal fuzzy sets, fuzzy singleton, alpha cut and a strong alpha cuts.

(Refer Slide Time: 00:48)

So, in the fuzzy set theory these terms are very very important once we know these terms we
you know when we are going through the literatures understanding of these literatures. The
literature is related to fuzzy set theory becomes very easier to understand.

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(Refer Slide Time: 01:24)

In today’s lecture we will cover the remaining nomenclatures that are used in fuzzy set
theory. And that are convexity, cardinality, fuzzy number, bandwidth, fuzzy symmetry, open
left and open right fuzzy sets.

(Refer Slide Time: 01:49)

Now, we have another term which is convexity of a fuzzy set in the nomenclature. And
convexity of a fuzzy set is also very important characteristic of a fuzzy set. And this is used
very commonly in fuzzy systems theory. So, convexity of fuzzy set can be checked or can be

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defined by this relation. So, this relation if this is satisfied if the if this relation is satisfied.
We can say a fuzzy set is a convex fuzzy set.

So, what is this condition? Condition here is that if we have a fuzzy set A and we choose any
two points as the generic variable values of this fuzzy set say x 1 and x 2. And we also choose a
parameter λ and this λ can be any value here, in between this λ is in between or it can be 0
and so, it can be 0 to 1. So, this is the interval of λ that is given.

So, this way and x 1 as I mentioned this generic variable values. So, x 1 and x 2 are any two
generic variable values. So, now we have λ x1, λ x2 and if we have these three values then if
we take μ A . So, please understand this A signifies the fuzzy set A, that is being undertaken
and with respect to this fuzzy set A only we are taking x 1, x 2. So, that is why A is mentioned

here. So, now if we take μ A ( λ x1 + ( 1−λ ) x 2 ) ≥ min ⁡[ μ A ( x 1 ) , μ A ( x2 ) ].

So, if we find minimum of μ A ( x 1 ) and μ A ( x 2 ). And then as I already mentioned that

μ A ( λ x1 +( 1−λ ) x 2 ). So, this μ A ( λ x1 + ( 1−λ ) x 2 ) ≥ min ⁡[ μ A ( x 1 ) , μ A ( x2 ) ].

So, if this condition is satisfied we can say a fuzzy set A is a convex fuzzy set. And this is a
very important property of a fuzzy set which many times we use while defining a fuzzy set.
So, a convex fuzzy set basically is the strictly monotonically increasing and then or in other
words I would say a convex fuzzy set is a fuzzy set whose membership function is strictly
monotonically increasing or monotonically decreasing.

So, this way we defined a convexity of a fuzzy set. So, we need to remember this criteria this

condition here like to repeat this condition, which is μ A ( λ x1 +( 1− λ ) x 2 ). And this value this
will be basically the value which will be some I mean right from 0 to 1 can be any value and
this should be either equal to or greater than min ⁡¿.

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(Refer Slide Time: 07:35)

So, this is how convexity of a fuzzy set is defined. And this can be understood even better by
taking this example. So, here we have two diagrams and in the first diagram here we have a
fuzzy set A, in this fuzzy set A if we are taking any two point any two points x 1 and x 2, and if
we choose any value of λ from 0 to 1. So, if we let us say we find after substituting the value
of λ and x 1 and x 1.

Let us say we find some value which is x 1 sorry x ' . So, we have x ' and then μ( x ' ) is equal to
this thing. So, we have computed μ A ( x ¿¿ ') ¿. And this must be either equal to this value
μ A ( x ¿¿ ') ¿ must be I can write it here μ A ( x ¿¿ ') ≥ min ⁡{μ A ( x 1 ) , μ A ( x 2 ) }¿. So, it means that we

have this value of this value is here μ A ( x 1 ) and this value here is the value which is μ A ( x 2 ), I
can write here A also because this for fuzzy set A.

So, this way you will take minimum, minimum of these two points should always be less than
or equal to μ A ( x ¿¿ ') ¿. So, if this condition throughout this fuzzy set for all the points all the
any points any generic variable value for this fuzzy set. If this satisfied this condition is
satisfied we can say this fuzzy set is a convex fuzzy set.

And if we take another example here where let us say this is a fuzzy set another fuzzy set
which is A. So, apply the same logic same condition we will not get the this condition
satisfied. So, μ A ( x ¿¿ ') ¿ that is coming out for this case the other case for non convex fuzzy

set case this will be less than the min of μ A ( x 1 ) and μ A ( x 2 ). So, for all non convex fuzzy sets,

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for all non convex fuzzy sets we for all non convex fuzzy sets we will not get this convexity
condition of a fuzzy set satisfied.

(Refer Slide Time: 11:40)

Just by looking at any set we can also comment on the convexity of the fuzzy set like if we
have any non convex fuzzy set. We will see that we have the fuzzy sets like this like the
fuzzy set here D fuzzy set is a non convex fuzzy set whereas, fuzzy set A, B, C are convex
fuzzy sets.

So, it means what the this fuzzy set D is not monotonically increasing or decreasing whereas,
fuzzy sets A, B, C are monotonically increasing or decreasing fuzzy sets. So, that is how we
can clearly by just looking at the fuzzy sets we can clearly make the make the distinction.

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(Refer Slide Time: 12:42)

Here we have a very important property of the you know convex fuzzy sets like if we have
two fuzzy sets; two convex fuzzy sets let us say A and B. So, if we take intersection of these
two fuzzy sets, which is coming out as this fuzzy sets. This is the fuzzy set which is which we
find by A ∩ B.

So, this fuzzy set will also be convex fuzzy sets. So, intersection of these two fuzzy sets how
we will be getting this will discuss this in coming lectures. So, in detail, but at this point we
need to know that if we have two convex fuzzy sets A and B, if we take intersection of these
two fuzzy sets that will also be a convex fuzzy set.

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(Refer Slide Time: 13:50)

So, we can also say here that a crisp set C that is defined in R n is convex if an only if for any
two points x 1 that belongs to c and x 2 that belonged to c their convex combination here is still
in C. Whereas I have already mentioned about lambda. So, λ will always be like this like λ
can take any value from 0 to 1.

Note that the definition of convexity of a fuzzy set is not as strict as the common definition of
function for comparison the definition of convexity of function fx is given as, so here we
have this condition as which checks the convexity of any function fx and the convexity of a
fuzzy set that we have just discussed is coming from this relation this condition.

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(Refer Slide Time: 15:19)

So, let us now take an example here to better understand the convexity of a fuzzy set. So, if
we take a discrete fuzzy set A whose universe of discourse here is X and if we are interested
in checking whether this fuzzy set is convex or not. So, here if we are interested in checking
we can choose any two points of this of the fuzzy set.

(Refer Slide Time: 15:59)

Okay, so now let us go to the cardinality of a fuzzy set. So, we all know. How do we define
how we do how do we know the cardinality of a fuzzy set of a crisp set, cardinality of a fuzzy
cardinality of a set first. So, what is the cardinality of a crisp set? Cardinality of crisp set is

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simply the number of elements that are present in the set. So, if we take an example here as A
set A here so, set Ais let us say A={5 , 10 ,15 , 20 }. So, this means here the cardinality of a
fuzzy set is 4 because here we have only 4 elements; however, for a continuous fuzzy set the
universe of discourse will have infinite elements. So, the cardinality of continuous fuzzy set
is infinite.

(Refer Slide Time: 17:24)

So, it means what, it means if we have a fuzzy set which is discrete. Let us say in this
example we have a fuzzy set A. Which is a discrete fuzzy set and this has 1, 2, 3, 4, 5, 6 6
elements. So, the cardinality of the fuzzy set here will be 6. Similarly the cardinality of the
fuzzy set B here will be 1, 2, 3, 4, 5, 6, 7, 8. So, the cardinality of the fuzzy set B will be 8.
And similarly we have the cardinality of a fuzzy sets C 11 because it has 11 elements. So, this
is how the cardinality of discrete fuzzy set we can quickly find.

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(Refer Slide Time: 18:23)

Now, if we have a continuous fuzzy set. So, since in a continuous fuzzy sets we have in the
fuzzy set we can have infinite elements present, infinite membership infinite generic variable
values present. So, that’s why we can say the cardinality of a continuous fuzzy set is always
infinite. So, that’s how we define the cardinality of a fuzzy set. Now we come to a term fuzzy
number.

What is a fuzzy number? Like crisp number we have fuzzy number. So, fuzzy number A is a
fuzzy set, A is simply a fuzzy set which follows two conditions. So, first condition is the
condition of normalized normality it means this fuzzy set should be a normal fuzzy set. And
then it should also follow the condition of convexity. So, this means that if A is a fuzzy
number fuzzy set any fuzzy set A is a fuzzy number. Then this fuzzy set should be this fuzzy
set A should be the normal fuzzy set number 1 and number 2 this A fuzzy set should follow
the condition of convexity.

It means the this fuzzy set A should be a normal fuzzy set and a convex fuzzy set. So, these
two conditions must be satisfied before we say that this fuzzy set is a fuzzy number. Most
non composite fuzzy sets used in literatures satisfy the condition of normality and convexity.
So, as I mentioned the fuzzy numbers are the most basic types of type of fuzzy sets.

So, we see here A is a fuzzy set and since this fuzzy set is a normal fuzzy set because it its
core is nonempty. And at the same time this is monotonically decreasing or increasing. So,

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this fuzzy set follows the criteria of convexity, so since this fuzzy set is normal fuzzy sets and
a convex fuzzy sets. So, this means this fuzzy set is qualified to be called as a fuzzy number.

Now, if we take another example here we have a B fuzzy set. So, we can clearly see here by
just looking at it we can clearly say that this fuzzy set is normal fuzzy set that is fine. So, this
is this says this one of the values are maybe we can say the core of this fuzzy set is non
empty. So, we can say this fuzzy set is normal fuzzy set.

However this fuzzy set is non convex fuzzy set. So, since this convexity criteria is not
satisfied means this fuzzy set is not a convex fuzzy set. So, B can be said as B is not a fuzzy
number.

(Refer Slide Time: 22:33)

So, here we have another diagram we have three fuzzy sets A , B ,C and these three fuzzy sets
if we look at we see A fuzzy number very clearly because this is normal and convex. If we
look at B fuzzy set B is again a normal fuzzy set and convex fuzzy set. So, B can be a fuzzy
number while if we talk of C, C is a subnormal fuzzy set core of the c fuzzy set is empty.

So, because none of the generic variable values have has the corresponding membership
value equal to 1. So, core of the fuzzy set C is empty. So, this means this is a subnormal sub
normal fuzzy set. Although this satisfies the criteria of convexity. So, this subnormal, but non
convex. So, that is why C is not a fuzzy set.

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So, fuzzy sets here fuzzy sets A and B are both normal and convex whereas, C is convex, but
subnormal. So, this way we can say A fuzzy set, fuzzy set A and B are fuzzy numbers,
whereas fuzzy set C is fuzzy set C is not a fuzzy number. So, this way we understand
whether a fuzzy set is a fuzzy number or not.

So, let me repeat here that fuzzy number should follow two criteria. One is the it should be
normal and then it should be the should follow the condition of convexity. So, it should be
convex fuzzy set. So, if n is fuzzy set which is a normal fuzzy set and a convex fuzzy set we
can say that this fuzzy set is a fuzzy number.

(Refer Slide Time: 26:01)

Now, another term that is bandwidth of normal and convex fuzzy set. So, bandwidth here is
defined by this expression, so for a normal and convex fuzzy set. So, please understand that
this bandwidth is defined only for a normal and convex fuzzy sets. So, it means that the
bandwidth is always found for a fuzzy number.

So, the bandwidth or width many times we use width also for bandwidth. So, either of these
two words are used. So, bandwidth our width here is defined as the distance between the two
unique crossover points and we know what are the crossover points of a fuzzy set. Crossover
points are basically the points whose corresponding membership values are equal to 0.5.

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So, this way we find the bandwidth of a fuzzy set and of course, as I already mentioned that
the this fuzzy set will be a fuzzy number because this follows two criteria. The first criteria is
this fuzzy set is normal and the other one is this fuzzy set is a convex fuzzy set.

(Refer Slide Time: 27:50)

So, here we have an example and with this we can understand better. So, here we have a
fuzzy set A and we have the x 1 and x 2, 2 points and these rest these respective membership
values are 0.5. So, if we take the distance between these two and these this distance will be
called as the bandwidth.

(Refer Slide Time: 28:25)

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And then if we take discrete fuzzy set. So, with this example we can understand the
bandwidth of discrete fuzzy set. So, if we take this fuzzy set here and with this we first find
the points for which the respective membership values are 0.5. So, we will find in this kind of
fuzzy set. The fuzzy set which is convex and normal. So, we will find only two points. So, we
find here we have two points and if we take the distance between these two will get the
bandwidth.

(Refer Slide Time: 29:19)

Now, for fuzzy symmetry. So, fuzzy symmetry of a fuzzy set is defined as that like a fuzzy
set is called symmetric, if its membership function is symmetric around a certain point x is
equal to c. It satisfies the condition given below for the universe of discourse X . So, this
means that if this condition is satisfied and then we can say the fuzzy set is a symmetric fuzzy
set. And this point c should be any point that is taken from the universe of discourse.

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(Refer Slide Time: 30:12)

So, let’s now understand this more clearly by taking this example. So, if we have two fuzzy
sets here A and B. And if we choose c 1 as the point for checking the symmetry. So, if we take
any point c 1 and if this condition is satisfied like μ A ( c +x ) = μ A ( c− x) then we can say the this
fuzzy set is symmetric. Similarly, for B we can check and these two will be found as a
symmetric fuzzy sets.

(Refer Slide Time: 31:02)

Another termed that we use here is open left, open right fuzzy set. So, if we take a fuzzy set
A and this will be called open left if we if this satisfies the condition that is limit of

143
lim μ A ( x ) =1. And lim μ A ( x )=0 . So, then this will be called as open left. Similarly a fuzzy
x →−∞ x →+∞

set here this is B and this is called as open right because the xlim μ A ( x ) =0 and lim μ A ( x )=1.
→−∞ x →+∞

So, this means that if we have fuzzy sets which are open on the right side are attaining the
value of the membership is close to 1 or equal to 1 and the right side is called as open right.
And if it is left side it is called open left. So, with this most of the terms that are used in fuzzy
systems theory are covered.

(Refer Slide Time: 32:52)

And in the next lecture we will discuss the set theoretic operations on classical and fuzzy sets
thank you.

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 Fuzzy Sets, Logic and Systems and Applications
Prof. Nishchal K. Verma
Department of Electrical Engineering
Indian Institute of Technology, Kanpur

Lecture – 10
Set Theoretic Operations on Fuzzy Sets

So, welcome to lecture number 10 of Fuzzy Sets, Logic and Systems and Applications. So,
this lecture will cover Set Theoretic Operations on Fuzzy Sets. In today’s lecture, we will
discuss following operations on fuzzy sets. These operations are complement, union,
intersection and difference.

(Refer Slide Time 00:44)

And before introducing these operations on fuzzy sets, let us first understand the notations
that we use in set theory.

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(Refer Slide Time 00:56)

And if you see here, these are the notations that are used in set theory; like if we have two
sets A and B here and which consists of the collection of some elements in the universe of
discourse capital X , the set theory notations are as follows.

So, if we see here x belongs to capital X this means x belongs to X . Similarly, here this is for
x belongs to A and here x does not belong to A. Here A is fully contained in B and here A is
contained in B and equivalent to B. And of course, here A is equal to B. ϕ here represents the
null or empty set.

(Refer Slide Time 02:15)

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So, before we move to set theoretic operations, we have two notions of two notions that we
will be discussing here. The first notion is equality and the second notion here is the
containment of sets or subset.

(Refer Slide Time 02:46)

So, let us now first discuss the equality. So, equality of fuzzy sets is very important like when
can we say two fuzzy sets are equal. So, here the point that needs to be noted is that even
when we have the any fuzzy set the generic variable values remains the same, in both the sets
may not be equal, because for equality their membership their corresponding membership
values also need to be equal. So, that is how it is written here.

If A and B are two fuzzy sets and if we say these are equal A and B; so, if they we can say
they are equal. So, this equality can be possible only when their corresponding membership
values are also equal.

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(Refer Slide Time 04:15)

So, let us look at this example to better understand. So, if we have two fuzzy sets A and B
and these fuzzy sets are discrete fuzzy sets. This A fuzzy set which is discrete fuzzy set, this
B fuzzy set which is again a discrete fuzzy set. So, let us now look at these fuzzy sets and
their you know the fuzzy elements. So, it means the generic variable values and the
membership values; corresponding membership values. So, let us now look at these and see
whether A and B both are equal or not.

So, if we see here for A; generic variable A, the first generate variable is 1 and the
corresponding membership value is 0.7. So, now, if we see the first element here in B also
this is same, means we have generic variable 1 and the corresponding membership value is
0.7. So, it means, we have the generic variable values; for the same generic variable values,
we have the same membership values.

Similarly, if we look at all the elements of A and then we look at all the elements of B, so we
see that we have exactly equal membership values for the generic variable values. So, this
way; we find this relation is satisfied, means that you know μ A ( x ) =μ B ( x ). And again this is
needless to say that all x, all the generic variable values must belong to the universe of
discourse within which we are working. So, this way fuzzy set A and fuzzy set B both can be
said to be equal.

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(Refer Slide Time 06:44)

Now, if we take two continues fuzzy sets, and let us see whether these two fuzzy continues
fuzzy sets are equal not. So, here we have this for you set A and the other one is B. So, we
see that although, A and B; in between A and B, the generic variable values remain the same,
but the corresponding, but their corresponding membership values are not same; not equal.

So, if we take any point if we take a point b 1 in between a ,c and then if we take some point
b 1 here in between a ,c. So, we see that here the corresponding membership value
corresponds; the membership value corresponding to b 1 in fuzzy set B is not equal to 1; is not
equal to 1 means the there is the membership value corresponding the generic variable value
b 1 whereas, here we can write this as μ B ( x). So, we can clearly see that for the same generic
variable values b 1 in both the fuzzy sets the μ A ( x) is not equal to μ B ( x). So, instead of x here,
because we are taking x is equal to b 1. So, I think we need to know we need to write here b 1.

So, this way we can clearly see that, the these two fuzzy sets are not equal. And if we take a
point b 2 here in fuzzy set B and if we take the same fuzzy set here, we take a same point b 2
here in between a to c. So, we see here that in fuzzy set A, the corresponding membership
value is less than is less than 1 or we can say is not equal to 1. So, this is μ B ( x).

So, this way we see for the same generic variable values in both the sets, their corresponding
membership values are not equal. So, that way we can say a fuzzy set A ≠ B.

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(Refer Slide Time 10:18)

Now, the containment or sub or subset of a fuzzy set. So, fuzzy set B; if we take the same
example, if we take two fuzzy sets A and B, so the containment can be defined as the; if we
have in fuzzy sets A and B, we follows the membership values follow this condition; we can
say the fuzzy set B contains fuzzy set A. So, if we have any two fuzzy set A and B and their
corresponding membership values follow this condition; means, μ A ( x) is less than or equal to
μ B ( x) for every x that belongs to the universe of discourse capital X .

So, then if this case is if this is the case if this condition is followed; we can say that, A is
contained in B. So, the fuzzy set A is contained in B.

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(Refer Slide Time 11:48)

Now, let us understand this a little bit more clearly with this example. So, we have here a
diagram and in this diagram, we have three fuzzy sets. So, here we have the fuzzy set A,
which is shown by the green line I mean in green colour. The B fuzzy set is shown in blue
colour and C fuzzy set is shown in the red colour.

So, by just looking at it, we can clearly say that all the membership values corresponding to
the generic variables in A, in C is less than their corresponding membership values less than
the membership values corresponding to the generic variable values of B and A. So, this
way, we can say that fuzzy set A contains B C, B and C fuzzy set this is also, this is fuzzy set
this is for set these two are fuzzy sets.

And, we can also say that here that; fuzzy set B does not contain A. So, if we compare fuzzy
set B and fuzzy set A, we can clearly see here that the membership values corresponding to
their generic variable values in both the cases. If we compare, then the membership values of
B fuzzy set are less than membership values of A. So, that way we can say fuzzy set B does
not contain A.

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(Refer Slide Time 14:08)

So, let us now take an example here to understand the containment or the subset of a fuzzy
set better. So, in this example, here we have two fuzzy sets. The first fuzzy set is the two
discrete fuzzy sets basically. And the first fuzzy set is A and the second fuzzy set is B.

So, these two fuzzy sets we see, we have the elements here and the generic variables values
that generic variable values 1 ,2 , 3 , 4 and their corresponding membership values. So, in both
the fuzzy sets, we have same generic variable values 1 ,2 , 3 , 4. But, the membership values
are different.

So, if we see clearly the membership values, so membership values of A fuzzy sets are lesser
than that of B for respective membership generic variable values. So, that is how we can say
μ A ( x ) ≤ μ B ( x). So, this means the the membership values in fuzzy set A are less than the
membership values in fuzzy set B for corresponding generic variable values. And of course,
this is needless to say that all these x, the generic variable values they all belong to the
universe of discourse.

So, this way we can say that A is contained inside B or we can say A is contained in B.

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(Refer Slide Time 16:38)

Here, we have another example and we see again two discrete fuzzy sets A and B. And just
by looking at the generic variable values and their corresponding membership values, we see
that, μ A ( x) ≰ μ B ( x) for the generic variable value x is equal to 3. So, if we look at B here; B
member; B fuzzy set and A fuzzy set, so if we look at these elements, we compare these
elements generic variable values remain again the same 1 ,2 , 3 , 4 in both the fuzzy sets, but
their corresponding membership function values are membership values are different.

So here, for x is equal to 3 for the membership for the generic variable value 3, the
membership value in the corresponding membership value in fuzzy set A is more than that of
the membership value in B. So, that that is why, we can say that; this condition is not
satisfied means, μ A ( x) ≰ μ B ( x) for all x’s and this is one of the generic variable value x is
equal to 3 which violates. Otherwise, if for x is equal to 3, these value the corresponding
value of membership would be equal are the less in A then B, we would have set that A B is
containing A or A is contained in B.

So, since at x is equal to 3, it is violated. We can clearly say that, A is not contained in B.
And mathematically, we can represent this by A is equal to A is not contained in B.

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(Refer Slide Time 19:34)

Now, let us discuss the set theory op set theoretic operations on fuzzy sets and these set
theoretic operations are complement, union, intersection and difference. So, since these
operations we are interested in applying on fuzzy sets. So, before we move to these
operations on fuzzy sets, we would be discussing these operations on crisp sets or classical
sets and then we will transition from classical sets to the fuzzy sets with respect to these
operations, set theoretic operations to understand the set theoretic operations and fuzzy sets
better.

(Refer Slide Time 20:33)

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So, let us now first understand; what do we mean by a complement of classical set. So, if we
have a classical set it is represented by A, we represent a complement of this classical set by
Á . And this can be represented here mathematically as Á={x∨ x ∉ A∧ x ∈ X .

(Refer Slide Time 21:19)

So, let us now understand the complement of a classical set. So, in this diagram here, we see
a classical set which is continuous and this is right from 1 to 3. So, as it was just defined if we
are interested in finding the complement of a classical set A, we represent this by A bar and
this Á will be the collection of all the points that are in the universe of discourse excluding
the points that are coming into excluding the elements that are there in set A.

So, if we look at this set which is a classical set. So here, we see that they in this is a
continuous set. So, here there are winds which are right from starting right from 1, 2, 3 are
covered and if we have some universe of discourse, so we exclude these points and we take
all the points which are contained which are there in the universe of discourse.

So, if we do that, we will get the Á like this and we see that this is the points which are there
in the A are excluded. So, this is it is this part is excluded. So, this way we find the
complement of classical set A.

155
(Refer Slide Time 23:20)

Now, if you are interested in finding a complement of fuzzy set, so the complement of a set A
is fuzzy set, that is also represented by Á . And of course, here that we have a universe of
discourse X and in fuzzy set also we take all the points all the elements you know, but with
this condition which is mentioned here. So, fuzzy set, if we have a fuzzy set A and then the
complement of fuzzy set A will have its membership values corresponding to the generic
variable values which will be defined as μ Á ( x).

So, you see here μ Á ( x) and all the corresponding membership values will be subtracted from
1. So, all the corresponding membership values of A fuzzy set will be subtracted from 1. So,
this way if we follow, we get the membership values corresponding to complement fuzzy set
and we can say that complement of a fuzzy set should contain all the elements along with
their membership values from the universe of discourse.

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(Refer Slide Time 25:12)

So, to understand the complement of a fuzzy set better, let us take this example. Here, we
have a fuzzy set A and please note that this fuzzy set here, this fuzzy set is a continuous fuzzy
set, is a continuous fuzzy set. So, we have here this 0. So, right from 0 to 3, this fuzzy set is
spreading. And we have varied membership values also like at 0, we have membership value
1 and at 3, we have membership value 0.

So, if we follow this criteria like complement fuzzy set Á , its membership values will be
computed as 1 minus μ A ( x), for every x belonging into the universe of discourse. So, if we
do that we are going to get Á like this. So, we can clearly see here that at 3 at the
generic generic variable value 3 in A, we have μ ( x )=0.

So, instead of μ ( x ), I will be writing μ ( 3 ), μ ( 3 ). And since this belongs to the fuzzy set A
we’ll write μ A ( 3 ) =0. Now, if we look at the fuzzy set Á which is complement of a fuzzy set.
So, we will look at this general variable value 3 so, this is μ B ( 3 ) =1. So, likewise, if we check
at all the points, all the generic variable values we’ll get to see that this follows, the
membership value follows membership values of Á follow this condition, 1−μ A ( x). And this
is for all the for every x belonging to the universe of discourse.

157
(Refer Slide Time 28:04)

Similarly, we take an example with two discrete fuzzy sets. So, if we have a fuzzy set A,
which is here and if we are interested in finding the complement of this fuzzy set. We can
clearly use the criteria as I just mentioned and we get the complement of a fuzzy set. So, this
is complement of A. This is complement of fuzzy set A.

Similarly, this is complement of fuzzy set A, and similarly if we take this fuzzy set B, we can
get the complement of fuzzy set here. And please note that the universe of discourse here is
given and this is this contains 1 ,2 , 3 , 4. So, we are excluding all other values except the other
than 1 ,2 , 3 , 4. So, this way we find complement of fuzzy set B. So, this will be the
complement of fuzzy set B.

158
(Refer Slide Time 29:43)

So, far in this lecture, we have covered the notions that means the equality and containment
or subsets used in the fuzzy set theory and complement operations on fuzzy sets. So, in the
remaining fuzzy set theoretic operations will be discussed in the next lecture.

159
 Fuzzy Sets, Logic and Systems and Applications
Prof. Nishchal K. Verma
Department of Electrical Engineering
Indian Institute of Technology, Kanpur

Lecture - 11
Set Theoretic Operations on Fuzzy Sets

Welcome to lecture number 11 of Fuzzy Sets, Logic and Systems and Applications. So, this
lecture is in continuation to the lecture number 10. In this lecture, we will be continuing the
discussions on Set Theoretic Operations on Fuzzy Sets.

(Refer Slide Time: 00:37)

So, in the previous lecture, we have covered the notions; that means, the equality and
containment or subset used in fuzzy set theory and complement operations on fuzzy sets. In
today’s lecture, we will be discussing the following operations on fuzzy sets. The first one is
the union and then the second one is intersection and then the third one is the difference of
fuzzy sets.

160
(Refer Slide Time: 01:09)

Now, let us talk about the union of classical sets. We all know the union of classical sets and
we can clearly see here if we have two classical sets. So, let this be a classical set and then
here B is also a classical set and we are interested in taking the union of these two classical
sets in the universe of discourse.

So we take, we follow this condition and the union is represented by A ∪ B here,

A ∪ B={x∨x ∈ A ∨x ∈ B }

So, this way we see here the shaded portion and shaded A and B and these two are together
gives us the union of classical sets.

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(Refer Slide Time: 02:41)

So, this can be clearly understood by this example and where we have two classical sets one
is A and the other one is B and when we are interested in finding the A union B we can
clearly see that all of the elements of A and B both are you know included in A ∪ B and this
way we get the union of classical set classical sets A and B.

(Refer Slide Time: 03:19)

Let us now understand the union of fuzzy sets which are little bit different from the union of
classical sets A and B. So, if we have fuzzy sets A and B and we are interested in finding the
union of these two, we must know that not only the elements which are present in these two

162
sets, the membership values are also important and they place they play a very important role
in managing in finding the union of fuzzy sets.

So, let us first look at the condition which we follow in finding the union of two fuzzy sets.
So, as I mentioned that it is the membership values which are very important for the
corresponding generic variable values and in case of a union of the two fuzzy sets

μ A ∪B ( x )=max [ μ A ( x ) , μ B ( x ) ] ∀ x ∈ X

So this is the condition for union of fuzzy sets A and B. So, this can be clearly understood by
the two continuous fuzzy sets. So, let this be fuzzy set A and this be another fuzzy set B. So,
let us now find the union of these two fuzzy sets. So, if we apply this condition; in this
condition says that at each and every generic variable value find, take the maximum of the
corresponding membership values. So, if we do that we see that we are getting this as the
A ∪ B means the union of these two fuzzy sets A and B. So, the maximum values if we plot,
we’ll be getting as this is shown by the red line the red color and this is going to be the A ∪ B.

(Refer Slide Time: 06:33)

Similarly, if we are taking two discrete fuzzy sets A and B and we can use the same condition
to find out A intersection, A ∪ B and this way here if we have A as the discrete fuzzy set, B
here as another discrete set. So if we apply this condition as A ∪ B, the we take the union of
these two fuzzy sets. We take the respective, maximum of the respective membership values.

163
So, here it is very clear that we have the A ∪ B which is coming out to be like this. So, this
way we can quickly find the union of two fuzzy sets; two discrete fuzzy sets as well.

(Refer Slide Time: 07:39)

Now let us discuss the intersection of two fuzzy sets. So, before we move to intersection of
two fuzzy sets or I would say intersection of fuzzy sets, let us first understand what happens
in the intersection of classical sets. So, in the intersection of classical sets we follow this
condition and this is represented by A ∩ B. So, A ∩ B basically contains all the elements
which are present in both the sets A and B. So, if we apply this condition A intersection will
be represented by this section. So, it means the elements which are common in both the
classical sets are regarded as A ∩ B.

164
(Refer Slide Time: 09:08)

So, now this can also be understood by this example where we have two classical sets here A
and B and A ∩ B will be the common portion of this two continuous classical sets.

(Refer Slide Time: 09:32)

Now let us understand the intersection of fuzzy sets and the intersection of two fuzzy sets A
and B represents all the elements for which the corresponding membership values can be
computed as

μ A ∩ B ( x ) =min [ μ A ( x ) , μ B ( x) ] ∀ x ∈ X

165
So, let us take an example here to better understand the intersection of two fuzzy sets. So,
let’s let this be a fuzzy set A and this be another fuzzy set B and when we apply this
condition and take the min of the corresponding membership values. We well get this red
portion as the A ∩ B and this will also be a fuzzy set. So, if we take the intersection of fuzzy
set A and fuzzy set B, we will get another fuzzy set which is shown by a red line a red color
and this comes here the out of the intersection of two continuous fuzzy sets.

(Refer Slide Time: 11:08)

So, if we are interested in knowing what is happening to the intersection of two fuzzy sets
when we take two discrete fuzzy sets. So, here we have two discrete fuzzy sets; the fuzzy set
A and fuzzy set B. Here we have two fuzzy sets, fuzzy set A and then fuzzy set B. So, if we
apply the same condition, we get A ∩ B which is you know for corresponding generic
variable value this is x=1 and then for this we have if we take min of the corresponding
membership values like in one case for a fuzzy set A we have 0.7 and then we have for the
other fuzzy set we have 0.

So, the minimum of this will come out to be 0. So that is why 0 has been mentioned here
although 0 is never written when we express when we represent a fuzzy set. So, this can be
neglected here. So, the outcome will be A ∩ B=0/ 1+0.5/ 2+0.1/ 3+0 / 4. So, this 0 has been
written here this term the first and last term is included just to make you understand as to
what is happening and why this generic variable values is not included in this outcome.

166
(Refer Slide Time: 12:53)

Now, let us discuss a difference of classical fuzzy set. So, the difference of a set A with
respect to B is defined as A∨B={x ∨x ∈ A∧x ∉ B }

So, this is with respect to the classical set, it is denoted by A, A straight line B so, A∨B and
this is a set which includes all the elements which are present in A, but not in B. So, we can
see here in the Venn diagram that here we have the difference of A and B and here we see the
difference of B and A.

(Refer Slide Time: 14:08)

167
Now, here we have another example to understand the difference of classical sets. So, we are
interested in the difference in A and B we follow this criteria, we are interested in finding the
difference in B and A we follow this criteria and these diagrams will show us will give us the
difference in A and B. So, A is this fuzzy set which is shown by the black line, the black
color and B is the another classical set which is shown by the blue color. So, if we are
interested in finding the A difference B, we represent first of all by this A oblique B. And
then if we apply the condition that was mentioned we get only this portion this portion and
this way we get A the difference of A and B and these sets are the classical sets.

(Refer Slide Time: 15:26)

Now let us understand the difference of fuzzy sets, on the same lines as we have already done
the other set theoretic operations where we have seen that the it’s the membership values
which are playing an important role in the set theoretic operations. So, here also we see that
when we are interested in the difference of fuzzy sets let us say difference of fuzzy sets A and
B, we try to first you know compute the their membership values respective membership
values and the these conditions are the membership values are represented by

μ A ∨B ( x ) =min [ μ A ( x ) , μ B ( x) ] ∀ x ∈ X

And then if we are interested in the difference in B and A fuzzy set B and A, we find the

respective membership values as μ B∨ A ( x ) =min [ μ B ( x ) , μ A ( x) ] ∀ x ∈ X

168
(Refer Slide Time: 16:55)

So, let us now understand the difference of fuzzy sets here and if we apply these two
conditions to the fuzzy sets that are given here is fuzzy set A fuzzy set B. And if we apply the
condition here as the we are interested in finding the difference between fuzzy sets A and B.
So if we are interested in the difference of fuzzy sets A and B, we have to find the mu B bar.
So, for finding this we have to first find the compliment of B. So, this is what is the
compliment of B fuzzy set and when we have this, we can compute the respective we can
find the respective membership values.

So, if we have B fuzzy set here we can say this is the compliment of B. So, this is
compliment of B fuzzy set and then when we apply this criteria this condition we find fuzzy
set the difference between fuzzy set A and B which is coming out to be represented in red
color. So, we take all the minimum values so, if we plot here the minimum value everywhere
wherever we see these conditions are satisfied in this figure and if we take at 0.3 at generic
variable x=3. So, we see that x has for A fuzzy set we have a 0 membership value and B bar
also has membership value 0. So, if we take min we are getting 0 here as well, all right.

169
(Refer Slide Time: 19:15)

So, we take some more examples here to make to understand the difference of fuzzy sets
better and here we have two discrete fuzzy sets A and B. And if you apply these conditions,
we are we can get this as the outcome of the computations and this is fuzzy this is the
difference between two fuzzy sets A and B. So, we have in this lecture we have covered,
these set theoretic operations and in the next lecture we will study the properties of fuzzy
sets.

Thank you.

170
 Fuzzy Sets, Logic and Systems and Applications
Prof. Nishchal K. Verma
Department of Electrical Engineering
Indian Institute of Technology, Kanpur

Lecture - 12
Properties of Fuzzy Sets

So, welcome to the lecture number 12 of Fuzzy Sets, Logic and Systems and Applications.
This lecture will cover the Properties of Fuzzy Sets.

(Refer Slide Time: 00:47)

So, in a way here we have the set properties and we’ll see whether fuzzy sets also follow
these properties or not. And we already know that the classical sets follow these properties
and these properties are Law of a Contradiction, Law of Excluded Middle, idempotency,
involution, commutativity, associativity, distributivity, absorption, absorption of complement,
DeMorgan’s laws. So, these are the properties that we know that a classical sets follow, but
let’s see whether the fuzzy sets also follow these properties or not.

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(Refer Slide Time: 01:39)

So, if we compare the crisp sets and fuzzy sets, we see that three properties are not followed
by the fuzzy sets, like we have a law of contradiction. So, when we use fuzzy sets instead of
crisp sets, we see that there is a contradiction and so that is why the law of contradiction
comes into picture and similarly the law of excluded middle also comes.

And another one the third one the absorption of complement that is not followed by the fuzzy
sets and as we move further in this lecture, we’ll verify these properties for both continuous
and discrete fuzzy sets through various examples.

(Refer Slide Time: 02:53)

172
So, let us now discuss the set theoretic properties one by one and the first one is the law of
contradiction. So, let us take first crisp set and see what is happening with the crisp set if we
take A ∩ Á . So, if we take any crisp set A, A ∩ Á =ϕ.

Now, if we take the case of a fuzzy set, so let’s say we have a fuzzy set A and we do the
same intersection, we take the same intersection on this fuzzy set A and its complement. So,
this intersection is never a null set. So, we clearly see that crisp set, there is a difference in the
intersection of crisp set and its complement and then intersection between fuzzy set and its
complement. And this has been clearly mentioned here that if we have a crisp set A,
intersection between its complement is a null set. And if A is a fuzzy set, A ∩ Á ≠ ϕ; that
means, a not a null set.

So, we’ll take example here one example here and we’ll see how if a fuzzy set we have and
then how what happens when we take the intersection of fuzzy set and its complement.

(Refer Slide Time: 05:13)

So, and of course, this is since this is not followed so, this contradicts right. So, there is a
contradiction. So, contradiction means if we take a crisp set, we get null set as a result and if
we take fuzzy sets we get a set which is not a null set as a result. So, that is why there is a
contradiction and that is how this is called the Law of Contradiction.

173
So, if we take an example here, we take a fuzzy set A which is a triangular fuzzy set and this
fuzzy set is defined by a triangular membership function. So, this is a triangular membership
function here, but the whole thing is called the A fuzzy set fuzzy set A fuzzy set A.

(Refer Slide Time: 06:35)

So, as we are trying to verify whether A ∩ Á =ϕ or not so, let us now first have A here and
then try to find Á . So, Á is the complement of A. So, we see here is a complement of A fuzzy
set which is represented by the green color. So, this is we can say the complement of fuzzy
set A and this is this will be denoted by Á which is complement of fuzzy set A and this is
fuzzy set A fuzzy set A.

So, if we take intersection of these two fuzzy sets so, we already know if we are taking the
intersection of any two fuzzy sets, we follow the criteria; we follow the condition as the min
we take the minimum of the membership values of the corresponding generic variable values
at each and every point. And this is needless to say that all of these generic variable values
must belong to the universe of discourse.

174
(Refer Slide Time: 08:11)

So, here we have the A ∩ Á . So, these two A fuzzy set and Á have been overlapped or super
imposed to find the intersection of these two fuzzy sets.

(Refer Slide Time: 08:35)

So, it’s very clear here if we apply this criteria, this condition we see that we are getting this
red the portion which is marked by red color. So, this portion is basically the A ∩ Á which is
represented here by this separately by this membership function.

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(Refer Slide Time: 09:03)

So, this is nothing but A ∩ Á and we clearly see that this is not a null set means, we are
getting something here. We are getting a fuzzy set here, so obviously, this is not equal to ϕ
means this not equal, this is not a null set. So, we can clearly say that the law of contradiction
is verified.

(Refer Slide Time: 09:43)

Now, if we take; so, this was the example when we took a triangular fuzzy set. So, obviously,
it was a continuous fuzzy set. But if we take the discrete fuzzy set just to see whether this law
of contradiction is verified for discrete fuzzy sets also. So, let us take an example here and

176
this example is with discrete fuzzy set. So, this is a discrete fuzzy set A, discrete fuzzy set
and if we try to find the Á of it the complement of the fuzzy set A is complement of fuzzy set
A and this is nothing but Á as it is written here.

So, if you take the intersection of these two fuzzy sets, again we see that we get something
here which is not equal to the null set. So, it means that here also we are getting a set which is
not a null set. So, we can say that the law of contradiction here also is verified.

(Refer Slide Time: 11:31)

Then we have another set theoretic property and this is called the Law of Excluded Middle.
So, if we take why is it called law of excluded middle, we will get to know in a moment. So,
if we have crisp set A and if we take the A ∪ Á . So, for crisp sets we always get the universe
of discourse as a result, whereas if we take a fuzzy set A and if we do the same operation
means we take the A ∪ Á this is never going to be the universe of discourse. So, here we have
clear cut difference in between the crisp set the union of crisp sets and fuzzy sets. So, we call
this discrepancy as this contradiction as the law of excluded middle.

So, here as I already mentioned that if we take crisp set A if we take union of A ∪ Á , we
should get the whole universe of discourse. Whereas, if we take fuzzy sets A, A ∪ Á is not
going to give you the universe of discourse. So, that is what is the difference and this is called
the law of excluded middle.

177
(Refer Slide Time: 13:41)

So, let us understand this law of excluded middle better by taking an example and here also
we have taken a continuous fuzzy set which is triangular.

(Refer Slide Time: 13:59)

So, if we take a fuzzy set A here and the complement of this fuzzy set A is here and if we
take the union of these two fuzzy sets so, here we have super imposed this fuzzy set which is
Á of this fuzzy set, right. So, if we apply the condition of the union the criteria which is here,
A ∪ Á=max [ μ A ( x ) , μ Á ( x ) ] ∀ x ∈ X

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So, what we get here is this which is represented by the red color.

(Refer Slide Time: 14:55)

So, we get if we take the max, we get this portion right and here we apply the max of the
corresponding membership values of the two fuzzy sets corresponding to the generic variable
values within the universe of discourse. So, if we separately represent this, it looks like this.
So, here is what we are going to get as result of fuzzy set A ∪ Á and which of course, is not
the universe of discourse. So, what do we mean by this statement when we are saying that
this is not equal to the universe of discourse? So, when this would have been the universe of
discourse; so, when we would have got this as the straight line like this?

So, this portion should not have been there, then we would have said that the A ∪ Á= X . But
since here this is not exactly what I mean the straight line we have this portion also. So, we
call this here as the A ∪ Á ≠ X .

So, this clearly gives us the idea as to how we check when we take the union of any fuzzy set
and its complement. So, this is how the law of excluded middle is verified for a fuzzy set.
Similarly we can take the example of a discrete fuzzy set here you see.

179
(Refer Slide Time: 17:21)

And then if we find the Á which is the complement of fuzzy set A which was taken here
which is taken here and we see that this is the complement of the fuzzy set A taken in this
example which is also discrete. And if you take A ∪ Á ≠ X . So, here also this is true that the
law of excluded middle is verified for any discrete fuzzy set.

(Refer Slide Time: 18:17)

Now, we have another set theoretic property which is idempotency. So, let’s now take a crisp
set first and see how this idempotency property is verified for a crisp set. So, A ∩ A means if
you take the intersection of the same set, we always get as a result the same set means if we

180
take the intersection of A ∩ A = A. So, which is here the same sets is a crisp set which was
taken.

Then again if we take the A ∪ A= A. So, this is known for crisp set which is you know which
we have already done so many times in the past. But if we take fuzzy set let us see what
happens. So, if we take fuzzy set here A. So, this is fuzzy set A and this is also fuzzy set A
fuzzy set and this is also fuzzy set. So, if we take intersection of the same fuzzy set, we are
going to get the fuzzy set same fuzzy set means A ∩ A or we say A ∩ A = A. Similarly we
take union it is going to return the same set and this is called the idempotency property.

So, what we see here is that we get the same set whether we take a crisp set or fuzzy set. So,
that’s how the idempotency property is verified for both the kinds of sets.

(Refer Slide Time: 20:37)

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(Refer Slide Time: 20:53)

Let us take an example to better understand this property. So, if you take a triangular fuzzy
set if we take a fuzzy set here as A, so since we would like to verify the idempotency
property. So, we have to have the A ∪ Aand A ∩ A and we see what we are getting as a result
of it. So, here we have A and then here we have A ∩ A = A. So, if we intersect these two
fuzzy sets, we are going to get the same set. Similarly if we are doing here if we are taking
A ∪ A, we are going to get the same set. So, this way we can say the idempotency property is
verified for fuzzy sets.

(Refer Slide Time: 21:45)

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So, if we take this as the fuzzy set here which is discrete and if we try to find the A ∪ A. So,
we see that A ∪ A= A. Similarly if we do this operation, we take A ∩ A = A. So, this way we
can say the idempotency property is satisfied or verified for fuzzy sets also because the crisp
set this property is satisfied.

(Refer Slide Time: 22:39)

So, here also for fuzzy sets, this property holds good. Then comes the involution property.
So, this means that if you take any Á . So, what does this mean is that if we take a crisp set A
we take the complement of it and then we further take the complement of it and this is going
to return us the crisp set which was originally taken. So, this is the crisp set, this is crisp set
which was originally taken. So, this is going this is going to return us the same set.

So, let us see this involution property holds good for fuzzy sets also. So, if we take a fuzzy set
A and if we take the Á , so it means that if we take the complement of fuzzy set and then we
take further complement of it here also this is going to return us the same fuzzy set same
fuzzy set which was used for taking the complement. So, it means Á= A and this is true for
both the cases means the, if we take crisp set or the fuzzy set both the cases, we are going to
get the same set.

So, but we are in here interested more on fuzzy sets. So, we can clearly say that if we take
fuzzy set A and then if we take the double complement of it, we are going to get the same
fuzzy set as a result.

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(Refer Slide Time: 24:47)

Let us know understand better this involution property fuzzy set. So, let us take an example
here and if we take a fuzzy set A.

(Refer Slide Time: 25:01)

Let us now take the complement of it. So, here we have the fuzzy set, the original fuzzy set
which we have taken and then the complement is here. Now if we take the double
complement see the double complement here and which is nothing, but the same set. If we
see here we are going to get the A, this set and this set is coming out to be the same. So,
double complement of any fuzzy set is going to return us the same fuzzy set. So, this way we

184
can say that the involution property is verified for continuous fuzzy sets. So, let us now
understand this for discrete fuzzy sets also.

(Refer Slide Time: 25:51)

So, if we take here an example of discreet fuzzy set which is here. So, let us take the
complement of it. So, this gives us the complement of complement of A that is Á . So, this is
represented by this discrete fuzzy set and if we take the double complement of is double
complement means further complement of; so, complement of Á and this is going to give us
Á and this is the Á and which if we see is the same set which we have taken, we have started
with.

So, we see that these two sets remain the same. So, we can clearly say that we are going to
get the same set if we are taking the double complement of it and this property is nothing, but
the involution property. So, this is verified.

185
(Refer Slide Time: 27:17)

So, in this lecture today we have studied the following properties of fuzzy sets and these
properties are law of contradiction, law of excluded middle, idempotency property, involution
property. So, but here we have so many other properties left which we will be discussing in
the next lecture.

186
 Fuzzy Sets, Logic and Systems and Applications
Prof. Nishchal K. Verma
Department of Electrical Engineering
Indian Institute of Technology, Kanpur

Lecture - 13
Properties of Fuzzy Sets

So, welcome to lecture number 13 of Fuzzy Sets, Logic and Systems and Applications. In this
lecture we will cover the remaining Properties of the Fuzzy Sets. So, we have already
discussed some of the properties of classical and fuzzy sets in the previous lectures and these
are the properties that are covered.

(Refer Slide Time: 00:45)

So, I will just mention here law of contradiction for fuzzy set is discussed and then law of
excluded middle is also discussed, idempotency property is discussed, involution is also
discussed. So, we see that we have covered so far 4 properties as mentioned here with respect
to fuzzy sets, now remaining properties will be discussed we will try to cover in this lecture.
So, we will start with the commutativity property.

187
(Refer Slide Time: 01:37)

So, as we know for crisp sets A and B, A ∪ B=B ∪ A and this is the commutativity property
for union. So, let us now see what is happening when we take fuzzy sets instead of crisp sets.
So, if we take two fuzzy sets A and B so, let’s see whether we get A ∪ B=B ∪ A or not, of
course it is written so we will be getting these two equal. So, let us see what is happening and
how are we getting this commutativity property for union satisfied.

(Refer Slide Time: 02:35)

So, if you take an example here where we take a two continuous fuzzy sets A and B, here
fuzzy sets A is a triangular fuzzy set, this fuzzy set A which triangular as we see and B fuzzy

188
set is a Gaussian fuzzy set here as we see. So, we call this as the as B fuzzy set. So, now let’s
see whether A ∪ B=B ∪ A or not.

(Refer Slide Time: 03:29)

So, if we take A ∪ B here, so we apply this condition where we take the max of all the
corresponding membership values from fuzzy set A and B with respect to their corresponding
generic variable values.

(Refer Slide Time: 04:03)

So, if we do that we find A ∪ B as this. So here, so this is nothing, but A ∪ B. So, we have
already discussed enough as to how we get the union of two fuzzy sets.

189
(Refer Slide Time: 04:25)

So, this way we get here after applying this condition we get A ∪ B as mentioned over here.
Now, let us see what are we getting as the outcome when we take B ∪ A.

(Refer Slide Time: 04:45)

So, we if we take B fuzzy set here first and then we take A fuzzy set.

190
(Refer Slide Time: 04:59)

And, then we see here that if we take B ∪ A we are going to get this fuzzy set as
B ∪ A =max [ μB ( x ) , μ A ( x ) ].

(Refer Slide Time: 05:21)

So, this way we see that what we are getting here is A union this A ∪ B and this B ∪ A. And,
we if we see here both the outcomes are same so, we can clearly say here that the
commutativity property for union is verified or satisfied for fuzzy set A and B. And, this is
written as A ∪ B=B ∪ A and please note that this commutativity property for fuzzy sets A
and B are satisfied.

191
(Refer Slide Time: 06:13)

Now, let us take another example where we have two discrete fuzzy sets. So, if we take here
A fuzzy set as a discrete fuzzy set and B also a fuzzy set which is discrete fuzzy set. So, let us
know try to see whether A ∪ B=B ∪ A or not; of course, this will this is going to be equal,
but let us see how are we going to get this verified.

(Refer Slide Time: 06:59)

So, here we have fuzzy set A discrete fuzzy set A discreet fuzzy set B and see here if we find
the A ∪ B of the two discrete fuzzy sets we are getting this and when we are taking B ∪ A we
are getting this as the outcome. So, we can clearly see that all the elements of A ∪ B=B ∪ A

192
are same. So, we can clearly say here that these two sets are these two fuzzy sets are equal.
So, when these two fuzzy sets are equal, we can very easily say or we can say that the
commutativity property for union is verified.

(Refer Slide Time: 08:05)

Now, on the same way we can define the commutativity property for intersection. So, here
when we talk of intersection, so, let us first see what is this for crisp sets. So, if we take crisp
sets A and B; so, when we take crisp sets A and B this property is satisfied means
A ∩ B=B ∩ A for crisp sets A and B. Now, let us see what is happening for fuzzy sets A and
B. So, here also it is equal means when we take two fuzzy sets A ∩ B=B ∩ A. So, let us see
how we are going to get this verified.

193
(Refer Slide Time: 09:09)

So, if you take an example here, in this example we have two fuzzy sets as we have taken in
the previous example.

(Refer Slide Time: 09:25)

So, we have taken the same example here also. So, we see that when we take A ∩ B using this
condition we take min of all membership values corresponding to the generic variable values
for A fuzzy set and B fuzzy set.

194
(Refer Slide Time: 09:45)

So, A ∩ B we are going to get like this. So, this is what is the A ∩ B which is represented by
the red color fuzzy set. So, this is what is the outcome and let me also make it very clear how
do we get that here.

(Refer Slide Time: 10:13)

So, you see here that we have this as B fuzzy set, this as A fuzzy set and when we take min of
the respective membership values for corresponding generic variable values we get here, the
portion which is represented by a red color plot.

195
(Refer Slide Time: 10:45)

So, we see that this is what is we are going to get as A ∩ B.

(Refer Slide Time: 10:51)

Now, when we take B first when we take B first and then A, of course this B and A are two
fuzzy sets. So, if we do that let us see what is happening.

196
(Refer Slide Time: 11:11)

So, here also we have B fuzzy set and then A fuzzy set and then when we apply this
condition the condition for intersection.

(Refer Slide Time: 11:29)

So, when we find this we are also we get again the same portion of the fuzzy sets.

197
(Refer Slide Time: 11:43)

So, B ∩ A= A ∩ B.

(Refer Slide Time: 11:49)

So, this way we can say that commutativity property for intersection is verified for fuzzy sets
A and B.

198
(Refer Slide Time: 12:03)

(Refer Slide Time: 12:11)

Now, again if you take the discrete fuzzy sets A and B, here also we see that if we take A ∩ B
, here for fuzzy set for discrete fuzzy set A and discreet fuzzy set B. So, we see that we have
A ∩ B as the outcome. So, this A ∩ B=0.7/1+0.3/2+0.1/3+0.5/ 4 as the outcome. Now, let
us see what are we getting when we take B ∩ A. So, this way we see that we are getting the
same outcome as we have gotten for A ∪ B.

So, all the elements remain same and then and hence we can always say that A ∩ B=B ∩ A
for discrete fuzzy sets also. And, hence we can say since we have already checked this

199
condition for continuous fuzzy set A and B and here in this case also for discrete fuzzy set we
have checked. So, we can say that commutativity property for intersection is verified.

(Refer Slide Time: 13:49)

Now, let us go to the associativity property for union. So, as we already know that for crisp
sets A, B and C we have this equality as the associativity for union, means
( A ∪ B)∪ C= A ∪(B ∪ C ). So, for fuzzy sets A, B and C this condition also is satisfied. So,
let us now see how is it happening.

(Refer Slide Time: 14:41)

200
So, if you take an example here again for the associativity property for union we take fuzzy
set A, continuous fuzzy set A, continuous fuzzy set B and then the third fuzzy set continues
fuzzy set C.

(Refer Slide Time: 15:07)

So, we see that if we take the A ∪ B here where A is this fuzzy set and B is this fuzzy set.

(Refer Slide Time: 15:21)

So, we are going to get A ∪ B like this. So, this is what we are going to get, means we are
going to get the max of all the respective membership values corresponding to the generic
variable values of both the fuzzy sets.

201
(Refer Slide Time: 15:47)

So, this way we can say here that we are getting A ∪ B like this.

(Refer Slide Time: 15:55)

Now, since we already have A ∪ B, now take the we use this and take the ( A ∪ B)∪ C. So,
let’s see what is happening. So, we have C fuzzy set here; so, ( A ∪ B)∪ C means ( A ∪ B)∪ C
we are going to get it like this.

202
(Refer Slide Time: 16:23)

So, if we apply the same criteria, we are going to get this portion as the ( A ∪ B)∪ C by
applying the same max criteria.

(Refer Slide Time: 16:55)

Now, if we take B the union of fuzzy set B and C, we are going to get we are going to get this
as we are going to get this as the outcome means B ∪ C is this outcome.

203
(Refer Slide Time: 17:13)

And, if you take the ( A ∪ B)∪ C we are going to get here this as the outcome by applying the
same max criteria.

(Refer Slide Time: 17:21)

So, we can clearly see here that these two outcomes are same as same in both the cases,
means either we take A ∪ B ∪ C or we take ( A ∪ B)∪ C.

204
(Refer Slide Time: 17:53)

So, we can clearly say that the associativity property for union is satisfied or verified for
fuzzy set for fuzzy sets A, B and C.

(Refer Slide Time: 18:09)

Now, the same can be tested, same can be checked by taking the discrete fuzzy sets A, B and
C.

205
(Refer Slide Time: 18:29)

So, if we take these three fuzzy sets A, B and C so, we find here if we directly go to the
outcome of ( A ∪ B)∪ C we are going to get this as the outcome. So, this is what is the
outcome, this is a fuzzy set a discrete fuzzy set. So, let me read the element of this fuzzy set
( A ∪ B ) ∪ C=0.8/ 1+0.7/ 2+0.7 /3+0.9/ 4.

And, when we find the A union the fuzzy set which we have got as a result of B ∪ C, if we
take this thing we get the outcome here as 0.8/ 1 ,0.7/ 2 ,0.7 /3 , 0.9/ 4. So, if we look at these
two fuzzy sets these two outcomes, we see that these two are equal these two are same. So,
then it is very clear that ( A ∪ B)∪ C= A ∪(B ∪ C ) means the associativity property for union
is satisfied or verified. So, this way you have checked the associativity property for union.

206
(Refer Slide Time: 20:11)

Now, let us do the same thing, let us now check the associativity property for intersection. So,
for crisp sets A, B and C this property, the associativity property for intersection is satisfied
and for fuzzy sets A, B and C are also satisfied, but we need to verify by taking some
examples.

(Refer Slide Time: 20:51)

So, let us here also take few examples, two examples: first example is with the continuous
fuzzy sets and the second example is with discrete fuzzy sets. So, the first example is here
and here we take three fuzzy sets A , Band C.

207
(Refer Slide Time: 21:21)

And, let’s now try to see what are we getting when we do A ∩ B ∩ C. So, what does this mean
when we say A ∩ B ∩ C? It means that we are going to get a fuzzy set as the outcome of
A ∩ B and then whatever we are getting as the outcome, we take the intersection of this fuzzy
set with the third fuzzy set C.

(Refer Slide Time: 22:01)

So, A ∩ B is here; so, this is what we are getting as a result by applying this condition, the
min condition. So, this is A ∩ B; so, this is a fuzzy set basically which is by just looking at it
we can say that this fuzzy set is a sub normal fuzzy set.

208
Refer Slide Time: 22:31)

And, when we take the intersection of this fuzzy set with C fuzzy set, the third one the third
continuous fuzzy set which is C, so it is here and if we do that let us see what are we getting.

(Refer Slide Time: 22:47)

So, if we apply this the min condition we are getting this fuzzy set which is again a sub
normal fuzzy set and this is the outcome of A ∩ B ∩ C. So, this is how we are getting the
outcome of A ∩ B ∩ C.

209
(Refer Slide Time: 23:23)

Now, let’s try to verify let us try to get B ∩ C first. So, B fuzzy set is here and please
remember that this is please note that this is a continuous fuzzy set, all these fuzzy set A, B ,
C are continuous fuzzy sets. So, B ∩ C we are going to get like this.

(Refer Slide Time: 23:53)

So, this is ourB ∩ C, now since this is a fuzzy set again by just looking at it we can say this is
sub normal fuzzy set. So, if we take this fuzzy set or we see if we take the intersection of this
fuzzy set and A, so let us see what are we going to get.

210
(Refer Slide Time: 24:29)

So, here we have A so, here we have to follow the sequence. So, we have A fuzzy set first,
we take the sub normal fuzzy set B ∩ C and when we find A ∩ B ∩ C, let’s see what are we
going to get.

(Refer Slide Time: 24:51)

So, when we apply the min condition we are going to get this as this as A ∩ B ∩ C. So, just by
looking at this we can clearly say that these two fuzzy sets are equal.

211
(Refer Slide Time: 25:23)

So, this way we can say the associativity property for intersection of three fuzzy sets A, B
and C are satisfied.

(Refer Slide Time: 25:35)

Now, let us see what is happening when we take the discreet fuzzy sets. So, here we have
three discrete fuzzy sets A and A , B and C is three are discrete fuzzy sets.

212
(Refer Slide Time: 25:29)

And, if we try to find here, I am going directly to the outcome of ( A ∩ B)∩ C. So, we are
going to get a fuzzy set a discrete fuzzy set which is ( A ∩ B)∩ C =0.4/1+0.3/2+0.1/3+0.5/ 4
. So, let me repeat it. So, we are going to get the outcome as 0.4 /1+0.3/2+0.1/ 3+0.5/ 4. So,
this is what is the outcome that we are getting as the A ∩(B ∩C ).

So now, when we compute the A ∩(B ∩C ), so this is what we are getting when we apply the
min criteria. So, if we look at all the elements of these two fuzzy sets, we find
A ∩ ( B ∩ C ) =0.4 /1+0.3/2+0.1/3+0.5/ 4. So, we see that all the elements are same as we
have here. So, this way we can say that if we take A ∩(B ∩C ) and if we take the ( A ∩ B)∩ C
both are equal.

So, this way we can say the associativity property for intersection is verified for all the three
discrete fuzzy sets also. So, this way we understand that the associativity property for
intersection are all satisfied for fuzzy sets also.

213
(Refer Slide Time: 28:41)

So, in today’s lecture we have so far covered the following properties of fuzzy sets: the
commutativity property for union, commutativity property for intersection, associativity
property for union, associativity property for intersection. So, we have covered all these four
properties for the fuzzy sets and we will stop here for this lecture. And, in the next lecture we
will cover the remaining properties of fuzzy sets that we have already seen listed.

Thank you.

214
 Fuzzy Sets, Logic and Systems and Applications
Prof. Nishchal K. Verma
Department of Electrical Engineering
Indian Institute of Technology, Kanpur

Lecture - 14
Properties of Fuzzy Sets

So, welcome to lecture number 14 of Fuzzy Sets, Logic and System and Applications. In this
lecture today, we will discuss the distributivity property with respect to Fuzzy Sets. So, here
is the list of all the properties that we have intended to cover.

(Refer Slide Time: 00:36)

So, so far we have covered the Law of Contradiction, Law of Excluded Middle, Idempotency,
Involution, Commutativity, Associativity. All these properties with respect to fuzzy sets have
been covered.

215
(Refer Slide Time: 01:09)

Now, today we will discuss distributivity property with respect to fuzzy sets and this,
distributivity property is divided into two parts. The first part will be the distributivity of
union over intersection and then the second part will be the distributivity of intersection over
union. And the distributivity property will be discussed with respect to the fuzzy sets, the
continuous fuzzy sets and the discrete fuzzy sets.

And we all know that for crisp sets A , B and C, this distributivity property of union over
intersection is valid. This means that when we take the A ∪(B ∩ C ), so if we take the union of
these two entities, what we get is here we get the ( A ∪ B ) ∩( A ∪C ). So, this is we know for
the crisp sets.

And let’s see what happens with the fuzzy sets. So, when we take fuzzy sets A , B and C, the
same applies same holds good. This means that when we take A ∪(B ∩ C ), this also is equal
to the ( A ∪ B ) ∩( A ∪C ), where in this case we have A , B ,C, all these are the fuzzy sets. So,
let us now understand by taking some examples.

216
(Refer Slide Time: 03:36)

So, here we have an example which has three continuous fuzzy sets. So, now let us see
whether for continuous fuzzy sets, the distributivity property of union over intersection is
satisfied or not. Of course, this has to be satisfied, but let us see how is it satisfied.

(Refer Slide Time: 04:01)

So, here we have to verify basically this thing.

217
(Refer Slide Time: 04:20)

So, for this we are first taking the B ∩ C and the B ∩ C, we can get these two fuzzy sets B and
C and when we overlap these two fuzzy sets, we get this portion by applying the min criteria.
So, this min criteria is going to give us a fuzzy set a sub normal fuzzy set as a result like this.

(Refer Slide Time: 04:49)

Now, what we have to get is what we have to obtain here is A ∪(B ∩ C ). So, now we take the
union of these two fuzzy sets, so this is what we have got as B ∩ C. Now we take the union of
these two, so let us see what are we going to get.

218
(Refer Slide Time: 05:15)

We overlap these two fuzzy sets here and when we take the union of these two fuzzy sets, of
course by applying the max criteria we are going to get this as the A ∪(B ∩ C ).

(Refer Slide Time: 05:33)

So, this way we get a triangular membership function which is a normal membership
function. So, when we are trying to see whether it holds good or not, so for this we have to
first find A ∪ B. So, when we take A ∪ B set here, so after overlapping these two fuzzy sets A
and B and when we apply the max criteria, when we take union of these two, we are going to

219
get this as A ∪ B, here this fuzzy set is a normal fuzzy set. So, we have got A ∪ B as shown
here by the red color fuzzy set.

(Refer Slide Time: 06:49)

And now we have to obtain A ∪C. So, when we have A fuzzy set here and C fuzzy set here,
so A ∪C where when we overlap these two, we get after taking the max criteria applied, we
are going to get A ∪C here. So, this is A ∪C. Now, we have to take the intersection of A ∪ B
and A ∪C. So, let us see what are we going to get out of these two.

(Refer Slide Time: 07:38)

220
So, we have A ∪ B which we have already just got and then we have A ∪C. Now let us see
what we are going to get as the intersection of these two fuzzy sets.

(Refer Slide Time: 07:57)

So, when we overlap these two fuzzy sets is the A ∪ B fuzzy set, this is A ∪C fuzzy set. So,
when we overlap these two fuzzy sets and since we are going to we are interested in the
intersection of these two, so we have to apply the min criteria. So, when we apply the min
criteria, min of the these two membership; min of these two fuzzy sets, so what we are going
to get is here. So, this is what is the outcome.

So, when we see this outcome is same as we have got here, so these two fuzzy sets are equal.
So, this means that if we have three fuzzy sets A , B and C, so if we take the union of this is
the union. So, if we take the union of A ∪(B ∩ C ), we are going to get this equal to the
( A ∪ B ) ∩( A ∪C ).

221
(Refer Slide Time: 09:20)

So, this is very clear, the results are same here; the outcomes are same. So, we can say here
that the distributivity property of union over intersection is verified or in other word we can
say distributivity property of union over intersection holds good for fuzzy sets as well.

(Refer Slide Time: 09:51)

So, now the same can be checked by taking the discrete fuzzy sets A , Band C. So, here we
have taken three discrete fuzzy sets and if we see here, we get B ∩ C as this discrete fuzzy set.

222
(Refer Slide Time: 10:11)

And then when we take the A ∪ ( B ∩C )=0.7 /1+0.5/ 2+0.3/ 3+0.6/ 4. So, this is what is the
outcome that we get here. Now, if we find here the ( A ∪ B ) ∩( A ∪C ) and this A ∪ B we get
from here and A ∪C we get from here.

So, if we take the intersection of these two, we take the intersection of these two like this, like
we have this and we have this and we take the intersection. So, if we take this intersection,
we are going to get discrete fuzzy set again as the outcome and this is
0.7/ 1+0.5/ 2+0.3/ 3+0.6/ 4 and which is same as this fuzzy set. So, it is clearly visible that
for discrete fuzzy sets also the distributivity property of union over intersection is verified.

223
(Refer Slide Time: 12:26)

So, now we can go ahead and define that the distributivity of intersection over union for crisp
sets, the distributivity property of intersection over union holds good and this we already
know. So, this means that A intersection or other words we can say the
A ∪ ( B ∩C )= ( A ∪ B ) ∩( A ∪C ).

So, this holds good for crisp sets and if we change this by fuzzy sets A , B and C, so this
distributivity property of intersection over union is here also satisfied. So, this mean that the
distributivity of intersection over union for fuzzy sets is also holding good. So as it is written
here, distributivity property of intersection over union and this holds good for fuzzy sets A , B
and C same as crisp sets.

224
(Refer Slide Time: 13:55)

Now, let us try to see how this is holding good for fuzzy sets A , B and C. So, let us take an
example here where we take three continuous fuzzy sets A , B and C. So, these are the fuzzy
sets you can see here.

(Refer Slide Time: 14:19)

Now, for verifying to verify this distributivity of intersection over union, we have to first get
the B ∪ C.

225
(Refer Slide Time: 14:41)

So B ∪ C, we have we are taking B fuzzy set and C fuzzy set and B ∪ C, we can get just by
overlapping these two fuzzy sets and apply the max criteria, we are going to get this as the
B ∪ C.

(Refer Slide Time: 14:56)

And now let us take a fuzzy set and then we take the A ∩(B ∪C ) which we have just got.

226
(Refer Slide Time: 15:16)

So, A ∩(B ∪C ) here. So, we can again overlap these two fuzzy sets and since we are taking
here intersection, we apply the min criteria and then on applying the min criteria, we are
going to get this as the result. So, this is represented by the red color fuzzy set. So, this is the
outcome of the A ∩(B ∪C ). So, this will keep this here this outcome here and then now we
will try to find the ( A ∩ B)∪( A ∩ C) and let us see if this is coming out to be same as this or
not.

(Refer Slide Time: 16:31)

227
So, let us now try to find the A ∩ B. So, A ∩ B again we take fuzzy set and fuzzy set B and we
try to we first overlap these two fuzzy sets and since we are taking the intersection, so we
have to apply the min criteria. So when we apply a min criteria, we get a fuzzy set here, a sub
normal fuzzy set and we can and this is A ∪ B, A ∩ B. Now, after this we have to find A ∩C.

(Refer Slide Time: 17:04)

So, when we take A fuzzy set and B , C fuzzy set, we take the intersection, we find the A ∩C.

(Refer Slide Time: 17:15)

228
We again do the same and we overlap these two fuzzy sets A and C together. We apply the
min criteria, after applying the min criteria we get the sub normal fuzzy set which is A ∩C
which is the outcome here.

So, after this what we have to do is we have to find the union of these two fuzzy sets. So, we
have now A ∩C and earlier we found A ∩ B. So, now we take the union of these two.

(Refer Slide Time: 17:58)

So, we had here A ∩ B and then we have A ∩C. Now, we have to find the union of these two
fuzzy sets. Let us see how does it look like.

(Refer Slide Time: 18:17)

229
So, we see here that we overlap these two fuzzy sets here. So, this fuzzy set and this fuzzy
set. So, what is this is A ∩ B and here we have A ∩C.

We have to take the union of these two fuzzy sets, it means that we have to apply the max
criteria and when we apply max criteria, we are going to get this as the result. So, we are
going to get here this as the union of A ∩ B and A ∩C. So, if we see these outcome is same as
this outcome is same as the previous outcome which is which we have got on taking the
A ∩(B ∪C ).

(Refer Slide Time: 19:22)

So, this way we can say here; this way we can see here say here that the distributivity
property of intersection over union is verified for fuzzy sets A , B and C which is here.

230
(Refer Slide Time: 19:45)

Now, the same can be verified by taking three discrete fuzzy sets. So, these three discrete
fuzzy sets we have taken here. So, we have A fuzzy set as 0.7/ 1+0.5/ 2+0.1/ 3+0.6 /4 and
then we have B as the discrete fuzzy set here as 0.8/ 1+0.3/ 2+0.7/ 3+0.5/ 4. And the 3rd one,
3rd fuzzy set 3rd discrete fuzzy set here is 0.4 /1 , 0.7/2 , 0.3/3 ,0.9/ 4. Let us now try to verify
the distributivity property of intersection over union.

So, for this we have to first find B ∪ C and then we take the if we find the A ∩(B ∪C ). So, let
us now go ahead and do that.

(Refer Slide Time: 21:04)

231
So, B ∪ C. is here and this is coming out to be after applying the max criteria, we are going
to get 0.8/ 1+0.7/ 2+0.7 /3+0.9/ 4 and this is nothing, but is again a discrete fuzzy set. So, all
these A , B ,C are also discrete fuzzy sets.

Now, let us take the intersection here A ∩(B ∪C ) here and then when we apply since we are
taking the intersection, we have to follow the min criteria here, we have to apply min criteria
here. So, when we do that we are getting 0.7/ 1+0.5/ 2+0.1/ 3+0.6 /4 as a discrete fuzzy set.
So, this way we have the outcome and now we need to find the A ∩ B and A ∩C. So if we
find A ∩ B, we are getting 0.7/ 1+0.3/ 2+0.1/ 3+0.5/ 4 by after applying the min criteria; min
criteria.

So here also A ∩C, we are getting after applying the min criteria 0.4 /1+0.5/2+0.1/ 3+0.6/ 4.
So now when we have found the intersections, we are getting two discrete fuzzy sets as A ∩ B
, A ∩C, now we have to take the union of these two fuzzy, discrete fuzzy sets. So, when we
are taking the union of these two fuzzy sets; so since we are taking union now what we have
to do is, we have to apply a max criteria. So when we apply max criteria, let us see what we
are going to get.

So when we apply max criteria, we are getting 0.7/ 1+0.5/ 2+0.1/ 3+0.6 /4. So, this way when
we compare the these two outcome, this was the first outcome, if I say here and this is the
second outcome here, so we see that the first outcome is equal to the second outcome or in
other words, we can say that the ( A ∩ B ) ∪ ( A ∩ C ) =A ∩( B ∪ C). So, this way we can clearly
see that the distributivity property of intersection over union is verified.

232
(Refer Slide Time: 25:08)

So, after verifying this we clearly we can say that the distributivity property of union over
intersection and distributivity property of intersection over union both are satisfied for fuzzy
sets. Here in our examples we have taken two fuzzy sets discrete as well as continuous and
we have seen that these two distributivity properties are verified. So, at this point we will stop
and in the next lecture we will discuss the remaining properties.

Thank you.

233
 Fuzzy Sets, Logic and Systems and Applications
Prof. Nishchal K. Verma
Department of Electrical Engineering
Indian Institute of Technology, Kanpur

Lecture - 15
Properties of Fuzzy Sets

So, welcome to lecture number 15 Fuzzy Sets, Logic and Systems and Applications. So, this
lecture is in continuation to our discussion on Properties of Fuzzy Sets. Here we have these
properties that we have intended to discuss or listed.

(Refer Slide Time: 00:38)

So, far we have discussed a law of contradiction, law of excluded middle, idempotency,
involution, commutativity, associativity, distributivity and these properties we have discussed
with respect to fuzzy sets. Now, the rest of the properties we will try to discuss and in this
continuation, we’ll be discussing absorption, absorption of complement and De Morgan’s
laws.

234
(Refer Slide Time: 01:22)

So, in today’s lecture we will try to discuss absorption and this absorption will be the
absorption of union over intersection and this is defined as A ∪( A ∩ B ). So, this is regarded
as the absorption of union over intersection and this is equal to A and then the absorption of
intersection over union and this is nothing but the A ∩ ( A ∪ B )= A.

So, let me tell you that when we discussed the absorption with respect to sets as we have
already discussed the absorption of union over intersection for crisp sets and absorption of
intersection over union for crisp sets. So, these hold good. Now, so let us we take fuzzy sets
A and B. So, these absorption of union over intersection, absorption of intersection over
union, both these absorptions here also hold good for fuzzy sets and then in this lecture we’ll
also discuss the absorption of complement.

So, when we say absorption of complement, again we have the absorption of complement for
union absorption of complement for intersection. So, here we see that the absorption of
complement for union hold good for the crisp set, but it doesn’t hold for the fuzzy sets.
Similarly, the absorption of complement for intersection hold good for crisp set, but this does
not hold good for the fuzzy sets. So, we will discuss these absorptions for fuzzy sets and
we’ll take some examples also to see as to how it works.

235
(Refer Slide Time: 03:53)

So, here is the absorption of union over intersection. So, as I have already mentioned that
when we take crisp set A and B and the absorption of union over intersection. So, we have
here the A ∪( A ∩ B ). So, here we have two entities: 1 is the entity A and other is the entity
( A ∩ B). So, when we take union of these two we get only A. So, this is for crisp set, same is
true when we take fuzzy sets and this is similar to the crisp set.

So, here also we see that when we take union of A fuzzy set, this is A fuzzy set this A fuzzy
set here and the A ∩ B fuzzy set. So, this is A ∩ B this is again A fuzzy set here. So, I will just
right it like this show it like this. So, you see here that we have two fuzzy sets, one is A fuzzy
set and other one is the fuzzy set which is the outcome of A ∩ B. So, when we take the union
of these two, we are getting the fuzzy set A. So, this means that the absorption property of
union over intersection, we can say that is satisfied or that holds good.

236
(Refer Slide Time: 05:48)

So, if we take an example here to understand this property better, we take two continuous
fuzzy sets, one fuzzy set is A here and then the other one is B fuzzy set. So, we see here this
two fuzzy sets, A and B they are defined within the universe of discourse right from 0 to 10.
So, both the fuzzy sets are defined in this range and let us see when we have the absorption of
union over intersection whether this hold good are not. Of course, this should hold good, but
let us see by taking this example how or in what way this is holding good.

(Refer Slide Time: 06:41)

237
So, we have here the fuzzy set A and then we have fuzzy set B and since we are interested in
this, the A ∪( A ∩ B ) and so for this we need to first find A ∩ B, A ∩ B. So, we have A fuzzy
set and B fuzzy set it’s very easy to find A ∩ B we have done it many times.

(Refer Slide Time: 07:12)

So, let us now find it. And to find this intersection we have to superimpose these two fuzzy
sets on each other and we see here that we have this as A fuzzy set and this as the B fuzzy set
and when we superimpose and then since we are finding the we are interested in getting the
intersection. So, what we have to do here is to take we apply the min of these two
corresponding membership values with respect to all the generic variable values within the
universe of discourse.

238
(Refer Slide Time: 07:57)

So, when we apply this min condition here, we find here this portion after applying the min
portion we get here A ∩ B. So, this is a fuzzy set, of course, this is a subnormal fuzzy set
because none of membership value is approaching 1 or other words we can say the core of
this fuzzy set is empty. So, that way here we were getting A ∩ B as the result and which is a
subnormal fuzzy set. So, now this outcome A ∩ B, let us take the union of this fuzzy set and
A.

(Refer Slide Time: 08:47)

239
So, here we are taking A fuzzy set and the union of these two. So, here we are taking the
union of these two fuzzy sets right and let us see what are we going to get here.

(Refer Slide Time: 09:03)

So, when we superimpose these two fuzzy sets on each other, we are getting here since we
are trying to get are we are interested in the union, so we have to apply the max criteria and
when we apply max criteria we are going to get a fuzzy set here which is like this, this is the
fuzzy set. So, we clearly see that this is when we apply max criteria we are going to get this
as a fuzzy set. Now, if we see the result and we compare we see that this same as A.

(Refer Slide Time: 09:51)

240
So, here, A is this and A ∪( A ∩ B ) both the fuzzy sets are same. So, we can clearly say here
that the outcome is A, and that is how we can say that the absorption property for union over
intersection is verified and of course these sets are the fuzzy sets. So, here we have taken two
fuzzy sets A and B and we applied absorption property of union over intersection and we
found that these two are verified.

(Refer Slide Time: 10:36)

Now, let us take another example where we take two fuzzy sets, A and B and these fuzzy sets
are discrete fuzzy sets. So, we have A fuzzy set here and then we have B fuzzy set here and
what we have to check is this property, where you see that this absorption of union over
intersection should come out like this. So, the above expression is defined as when, when we
apply the max correct criteria we can write it like this and then when we apply this to these
two discrete fuzzy sets, we are going to get as a result this after applying the max and min
criteria which is mentioned over here and when we see this, this is same as A.

So, we can clearly see here that this is same as A is you can say, you can see here that A was
0.7/ 1+0.5/ 2+0.1/ 3+0.6 /4 and the result is also coming out to be the same. So, we can say
here the result of A ∪( A ∩ B ) and which is coming out to be the same as A. So, we can
clearly say that this absorption of union over intersection is verified.

241
(Refer Slide Time: 12:28)

Now, let us discuss the absorption of intersection over union. So, you see earlier was
absorption of union over intersection, now we are going to discuss the absorption of
intersection over union. So, let us see this for crisp sets A and B. So, we have for crisp sets,
we have you see the A ∩( A ∪ B ) which is coming out to be A. Now when we take fuzzy sets
instead of crisp sets, so, we see that this property holds good for the fuzzy sets as well.

So, if we take A as fuzzy set if this is fuzzy set and here B also is fuzzy sets, so, when we
take union of these two fuzzy sets this will also be a fuzzy set. We can write it like this we
can show it like this, that these are the fuzzy sets, this A is a fuzzy set and then A ∪ B is also
fuzzy set and when we take the intersection of these two fuzzy sets we are going to A as a
result which is again a fuzzy set. So, A fuzzy set we are going to get as a result. So, we can
say that the absorption of intersection over union is also holding good. So, now let’s take
some examples and understand this better.

242
(Refer Slide Time: 14:26)

So, as we have seen previously by taking this by taking the example here we take three fuzzy
sets, three continuous fuzzy sets and we see that here we are taking fuzzy sets A and B for
this example. So, A is this and B is this.

(Refer Slide Time: 14:55)

And then since we have to verify that the A ∩( A ∪ B ), So, we have to first get the A ∪ B. So,
this is A ∪ B now let us find A ∪ B for A and B.

243
(Refer Slide Time: 15:14)

So, when we superimpose these two fuzzy sets on each other, we get to see after applying the
max criteria we are going to get this as the A ∪ B which is a fuzzy set again and now if we
take the intersection of this fuzzy set and A fuzzy set, here we have A fuzzy set and A ∪ B,
let us see what are we going to get.

(Refer Slide Time: 15:39)

(Refer Slide Time: 15:49)

244
So, when we superimpose again these two fuzzy sets on each other, we are going to get this
as the outcome which is again we see nothing but the same fuzzy set as A or we can say these
two fuzzy sets are equal. So, we can say that we are going to get here the outcome which is a
fuzzy set A. So, this way we can say absorption of intersection over union is satisfied for the
fuzzy sets A and B.

(Refer Slide Time: 16:30)

Please note that here in this example we have taken the continuous fuzzy sets A and B.

245
(Refer Slide Time: 16:39)

Now, we take another example were we take to discrete fuzzy sets A and B and let us see
how this is satisfying this is verifying the absorption of intersection over union. So, as we
have done in the previous example, we will find out first the A ∪ B and A ∪ B is here. So,
when we find A ∪ B, we get here this as the fuzzy set discrete fuzzy set and then when we
taken A ∩( A ∪ B ) we are going to get this thing which is nothing but same as A.

You can verify here this fuzzy set with this, this two fuzzy sets are equal here and this way
we can clearly say that the absorption property of intersection over union is verified. So, we
have seen that whether we take the discrete fuzzy set or the continuous fuzzy sets, for the
both the kinds of fuzzy sets the absorption of intersection over union is also verified.

246
(Refer Slide Time: 18:04)

Now, we come to the absorption of complement for union. So, absorption of complement for
union is here this is represented by the A ∪( Á ∩ B ). So, here we have the A and its
complement and we see that when we take the A ∪( Á ∩ B ) we see that the Á is, Á is not
there in the outcome.

So, this holds good for crisp sets A and B we all know this, let us now see what is happening
when we take fuzzy sets A and B. So, when we take fuzzy sets A and B here, we see that we
take when we take A ∪( Á ∩ B ). So, if we take you see here when we take Á ∩ B , this going to
be a fuzzy set this going to the result is a fuzzy set here, this also is a fuzzy set. So, if we take
the union of these two fuzzy sets and the result here is not going to be the A ∩ B as we were
getting this in the crisp set case.

So, here this the result of these Á ∩ B and if we take the A ∪( Á ∩ B ), we are not going to get
A ∪ B. So, this is to be noted here. So, it means that for crisp case, for when we have crisp
sets A and B, the absorption of complement for union holds good. If we take the fuzzy sets,
so, this absorption of complement for union doesn’t hold good. So, let us now take some
example and we understand little better.

247
(Refer Slide Time: 21:03)

And in this example, we take again two fuzzy sets A and B. So, first fuzzy set is A and
second fuzzy set is B let’s now check whether there is absorption of complement for union is
same as what we have discussed or not.

(Refer Slide Time: 21:23)

So, since we have to find here we have to use Á . So, we have A here.

248
(Refer Slide Time: 21:31)

Let us find Á . So, Á is this here, this is fuzzy set A and if we find Á will look like this and
then let us now take the Á ∩ B .

(Refer Slide Time: 21:49)

So, Á and then B fuzzy set, these two fuzzy sets if we take the intersection, we have to go for
superimposing this two sets on each other.

249
(Refer Slide Time: 22:02)

And which we are doing here and then since we are taking the intersection we have to apply
the min criteria. So, when we apply min criteria, we find a fuzzy set here we find a fuzzy set
as a result which is again you know A, the Á ∩ B which is by looking at it we can say a
subnormal fuzzy set. So, we have got this now we have to take the union of this fuzzy set
with A.

(Refer Slide Time: 22:44)

So, A is here and the Á ∩ B is also here, now let us take the union of these two fuzzy set.

250
(Refer Slide Time: 22:57)

Since this is the union we have to apply the max criteria. So, when we apply max criteria, we
are going to get this fuzzy set as a result of this thing and then if we take A ∪ B, A ∪ B is here
and we can clearly say that this is not same as this fuzzy set. So, here A ∪ ( Á ∩ B ) ≠ A ∪ B.

(Refer Slide Time: 23:43)

So, since these two outcomes are not equal for fuzzy sets A and B, we can clearly say that the
absorption of complement for union is not holding good when we take fuzzy sets A and B.

251
(Refer Slide Time: 24:11)

Now, when we take discrete fuzzy sets and see whether for discrete fuzzy sets also this
absorption of complement for union is satisfied or not. So, we take two fuzzy sets two
discrete fuzzy sets A and B here.

(Refer Slide Time: 24:40)

And see here we find A, the A ∪( Á ∩ B ) and we find that we are getting here
0.7/ 1+0.5/ 2+0.7 /3+0.6/ 4 and when we take the A ∪ B we see that we are finding we are
getting 0.8/ 1+0.5/ 2+0.7/ 3+0.6/ 4 and we see that this is not equal to the when we take the
A ∪( Á ∩ B ).

252
So, we can clearly say here also here also this absorption of complement for union does not
hold good. So, in both the cases we clearly see that the absorption of complement for union is
not satisfied or does not hold good.

(Refer Slide Time: 26:05)

Now, the absorption of complement for intersection: so, let us now see what is happening for
absorption of complement for intersection. So, of course, when we take crisp sets A and B,
we take the A ∩( Á ∪ B ), we see that we are getting A ∩ B. When we take fuzzy sets A and B,
this does not hold good and hence we write here the A ∩¿. So, we can clearly say that the
absorption of complement for intersection for fuzzy sets A and B does not hold good.

253
(Refer Slide Time: 27:14)

Let us now understand this better by taking few examples. So, here we take an example
which contains two continuous fuzzy sets A and B. So, A is here and B is here.

(Refer Slide Time: 27:37)

And first we are interested in finding A ∩( Á ∪ B ). So, for this we need to find the Á ∪ B. So,
Á is this.

254
(Refer Slide Time: 28:00)

This is Á and then if we take the Á ∪ B , where B is this fuzzy set.

(Refer Slide Time: 28:10)

255
(Refer Slide Time: 28:17)

We see that after superimposing these two fuzzy sets on each other and since we are
interested in union, we have to apply the max criteria. When we apply max criteria what are
we going to get is here. So, this is the Á ∪ B which is again a fuzzy set. So, this is the
outcome which we are getting when we take the Á ∪ B . Now, we take the intersection of this
fuzzy set with A.

(Refer Slide Time: 28:59)

So, for this we have taken A here and we have taken a fuzzy set which is the union of Á ∪ B.

256
(Refer Slide Time: 29:25)

So, when we take the intersection of these two fuzzy sets when we take intersection of these
two fuzzy set what are we going to get is here. We superimpose these two fuzzy sets on each
other and then when we do that since we are taking the intersection we have to apply the min
criteria. When we apply min criteria when we apply min criteria we are going to get this
outcome as a fuzzy set. So, A, so, this is intersection of these two fuzzy sets and then let see
whether this is equal to A ∩ B or not.

(Refer Slide Time: 30:21)

257
So, when we take A fuzzy set here and B fuzzy set here and we take the intersection of these
two fuzzy sets, we apply the min criteria because again we are taking the intersection. So,
when we apply min criteria we are going to get this fuzzy set as the outcome of A ∩ B. And
when we compare these two when we compare this fuzzy set and this fuzzy set, so, these two
are not equal. So, this way we can say that the absorption of complement for intersection does
not hold good for fuzzy sets A and B.

(Refer Slide Time: 31:08)

(Refer Slide Time: 31:13)

258
Now, if we take the discrete fuzzy sets instead of continuous fuzzy sets and we see that here
also which is very clear from this example.

(Refer Slide Time: 31:33)

Here also when we try to find the A ∩( Á ∪ B ), we see here that we are getting
0.7/ 1+0.5/ 2+0.1/ 3+0.5/ 4. So, this is a discrete fuzzy set as the outcome and then when we
take the intersection of A and B here which is coming out to be 0.7/ 1+0.5/ 2+0.1/ 3+0.5/ 4
which is clearly not equal to this case.

So, this way we can say this is not equal to this. It means what? It means the absorption of
complement for intersection does not hold good for discrete fuzzy sets A andB.

259
(Refer Slide Time: 32:43)

So, this way we have checked, we have verified, the absorption of union over intersection,
absorption of intersection over union, absorption of complement for union, absorption of
complement for intersection and these have been verified these have been checked for fuzzy
sets which are continuous and discrete.

So, we will stop here in this lecture and in the next lecture will study will try to cover the
following. So, we will be discussing the De Morgan’s law of union and De Morgan’s law of
intersection for fuzzy sets.

Thank you.

260
 Fuzzy Sets, Logic and Systems and Applications
Prof. Nishchal K. Verma
Department of Electrical Engineering
Indian Institute of Technology, Kanpur

Lecture – 16
Properties of Fuzzy Sets

(Refer Slide Time: 00:17)

Welcome to lecture number 16 of Fuzzy sets, Logic and Systems and Applications. So,
here we are in the continuation of Properties of classical and Fuzzy Sets where we have
seen that the properties listed here all are holding good for classical sets. Whereas, we are
here seeing that some of the properties like law of contradiction, law of excluded middle,
absorption of complement, these three properties out of all mentioned here are not holding
good for fuzzy sets.

261
(Refer Slide Time: 01:08)

So, so far we have covered Law of Contradiction, Law of Excluded Middle, Idempotency
property, Involution, Commutativity, Associativity, Distributivity, Absorption,
Absorption of complement.

(Refer Slide Time: 01:37)

So, in the continuation in this lecture today, we will discuss DeMorgan’s law with respect
to fuzzy sets. And when we see DeMorgan’s law of union is defined here for crisp sets A
and B as when we take the complement of A union B, this is going to be the intersection
of A complement and B complement.

262
So, this is holding good for crisp sets we all know. Let us see if we take fuzzy sets whether
the DeMorgans law of union hold good for fuzzy sets or not. Yes it holds good for fuzzy
sets as well. So, if we take two fuzzy sets 𝐴 and 𝐵 and we take the complement of 𝐴 ∪ 𝐵,
we are going to get or we are getting the fuzzy set which is a complement 𝐴 ∩ 𝐵̅. So,
DeMorgan’s law for fuzzy sets also hold good. So, let us now understand DeMorgan’s law
for fuzzy sets better by taking a couple of examples.

(Refer Slide Time: 02:59)

So, here we have example which has two fuzzy sets 𝐴 and 𝐵, two continuous fuzzy sets 𝐴
and 𝐵; 𝐴 is here and 𝐵 is here.

263
(Refer Slide Time: 03:19)

So, let us now take the ̅̅̅̅̅̅̅

𝐴 ∪ 𝐵 which is mentioned over here and then let us see whether
this is equal to or this is exactly the same as the 𝐴̅ ∩ 𝐵̅. So, we first find the 𝐴 ∪ 𝐵. So,
since 𝐴 is here 𝐵 is here fuzzy set a fuzzy set 𝐵, let us find 𝐴 ∪ 𝐵.

(Refer Slide Time: 03:57)

So, to find 𝐴 ∪ 𝐵, we know we superimpose these two fuzzy sets on each other and we
apply max criteria and when we applied max criteria we get 𝐴 ∪ 𝐵 as a result here.

264
(Refer Slide Time: 04:04)

So, this is 𝐴 ∪ 𝐵. Now, we have to find the complement of this fuzzy set which will be the
̅̅̅̅̅̅̅
𝐴 ∪ 𝐵.

(Refer Slide Time: 04:28)

265
(Refer Slide Time: 04:33)

So, let us see how does it look like. So, when we have ̅̅̅̅̅̅̅
𝐴 ∪ 𝐵 , we just subtract this, we
apply the criteria mentioned here, we subtract this from one. So, we apply for getting A
the ̅̅̅̅̅̅̅
𝐴 ∪ 𝐵, we get 1 minus max of all the corresponding values of membership functions
of A and membership function of fuzzy set 𝐵 for respective generic variable values in the
universe of discourse of course. So, we see that we have the complement of A union B like
this. Now let’s find out the 𝐴̅ here, 𝐴̅ here and 𝐵̅ here.

(Refer Slide Time: 05:48)

266
Let us find out from these two fuzzy sets and see what are we going to get when we take
the intersection of these two fuzzy sets.

(Refer Slide Time: 05:56)

(Refer Slide Time: 05:58)

So, 𝐴̅ is here is complement of A, I mean 𝐴̅ and then complement of 𝐵 that is 𝐵̅ which is

here.

267
(Refer Slide Time: 06:08)

So, the 𝐵̅, this is 𝐴̅ we are interested in finding the intersection of these two complements.

(Refer Slide Time: 06:20)

And let us see what are we going to get when we take the intersection of these two.

268
(Refer Slide Time: 06:30)

So, we super impose these two fuzzy sets on each other here as we have done and since
we are taking intersection when we apply main criteria we are going to get here this fuzzy
̅̅̅̅̅̅̅
set as the outcome. So, we clearly see here that when we take the 𝐴 ∪ 𝐵 and when we take
𝐴̅ ∩ 𝐵̅, we see that these two are same.

(Refer Slide Time: 07:06)

So, this way we can since the outcome both the outcomes are equal. So, we can say that
DeMorgan’s Law of Union holds good for fuzzy sets 𝐴 and 𝐵 and that’s how this relation
is valid for fuzzy sets.

269
(Refer Slide Time: 07:30)

Now, let us take an example here with discrete fuzzy sets and see what is happening. So,
here also when we take fuzzy set 𝐴 and fuzzy set 𝐵, and these two fuzzy sets are discrete
fuzzy sets. And when we compute the complement of 𝐴 ∪ 𝐵 as it is shown here.

(Refer Slide Time: 07:53)

̅̅̅̅̅̅̅
𝐴 ∪ 𝐵 the 𝐴 ∪ 𝐵 we are getting the fuzzy set point 2/1 + 0.5/2 + 0.3/3 + 0.4/4 as a
result. And when we find the 𝐴̅ ∩ 𝐵̅ so, we find here as a result 0.2⁄1 + 0.5 /2 + 0.3/3 +
0.4/4.

270
 ̅̅̅̅̅̅̅
So, we see that if we take this as the outcome here of the 𝐴 ∪ 𝐵 and we see that this is
exactly equal to the same as when we take the 𝐴̅ ∩ 𝐵̅. So, we see that the DeMorgan’s Law
of union here also is valid when we take the discreet fuzzy sets 𝐴 and 𝐵.

(Refer Slide Time: 09:19)

Now, let’s so so, earlier what we were seeing was the De Morgan’s law of union and this
we verified by taking examples of fuzzy sets 𝐴 and 𝐵. Now we are extending this to three
̅̅̅̅̅̅̅̅̅̅̅̅̅
fuzzy sets. So, we all know that for crisp sets also when we take the 𝐴 ∪ 𝐵 ∪ 𝐶 , this is
equal to the 𝐴̅ ∩ 𝐵̅ ∩ 𝐶̅ .

So, this is same as when we use instead of crisp sets we take fuzzy sets 𝐴, 𝐵 and 𝐶, we see
that the DeMorgan’s law of union is valid here or holds good here. So, in other words we
can say for fuzzy sets 𝐴, 𝐵, 𝐶 when we take the complement of 𝐴 ∪ 𝐵 ∪ 𝐶 = 𝐴̅ ∩ 𝐵̅ ∩ 𝐶̅ .

So, these two are valid. So, this is valid for crisp here means the DeMorgan’s law of union
is valid for crisp as well as fuzzy sets when we take more than two fuzzy sets. For two
fuzzy set we have already seen. So, when we increase the number of fuzzy sets here also
this is valid.

271
(Refer Slide Time: 10:59)

So, let us now take an example and understand better. So, see here we have 𝐴 fuzzy set, 𝐵
fuzzy set and 𝐶 fuzzy set.

(Refer Slide Time: 11:08)

And let us now, take the complement of 𝐴 ∪ 𝐵 ∪ 𝐶. So, for getting this, we super impose
these 𝐴, 𝐵, 𝐶 fuzzy sets on each other.

272
(Refer Slide Time: 11:27)

And when we have done this, we see that this looks like this. So, we have to apply the max
criteria.

(Refer Slide Time: 11:39)

And when we apply max criteria, we are going to get this as which is shown by the red
color. So, this is 𝐴 ∪ 𝐵 ∪ 𝐶.

273
(Refer Slide Time: 11:58)

Now when we take the complement of this, so this is exactly the 𝐴 ∪ 𝐵 ∪ 𝐶. And when we
take the complement of this, we are going to get this as the fuzzy set.

(Refer Slide Time: 12:03)

Now, let’s take the 𝐴 ∩ 𝐵 ∩ 𝐶. So, we have 𝐴 here as fuzzy set 𝐵, 𝐶.

274
(Refer Slide Time: 12:24)

(Refer Slide Time: 12:27)

So, let us take the intersection of 𝐴̅. So, A complement we see here, this is 𝐴̅.

275
(Refer Slide Time: 12:30)

(Refer Slide Time: 12:32)

And then we have 𝐵̅ and then we have here 𝐶̅ .

276
(Refer Slide Time: 12:35)

So, now we are interested in finding the intersection of these compliments. Let us
superimpose these three compliments on each other which is here.

(Refer Slide Time: 12:47)

And since we are interested in intersection, we have to apply the min criteria.

277
(Refer Slide Time: 12:58)

When we apply min criteria, we see the result which is shown by red color which is here.

(Refer Slide Time: 13:04)

So, we see here very clearly that these two outcomes are equal, means when we take
̅̅̅̅̅̅̅̅̅̅̅̅̅
the𝐴 ∪ 𝐵 ∪ 𝐶 = 𝐴̅ ∩ 𝐵̅ ∩ 𝐶̅ .

278
(Refer Slide Time: 12:34)

So, this way we can say that DeMorgan’s law of union is holding good for fuzzy sets 𝐴, 𝐵
and 𝐶. Now the same can be checked by taking three discrete fuzzy sets 𝐴, 𝐵 and 𝐶.

(Refer Slide Time: 14:00)

So, let’s now see what is happening when we take three discrete fuzzy sets. So, let me just
write it just represent this fuzzy set by I here and then by II here is by III here. So, these
three are three discrete fuzzy sets.

279
(Refer Slide Time: 14:22)

And we are interested in the ̅̅̅̅̅̅̅̅̅̅̅̅̅

𝐴 ∪ 𝐵 ∪ 𝐶 first. So, when we find this as this is mentioned
̅̅̅̅̅̅̅̅̅̅̅̅̅
here so, A the 𝐴 ∪ 𝐵 ∪ 𝐶 = 0.2/1 + 0.3/2 + 0.3/3 + 0.1/4. So, this is what is the
̅̅̅̅̅̅̅̅̅̅̅̅̅
discrete fuzzy set which is outcome of the 𝐴 ∪ 𝐵 ∪ 𝐶 . Now let us find out the 𝐴̅ ∩ 𝐵̅ ∩ 𝐶̅ .

So, first we need to find out the 𝐴̅. So, this 𝐴̅, then this is our 𝐵̅ and this is our 𝐶̅ . So, these
three are the fuzzy sets or the complements of 𝐴 fuzzy set 𝐵 fuzzy set 𝐶 fuzzy sets
respectively. Now if we take their intersection their intersection means the 𝐴̅ ∩ 𝐵̅ ∩ 𝐶̅ . So,
means we have three fuzzy sets, these three fuzzy sets are complements of 𝐴, 𝐵 and 𝐶
separately. And when we take intersection we of course, we apply the min criteria and then
when we apply min criteria, we are getting a fuzzy set which is A discreet fuzzy set and
this is 0.2/1 + 0.3/2 + 0.3/3 + 0.1/4 and when we see this fuzzy set is exactly same as
we have got here.

̅̅̅̅̅̅̅̅̅̅̅̅̅
So, this fuzzy set is same as the fuzzy set which we have got out of the 𝐴 ∪ 𝐵 ∪ 𝐶 . So, this
way we can say that the DeMorgan’s Law of union for discrete fuzzy set A, B and C is
holding good.

280
(Refer Slide Time: 17:02)

So, first when we take crisp fuzzy set we all know that DeMorgan’s law of intersection is
valid holding good and when we take fuzzy sets 𝐴 and 𝐵, here also this is holding good
means this is valid. So, when we take the ̅̅̅̅̅̅̅
𝐴 ∩ 𝐵 = 𝐴̅ ∪ 𝐵̅ . So, this is called the DeMorgan’s
law of intersection and let us now understand this better by taking couple of examples. So,
first we will take two fuzzy sets 𝐴 and 𝐵 here. We have A here 𝐴 fuzzy set here and 𝐵
fuzzy set here.

(Refer Slide Time: 17:49)

And let us see how this DeMorgan’s law of intersection is holding good.

281
(Refer Slide Time: 18:06)

So, to check that let us first find the complement of 𝐴 ∩ 𝐵. So, here first find the intersect
𝐴 ∩ 𝐵. So, let us first super impose A and B on each other as it is shown here.

(Refer Slide Time: 18:25)

282
(Refer Slide Time: 18:28)

And then since we are taking the intersection of course, we will be applying the min criteria
and then we see here we are getting this fuzzy set as 𝐴 ∩ 𝐵.

(Refer Slide Time: 18:53)

Now, let us take the complement of this outcome; let us take the ̅̅̅̅̅̅̅
𝐴 ∩ 𝐵. So, let us see what
are we going to get we had to subtract the respective membership values from one and if
we do that we are going to get this fuzzy set as the ̅̅̅̅̅̅̅
𝐴 ∩ 𝐵.

283
(Refer Slide Time: 19:00)

Now, let us find the 𝐴̅ and 𝐵̅ and then take their union and let us see whether this equal to
the ̅̅̅̅̅̅̅
𝐴 ∩ 𝐵 or not. So, let us now start. So, 𝐴 fuzzy set is here and 𝐵 fuzzy set is here. Let’s
take the 𝐵̅.

(Refer Slide Time: 19:47)

284
(Refer Slide Time: 19:48)

Let us get the 𝐴̅, 𝐴̅ is here. This is 𝐴̅ and 𝐵̅ is here, now we are interested in the 𝐴̅ ∪ 𝐵̅.

(Refer Slide Time: 20:08)

285
(Refer Slide Time: 20:10)

It means we have to super impose 𝐴̅ and 𝐵̅ on each other like this and then since we are
interested in the union, we have to apply the max criteria and when we apply max criteria,
we are going to get here this as the outcome.

(Refer Slide Time: 20:20)

So, this is the outcome of the 𝐴̅ ∪ 𝐵̅ and if we see it is very clear that these two outcomes
are same. So, what does this mean? This means that the ̅̅̅̅̅̅̅
𝐴 ∩ 𝐵 = 𝐴̅ ∪ 𝐵̅ .

286
(Refer Slide Time: 20:53)

So, this way you see here we can say that the DeMorgan’s law of intersection holds good
for continuous fuzzy sets. Now let us check the same for discrete fuzzy sets.

(Refer Slide Time: 21:05)

So, we see here if we take two fuzzy sets 𝐴 and 𝐵 and we see that we have on the same
lines as we have done in the previous example, we take the ̅̅̅̅̅̅̅
𝐴 ∩ 𝐵. So, 𝐴 ∩ 𝐵 is here ̅̅̅̅̅̅̅
𝐴 ∩ 𝐵.

287
(Refer Slide Time: 21:31)

So, this is coming out as 0.3/1 + 0.7/2 + 0.9/3 + 0.5/4. So, this is what we are getting
as the fuzzy set discrete fuzzy set which is coming as the outcome when we take the ̅̅̅̅̅̅̅
𝐴 ∩ 𝐵.
Now let us take the union of the complements the complements of A and B. So, we have
𝐴̅ here 𝐵̅ here and I can show it like this by this arrow 𝐴̅, 𝐵̅.

And when we take union of these two fuzzy sets when we take union of these two fuzzy
sets, so we see that we are getting0.3/1 + 0.7/2 + 0.9/3 + 0.5/4. If we see here this
outcome is same as the outcome of the ̅̅̅̅̅̅̅
𝐴 ∩ 𝐵.

(Refer Slide Time: 23:01)

288
Now, if we take more than two fuzzy sets so, let us see what is happening of course, here
also the DeMorgan’s law of intersection is holding good. So, and if you talk of crisp sets
𝐴 and 𝐵, 𝐴, 𝐵 and 𝐶 these crisp sets the DeMorgan’s law of intersection is satisfied. So,
we do not have to care about crisp sets because when we for crisp sets when we see the
DeMorgan’s law of intersection this is valid. So, so this valid when we say valid means
̅̅̅̅̅̅̅̅̅̅̅̅̅
when we take the 𝐴 ∩ 𝐵 ∩ 𝐶 = 𝐴̅ ∪ 𝐵̅ ∪ 𝐶̅ .

So, this is valid for crisp sets and when we take fuzzy sets 𝐴, 𝐵 and 𝐶 here also this is valid
this DeMorgan’s law of intersection is valid. So, this means that if we have 𝐴 fuzzy set 𝐵
̅̅̅̅̅̅̅̅̅̅̅̅̅
fuzzy set 𝐶 fuzzy set and if we take the 𝐴 ∩ 𝐵 ∩ 𝐶 this is going to be same as the
complement of 𝐴̅ ∪ 𝐵̅ ∪ 𝐶̅ and this is called the DeMorgan’s law of intersection as I just
mentioned.

(Refer Slide Time: 24:39)

So, let us now understand this also better by taking one example here and here we take
again same as other examples we take the three fuzzy sets 𝐴 and 𝐵, 𝐴, 𝐵 and 𝐶 fuzzy sets.
These fuzzy sets are continuous fuzzy sets. So, here this 𝐴 fuzzy set 𝐴 continuous fuzzy
set 𝐵 continuous fuzzy set and 𝐶 continuous fuzzy sets 𝐴 is triangular 𝐵 is Gaussian fuzzy
set and 𝐶 is trapezoidal fuzzy set.

289
(Refer Slide Time: 25:21)

̅̅̅̅̅̅̅̅̅̅̅̅̅
So, our intention now is to find the 𝐴 ∩ 𝐵 ∩ 𝐶 , how do we get that? So, for getting this
first, we have to find 𝐴 ∩ 𝐵 ∩ 𝐶 and then we take the complement of the outcome.

(Refer Slide Time: 25:45)

So, if we have 𝐴 here; 𝐴 fuzzy set here 𝐵 fuzzy set here 𝐶 fuzzy set here, we can super
impose these three fuzzy sets on each other which is here and which is shown here and
then apply the min criteria.

290
(Refer Slide Time: 26:06)

And when we apply min criteria, we get here fuzzy set as the outcome. So, this is nothing,
but the intersection of 𝐴, 𝐵 and 𝐶 fuzzy sets.

(Refer Slide Time: 26:09)

291
(Refer Slide Time: 26:29)

Now, we are interested in complement of it. So, we have to take the complement when we
take the complement, we are going to subtract all respective values of membership values
all respective membership values from one. So, when we do that, we are going to get this
̅̅̅̅̅̅̅̅̅̅̅̅̅
fuzzy set as the outcome. So, this is nothing, but the 𝐴 ∩ 𝐵 ∩ 𝐶 . Now let us find out the
𝐴̅ ∪ 𝐵̅ ∪ 𝐶̅ . So, here we have 𝐴̅ which we have got from the fuzzy set 𝐴.

(Refer Slide Time: 27:07)

So, we have 𝐴̅ here and then let us find 𝐵̅.

292
(Refer Slide Time: 27:14)

(Refer Slide Time: 27:23)

So, 𝐵̅ is here, we already know as to how we can find the complements. We subtract the
respective membership values from one throughout the universe of discourse.

293
(Refer Slide Time: 27:47)

(Refer Slide Time: 27:52)

294
(Refer Slide Time: 27:53)

We get the complement and so, when we have 𝐵̅, now we see 𝐶̅ . So, 𝐶̅ is here. Now we
have 𝐴̅, we have 𝐵̅, we have 𝐶̅ .

We are interested in their union. So, let us find the union of these three compliments. How
do we do that? So, we super impose each of these complements, each of these fuzzy sets
the complement fuzzy sets on each other and since we are interested in the union, we apply
the max criteria.

(Refer Slide Time: 28:31)

295
So, we can clearly see here that in this plot here, we have super imposed; we have super
imposed all the three complements.

(Refer Slide Time: 28:48)

And then when we apply the max criteria, we find the fuzzy set which is mentioned by
which is denoted by the red color which is shown in red color.

(Refer Slide Time: 29:03)

296
So, what we are getting here as the outcome of the 𝐴̅ ∪ 𝐵̅ ∪ 𝐶̅ , we see that this is exactly
the same as what we have got. This is the fuzzy set exactly same as what we have got when
̅̅̅̅̅̅̅̅̅̅̅̅̅
we have taken the 𝐴 ∩ 𝐵 ∩ 𝐶.

(Refer Slide Time: 29:35)

So, this way we can say since the outcome is equal here for the continuous case when we
have taken three fuzzy sets three continuous fuzzy sets, we can say that the DeMorgan’s
law of intersection is holding good for continuous fuzzy sets. So, in other words I can
repeat here in other words we can say that if we have to find the union of the complements
of 𝐴, 𝐵 and 𝐶 we can do this, we can find this by simply taking the ̅̅̅̅̅̅̅̅̅̅̅̅̅
𝐴 ∩ 𝐵 ∩ 𝐶 . Or in other
̅̅̅̅̅̅̅̅̅̅̅̅̅
words again 𝐴 ∩ 𝐵 ∩ 𝐶 = 𝐴̅ ∪ 𝐵̅ ∪ 𝐶̅ . So, either way we can say.

297
(Refer Slide Time: 30:38)

Now, the same can be checked with three discrete fuzzy sets. So, we have taken here three
discreet fuzzy sets. This is first fuzzy set, this is second fuzzy set is 3, this is third discrete
fuzzy set. So, DeMorgan’s law of intersection, let us now try to see what is happening
when we take these three discrete fuzzy sets and see whether the DeMorgan’s law of
intersection is holding good or not.

(Refer Slide Time: 31:18)

̅̅̅̅̅̅̅̅̅̅̅̅̅
So, for this we have to first find the 𝐴 ∩ 𝐵 ∩ 𝐶 . So, when we see that what we are getting
here is the 0.6/1 + 0.7/2 + 0.9/3 + 0.5/4 here.

298
 ̅̅̅̅̅̅̅̅̅̅̅̅̅
So, I can repeat the outcome here 𝐴 ∩ 𝐵 ∩ 𝐶 is coming out to be 0.6/1 + 0.7/2 + 0.9/3 +
0.5/4. Now let us take the 𝐴̅ ∪ 𝐵̅ ∪ 𝐶̅ . So, when we take this union, we see that we are
getting here as the 0.6/1 + 0.7/2 + 0.9/3 + 0.5/4.

So, if we compare this fuzzy this discrete fuzzy set with this fuzzy set. So, we see that this
is same as these two fuzzy sets are same as you know they both are same they both are
these two fuzzy sets are equal. So, we can right here this fuzzy set is equal means 𝐴̅ ∪ 𝐵̅ ∪
𝐶̅ = 𝐴
̅̅̅̅̅̅̅̅̅̅̅̅̅
∩ 𝐵 ∩ 𝐶 . So, we can clearly say that the DeMorgan’s law of intersection is holding
good for discrete fuzzy sets 𝐴, 𝐵 and 𝐶.

(Refer Slide Time: 33:19)

So, this way we have seen that the DeMorgan’s law is very much holding good for fuzzy
sets as well. Now let us go through all the properties of sets once again. So, we have a
table here and in the first column, we have listed all the properties and then the second
column, we have the classical sets third column, we have fuzzy sets. So, this way we can
say that out of all these properties that are listed here for classical sets and fuzzy sets, only
three namely law of contradiction, law of excluded middle and absorption of complement.

These three they behave differently for classical sets and fuzzy sets rest all others are same
as I mean in both the cases for classical set and fuzzy sets, they behave the same means
they are holding they are valid for classical sets and fuzzy sets.

299
(Refer Slide Time: 34:44)

So, with this I stop here and in the next lecture, we will discuss the concept of fuzzy
distance.

Thank you.

300
 Fuzzy Sets, Logic and Systems and Applications
Prof. Nishchal K. Verma
Department of Electrical Engineering
Indian Institute of Technology, Kanpur

Lecture – 17
Distance between Fuzzy Sets

So, welcome to lecture number 17 of Fuzzy Sets, Logic and Systems & Applications. So,
in this lecture today we will be discussing the distance between two fuzzy sets. That means,
if there are two fuzzy sets 𝐴 and 𝐵 and then, we learn to find the distance between these
two fuzzy sets 𝐴 and 𝐵.

(Refer Slide Time: 00:45)

So, if we have two fuzzy sets say 𝐴 and 𝐵 means there are two fuzzy sets 𝐴 and 𝐵, within
the universe of discourse 𝑋, so the distance which is represented by 𝑑 here the distance
𝑑 (𝐴, 𝐵). So, here the distance between these two fuzzy sets can be defined by the
extension principle and this distance can be given by this formula here and here we have
two formulae like one is for discrete fuzzy sets 𝐴 and 𝐵. Let’s say we have discreet fuzzy
sets then we use this formula to find the distance between two fuzzy sets 𝐴 and 𝐵.

So as we all know that we represent the discreet fuzzy set by this representation.

𝑑(𝐴, 𝐵) = ∑ 𝜇𝑑(𝐴,𝐵) (𝛿)/𝛿

𝑋

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I will tell you what the 𝛿 is, so let us first understand here that this is the formula for finding
the discreet fuzzy set 𝐴 and 𝐵 the distance in between.

So, here if we have let’s say two continuous fuzzy sets A and B like here in this case,
where we will be having the distance between A and B fuzzy sets represented by

𝑑(𝐴, 𝐵) = ∫ 𝜇𝑑(𝐴,𝐵) (𝛿)/𝛿

𝑋

And since we are using here, so let me tell you first that we have the resultant and this
resultant in case of discreet fuzzy set we have the discreet fuzzy set as a result. Means the
distance will be a discreet fuzzy set. So, in a way we can say, the distance is itself a fuzzy
set. So, this distance between 𝐴 and 𝐵 or in other words if I say the distance between two
discreet fuzzy sets 𝐴 and 𝐵 is going to give us a fuzzy set which is again a discreet fuzzy
set. And this is represented by ∑𝑋 𝜇𝑑(𝐴,𝐵) (𝛿)/𝛿 .

And similarly in case of a continuous fuzzy sets A and B we have the resultant, the distance
that we compute we find is nothing but again a continuous fuzzy set. And this is
represented by the fuzzy set which is described as ∫ 𝜇𝑑(𝐴,𝐵) (𝛿)/𝛿 .
𝑋

So, in this two expressions, one is for the discreet and other one is for continuous both
includes 𝜇𝑑(𝐴,𝐵) (𝛿). So, how to find this 𝜇𝑑(𝐴,𝐵) (𝛿) is here. So, if we see here this
expression is helping us in finding the 𝜇𝑑(𝐴,𝐵) (𝛿) =
𝑚𝑎𝑥𝛿=𝑑(𝑥 𝐴 ,𝑥 𝐵 ) [min(𝜇𝐴 (𝑥 𝐴 ), 𝜇𝐵 (𝑥 𝐵 ))]∀𝛿 ∈ ℝ+ . Of course, because we have that we are
computing the distance and that is how this delta is coming out to be the real positive value.
So that is why this delta is coming from the set of positive real value.

So, it is also mentioned here that, the delta because we already have seen that how we can
find the 𝜇𝑑(𝐴,𝐵) (𝛿), which is needed when we are computing the distance between two
fuzzy sets. Now what is delta? So, delta is nothing but it is the difference between the
corresponding generic variables of a two fuzzy sets 𝐴 and 𝐵 that we have taken for a
generic variable.

And please note that we have 𝑥𝐴 and 𝑥𝐵 and these are the generic variables of the fuzzy
sets 𝐴 and 𝐵 respectively. So, we can say for every 𝑥𝐴 and 𝑥𝐵 and these values for every

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𝑥𝐴 , 𝑥𝐵 ∈ 𝑋 which is nothing but the universe discourse. So, I hope this is alright to you
and with this we can manage to find the distance between any two fuzzy sets.

So, now we know that we can compute the membership values of the corresponding
generic variable values by this equation by this relation and delta as I already mentioned
that this is nothing but the difference between the corresponding generic values of two
fuzzy sets for a generic variable. So, let us now make this thing clear or more
understandable by taking one example here. And this way we will be able to understand
as to how we can find the distance between two fuzzy sets.

(Refer Slide Time: 08:34)

So, here I am going to take two discrete fuzzy sets 𝐴 and 𝐵. So, we have the first fuzzy set
here as 𝐴 and then the second is 𝐵 these two fuzzy sets are discreet. As we already we can
see here and we can represent it this fuzzy set like this.

So let us first take fuzzy set 𝐴 here. So, A discreet fuzzy set is having three elements, so
first element we have here is the 1, 0.5. So, 1, 0.5 means, we have 1 as the generic variable
value and 0.5 here is the corresponding membership value. Similarly, the second element
here of the discreet fuzzy set 𝐴 2, 1 so, this 2 is a generic variable value and 1 is the
corresponding membership value.

The third element here is 3 , 0.3, so 3, 0.3 is nothing but we have three as the generic
variable value and 0.3 as the it is corresponding membership value. Now let us draw this

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fuzzy set here and this will look like this. So, we clearly see here that we have three
elements in the discreet fuzzy set 𝐴 and then we draw this fuzzy set we have add 1 as the
generic variable value; we have 0.5 here this is 0.5, so 0.5 here is the corresponding
membership value. So, it means that at 1 we have 0.5 membership value level. So, that is
how we have a drawn a line over here at the level 0.5. And then for the second element
here and second element has 2 as generic variable value and for this we have its
corresponding membership value as 1. So, that is how we have its level as 1.

So, please note that this is a discreet fuzzy set and for representing this we have a taken
range plus minus 0.5 as the generic variable value. So, that is how at one we have at 1 we
can see here at 1 here we have 0.5 and it goes this side and this side both side 0.5 at the
same level. This is just for the representation purpose and this will help us in understanding
this computation of distance between two fuzzy sets a little better.

So, this 0.5 plus minus 0.5 variation has been taken for all the points all the elements of
the discrete fuzzy set. And this way when we see here for the second element also here we
have its membership value 1 and you see here this side also 0.5 and this side also 0.5 it
remains the same here.

Similarly for the third element of the fuzzy set discrete fuzzy set 𝐴; we have the 0.3, here
the 0.3 as the membership value and here also we have 0.5 plus side and minus side means
left side and right side both. Or in other words we can say the width of this column is what
is taken as 1 and mean of this has been taken as the generic variable value. Similarly this
logic will is applied to plot all the discreet fuzzy sets for a distance computation purpose.

So here also if you see for fuzzy set 𝐵 here, fuzzy set 𝐵 we have three elements again and
these three elements are this is the first element and this is the second element and third
element here. So, in the first element we see we have 2, 0.4, 2 is the generic variable value
and its corresponding membership value is 0.4. So, we can clearly see here that we have
at 2 we have 0.4 as the membership value. Similarly the second element we have 3 and it
is a corresponding membership value is 0.4.

So, 3 here is the generic variable value and 0.4 again means there is both 2 and 3 are the
same level. So, 3 here will have 0.4 as it is corresponding membership value. Similarly
the third element here is 4, 1. So, this 4 is the generic variable value and the corresponding

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membership value here is 1. So, we can see and with the same logic that each bar each
column here is of width 1 and keeping its mean add the corresponding generic variable
value. So, this means that we have this point as point 4.5 and this point we will have 3.5.
So here also the same logic is applied.

So now, we can clearly see that we have two fuzzy sets 𝐴 and 𝐵 and each fuzzy sets; I
mean both the fuzzy sets 𝐴 and 𝐵 are discreet and these two fuzzy sets have three elements
each. So now, our aim is to find the distance between these two fuzzy sets these two
discrete fuzzy sets, let us see how we can manage to find the distance 𝑑(𝐴, 𝐵).

So, for this as I have already explained how to find the 𝜇𝑑(𝐴,𝐵), means the membership
values of the resulting discrete fuzzy set here and then the corresponding generic variable
value which is nothing but delta. So, let,s now go to the solution and for computing the
fuzzy distance of the these two fuzzy these two discrete fuzzy sets 𝐴 and 𝐵.

(Refer Slide Time: 17:07)

So, we first need to find the 𝛿 and let us see how we can manage to find the 𝛿 in each and
every case. And as I mentioned here already that when we consider two fuzzy sets let us
say 𝐴 and 𝐵 and both these are the discreet fuzzy sets, we represent the fuzzy set here A
is equal to its generic variable value 𝑥𝐴 . So, 𝑥𝐴 is nothing but this is the generic variable
values taken from the fuzzy set 𝐴. Or in other words again we can say that 𝑥𝐴 or all those
generic variable values which are present in fuzzy set 𝐴 and 𝜇𝐴 (𝑥𝐴 ) here is the
corresponding membership values.

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So similarly if we apply the same logic, we represent here the fuzzy set 𝐵 and both these
fuzzy sets are discrete fuzzy set which is very clearly visible. So, these both these fuzzy
sets are discrete fuzzy sets. So, let us now understand this also that the resulting fuzzy set
here that which is nothing but the distance 𝑑(𝐴, 𝐵) will have two components one is the
generic variable value.

This is generic variable value which is here delta and it’s corresponding membership value.
So, we can say here this 𝜇𝑑(𝐴,𝐵) (𝛿) is a membership value. But if we if we have multiple
values or if this is a general function we can say its a membership function.

(Refer Slide Time: 19:45)

So, now let us move ahead and find the distance between two discreet fuzzy sets 𝐴 and 𝐵.
So, as we have already seen that we have two for fuzzy sets 𝐴 and 𝐵 like this; in a fuzzy
set, we have three elements one is 1,0.5. The second element we have 2,1 and the third
element here is 3,0.3. Similarly, we discrete fuzzy sets we have three elements the first
element 2,0.4 the second element is 3,0.4 the third element here is 4,1.

And as I mentioned we can find here a 𝛿 by this expression and 𝛿 is nothing but all possible
differences in the generic variable values of the discrete fuzzy sets 𝐴 and 𝐵. So, let us now
first find out the delta set, delta set is a set which has all the possible values of 𝛿 ∈ ℝ+
means positive real value. So, since we have two fuzzy sets 𝐴 and 𝐵, so let us take all the
generic variable values from 𝐴 and 𝐵 both so for computing the 𝛿. So, we see here that we

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have first generic variable here is 1 from 𝐴 fuzzy set and then the second fuzzy set 𝐵 we
have and the generic variable value here is we have generic variable value as 2.

(Refer Slide Time: 21:51)

So, in a way what we are doing here is, we are finding the difference in the corresponding
generic variable values and which are termed as the values of delta. Once again I would
like to mention that all the 𝛿 values will be positive values, because we are going to
compute the distance. So, let us first start from 0 as the difference, so let us first find out
what are the combinations of the generic variable values from A and B which are going to
give us the difference the 𝛿 = 0.

So we see that here we have for 0 we have here if we take 2, 2 and 3, 3 combination, 2, 2
means this is from 𝐴 and this is from 𝐵 this is from 𝐴 this is from 𝐵 fuzzy set. So, and we
have 2 here from 𝐴 fuzzy set and 2 here from 𝐵 fuzzy set. So, this is generic variable this
is my 𝑥𝐴 I can write here as 𝑥1 𝑥𝐴 1 and here I have 𝑥𝐵 1 and this value is going to give us
delta is equal to 0. So, we are interested here to first find the difference which is the least
and least difference is 0, because there cannot be any value of 𝛿 below 0. Because we are
interested in finding out the distance and distance will be only a real positive value. So,
the least difference is going to be 𝛿 = 0.

So, for 𝛿 = 0, we can have two combinations 2, 2 means this is 𝑥1𝐴 and here we have 𝑥1𝐵 .
Similarly here we have 𝑥𝐴 it is the second value and here we have 𝑥2𝐵 . So, or we can
otherwise we can represent these values, but here we are representing this way that the

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first value second value and both these the pair of these values are from set 𝐴 and set 𝐵
fuzzy set 𝐴 fuzzy set 𝐵.

So, there are two pairs of the generic variable values 2, 2 and 3, 3, these two are going to
give us this is the first pair and this is the second pair. These two pairs are going to give us
𝛿 = 0 and 𝛿 is nothing but the difference which is mentioned over here. So, this is mod of
𝑥𝐴 − 𝑥𝐵 . So, these two pairs are going to give us 𝛿 = 0 which is the least value of
difference. So, when we start the first and foremost step here is to start with finding the
delta is equal to 0. In many cases we may not have any such combinations for which we
have 𝛿 = 0.

So, if we do not find any such combination any such pair for which we get 𝛿 = 0, we can
simply move ahead for finding another or the incrementally the next least value of delta.
For example, if we are not getting here any pair for which we are getting 𝛿 = 0, we can
immediately move ahead and try for getting 𝛿 = 1.

So, similarly here also if we are not getting any pair any such pair for which we have 𝛿 =
1, then we can move ahead and we will try to find any such combinations for which we
have 𝛿 = 2. And this way we can keep on increasing our steps or moving forward to find
the next least values. But if we get here any pair like in this example we have 2,2, 3, 3. So,
two pairs of the generic variable values and these two pairs are resulting 𝛿 = 0. So now,
let us see what we should do and how should we account this value.

(Refer Slide Time: 27:42)

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So, for 𝛿 = 0 we have two pairs. So, that is how we are writing 𝛿 = 0 you can see here.
So, this is our 𝛿 = 0; we write here we have a table and this table will finally help us in
getting the resultant distance of the two fuzzy two discreet fuzzy sets. So, we have the 𝛿 =
0 we will write in delta column 0 and since we are getting two pairs we will write those
pairs here.

So, since we have got two pairs one pair is 2 2. So, we have written 2 here as 𝑥𝐴 and the 2
here also as 𝑥𝐵 . Why 𝑥𝐴 and why 𝑥𝐵 ? Because this 2 the second column 2 is from discreet
fuzzy set 𝐴 and the this 2 this which is in the third column is corresponding to the 𝐵 fuzzy
set, so we have written here 2,2.

Similarly, now we have another pair which is 3,3 for which we are getting 0. So, we will
write here 3,3 as 𝑥𝐴 and 𝑥𝐵 you can see here. So, let us now understand this as to how we
are getting these values of 𝑥𝐴 𝑥𝐵 for which the 𝛿 = 0. Now another important point here
is to list the corresponding membership values.

So, the corresponding membership value here 𝜇𝐴 (𝑥𝐴 ), we are getting here as 1 and 𝜇𝐵 (𝑥𝐵 )
we are getting 0.4. So, which all these values are coming from the fuzzy sets the discreet
fuzzy sets that are given to us. So, we need not worry about anything we are just we need
to just take the values from the given fuzzy sets and include in this table.

So, we have 2,2 as the generic variable values from the first pair and then corresponding
membership values. Now what next? Next is that, let us find the minimum of these two
membership values. So, minimum of these two membership values means here. So, we
need to find the minimum of membership values which is here. So, minimum of 1.0 and
0.4 so we have 0.4.

So, next is the second pair here 3, 3 we have corresponding membership values as 0.3 and
0.4 we can clearly see here its mentioned its encircled here. So, we see here that we have
0.3, 0.4 or the membership values. Now when we take the minimum of these two values
we are going to get what? We are going to get 0.3. So, this are the minimum of these two
minimum, so this is what we need to do.

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(Refer Slide Time: 31:44)

And then the next step here is, for 0 as the 𝛿, so we need to find the maximum of these
two. So, here please understand that we have two pairs of the generic variable values from
fuzzy set 𝐴 and fuzzy set 𝐵 and these two pairs are for 𝛿 = 0. So, when we have these 0.4
and 0.3 0.4 and 0.3 as the min of these two we take the max of these two and we find 0.4.
So, by now we have understood as to how we can compute the delta and this way we can
go ahead and we can compute delta is equal to 1, 2, 3 if these deltas exist.

(Refer Slide Time: 32:57)

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So, since the time is up. And, I will continue the rest of the discussion with respect to this
example in the next class.

Thank you very much.

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 Fuzzy Sets, Logic and Systems and Applications
Prof. Nishchal K. Verma
Department of Electrical Engineering
Indian Institute of Technology, Kanpur

Lecture - 18
Distance between Fuzzy Sets

So, welcome to lecture number 18 of Fuzzy Sets, Logic and Systems and Applications.
So, this lecture we will discuss the Distance between two Fuzzy Sets which is in
continuation to over previous lecture, lecture number 17, where we have discussed how to
find the distance between two fuzzy sets 𝐴 and 𝐵.

(Refer Slide Time: 00:45)

So, if we look at the fuzzy sets that we have taken these fuzzy sets were 2 discrete fuzzy
sets, 𝐴 and 𝐵 and this is how we had plotted these 2 discrete fuzzy sets 𝐴 and 𝐵.

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(Refer Slide Time: 01:08)

And in the previous lecture we also saw as to how we can tabulate the values of
𝛿, 𝑥𝐴 , 𝑥𝐵 , 𝜇𝐴 (𝑥𝐴 ), 𝜇𝐵 (𝑥𝐵 ) and then we took minimum of the 2, that means, minimum of
𝜇𝐴 (𝑥𝐴 ) and 𝜇𝐵 (𝑥𝐵 ). So, when we have found the minimum of 𝜇𝐴 (𝑥𝐴 ) and 𝜇𝐵 (𝑥𝐵 ), we
with respect to the previous example we found 2 values of minimum of 𝜇𝐴 (𝑥𝐴 ), 𝜇𝐵 (𝑥𝐵 )
which were 0.4 and 0.3. So, in the last column of this table, we had to take the maximum
of these values.

So, here we have 2 values 0.4, 0.3, when we take the maximum of the these 2 values we
get 0.4. So, this 0.4 is with respect to the delta value which is a 0 which was taken 0 and
correspondingly we have taken the 𝑥𝐴 , 𝑥𝐵 values. Similarly, now we will move ahead and
we will try to find if we have any such combinations which are leading to 𝛿 = 1.

It may be possible that we may not find any such pairs for which 𝛿 = 1 and if this is the
case then we will move ahead and we will try to find the pairs for which we get 𝛿 = 2.
And this way we will keep moving ahead till the highest value for which the 𝑥𝐴 , 𝑥𝐵 pairs
we are getting the highest value of 𝛿.

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(Refer Slide Time: 03:47)

All right. So, let us now quickly try for 𝛿 = 1 which is here. So, if we start finding the 𝑥𝐴 ,
𝑥𝐵 values from these 2 discrete fuzzy sets 𝐴 and 𝐵 which are given, so we see that we have
4 such combinations. We have 1 from fuzzy set 𝐴 which is here and 2 from fuzzy set 𝐵
and then we have another element another pair 2, 3 and this 2 is from fuzzy set 𝐴 and here
we have 3 which is from fuzzy set 𝐵.

Similarly, we have 3, 2 as the third element 3 is from discrete fuzzy set 𝐴 and 2 is from
the discrete fuzzy set B and the 4th pair here is 3, 4. So, all these 4 pairs are resulting 𝛿 =
1, no other pair is going to result us 𝛿 = 1. So, these pairs should be drawn from discrete
fuzzy set 𝐴 and 𝐵 which is given in the example.

So, let us now use these pairs for 𝛿 = 1 and tabulate the values of 𝑥𝐴 , 𝑥𝐵 ,𝜇𝐴 (𝑥𝐴 ), 𝜇𝐵 (𝑥𝐵 )
and then we take minimum of all these findings and then finally, we take max and let us
see how does it go. So, for the first pair 1, 2 which you see here it is encircled here and we
write here 𝛿 = 1.

So, this 𝛿 = 1 which is in the first column, so, please look at the first column here first
column of the table where we have entered in delta column which is first column of the
table 1 and then let us tabulate all the corresponding entries with respect to 𝛿 = 1.

So, first entry here will be for the pair 1, 2. So, when we take 1, 2 we write here as this is
𝑥1𝐴 and then we have 2 as 𝑥1𝐵 . This signifies that is the first generic variable of the discrete

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fuzzy set A and 𝑥1𝐵 signifies the generic variable value which is from fuzzy set 𝐵.
Similarly, all other values can be understood.

Now, here enter in the second column which is 𝑥𝐴 , we enter 1 and for 𝑥𝐵 we enter 2 here
and then let us find from the 𝐴 discrete fuzzy set that is given we have 0.5 with respect to
the generic variable value 𝑥𝐴 here. So, we are getting 0.5 and then we enter here 𝜇𝐵 (𝑥𝐵 )
which is 0.4. Now, in the 6th column here we take minimum of all these entries, minimum
of all these entries and this gives us 0.4. Now let us take other pairs for which pair getting
𝛿 = 1.

(Refer Slide Time: 08:05)

So, when we take the second pair which is 2, 3, we are getting we are first of all we are
entering these values 2, 3 𝑥𝐴 , 𝑥𝐵 and then we take the 𝜇𝐴 (𝑥𝐴 ), 𝜇𝐵 (𝑥𝐵 ) and we take the
minimum of these two which is coming out to be 0.4 and that is how we get here 0.4. Next,
we take the third pair and third pair for which 𝛿 = 1.

So, we have 3, 2 we have written here 3 and then 2 and then we have their corresponding
membership values which we are getting from the discrete fuzzy sets A and B. Now, when
we take minimum of these 2 we are obviously, getting 0.3.

So, that is how this row is enter and then we have a 4th element; 3, 4, 4th pair for which
we are getting 𝛿 = 1. So, here also we enter 3, 4 and then their corresponding membership
values and when we take min of these min of these 2 entries, we are going to get 0.3. Now,

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when we have found all 4 minimum of the corresponding membership values which are
these corresponding to 𝛿 = 1, then we take the max of all these.

(Refer Slide Time: 09:57)

So, here we take max and then this max is going to give us 0.4. So, now, this way we are
able to find all these values of 𝑥𝐴 , 𝑥𝐵 corresponding membership values and min of all
these and then finally, max we are getting 0.4. So, that is how for 𝛿 = 1, we have entered
all the corresponding values. Now, we will check for 𝛿 = 2.

So, the 𝛿 = 2 let us see how many pairs how many such pairs we are getting for which we
are getting 𝛿 = 2. So, here if we see both the discrete fuzzy set, we are getting 1, 3 and
3, 4. So, these 2 pairs are giving us 𝛿 = 2. So, now, let us on the same lines enter these
values. So, when we take 1, 3 we are getting 0.5, 0.4 as the their corresponding
membership values, when we take min we are getting 0.4 and similarly for the second pair
which is 2, 4.

So, when we take 2, 4, we are getting the corresponding membership values 1 and 1. So,
this way our all entries with respect to the 𝛿 = 2 are complete and then when we take max
we are getting one of course, this is the max of these 2 or 1.

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(Refer Slide Time: 12:02)

Next let us try to find all the pairs for which 𝛿 = 3. So, this 𝛿 = 3, we get only one such
pair and when we use these values 1, 4 as the pair elements. So, 1, 4 for 𝛿 = 3 and then
the corresponding membership values are entered over here and when we take minimum
of these we are getting 0.5 and since we have only one entry here, so, maximum of this is
going to be 0.5. So, this will be 0.5.

So, now, we understood as to how we fill this table for which is going to help us in finding
or making the fuzzy set which is giving us the distance between 2 discrete fuzzy sets 𝐴
and 𝐵.

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(Refer Slide Time: 13:13)

So, now, we can use this delta, when this table is complete. We use this table with the pick
all the delta values and we pick all the 𝜇𝑑(𝐴,𝐵) values 𝜇𝑑(𝐴,𝐵) (𝛿) values. So, when we take
these 2 entries the 𝛿 entries and corresponding membership values, so these 2 is the first
column and the last column. So, we see that when we take 0 as 𝛿, so, 𝛿 = 0 is is going to
give us 𝜇𝑑(𝐴,𝐵) (𝛿) = 0.4.

Similarly, if we take 𝛿 = 0 we are getting 𝜇𝑑(𝐴,𝐵) (𝛿) = 0.4. Similarly for 2, 𝛿 = 2 𝛿 = 3

we get the its corresponding 𝜇𝑑(𝐴,𝐵) (𝛿). And when we use this these entries because one
is the generic variable value which is delta which is nothing but the difference as we have
already discussed was 𝛿 is the difference between the corresponding generic variable
values of the 2 fuzzy sets 𝐴 and 𝐵 and then the last column gives us the it is corresponding
membership values.

So, we can make use of these 2 entries and we can construct here a fuzzy set which is
mentioned by 𝑑(𝐴, 𝐵) = 0.4/0 means at generic variable value 0 which is the difference
that is a 𝛿 = 0 this is 𝛿 = 0 and for this 𝛿 = 0, we are getting here 0.4 as its corresponding
membership value. Similarly here, we are getting another element as 0.4/1.

So, we see that 𝛿 = 1, we are getting 0.4 as its corresponding membership value. And then
for 𝛿 = 2, we are getting 1 and similarly for 𝛿 = 3 we are getting its corresponding
membership 0.5. So, this is how we are making use of this table and we are constructing

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fuzzy set which is nothing but the distance between the fuzzy sets A and B which is
represented by 𝑑(𝐴, 𝐵) which is this.

Now, let us plot this fuzzy set and if we make use of these values here the its elements we
can very easily plot or find the fuzzy set which is nothing but the distance between the 2
fuzzy sets. So, here we can clearly see that this is delta this x axis is here is delta and the
y axis is the ordinate here is the membership grade that is mu of delta.

So, at 0 we have its corresponding membership as 0.4 which is here, which is here and at
1 also we have 0.4 which is here this level and then at 2 we have 1 which is here and at 3
we have 0.5, see here 0.5 there is 0.5. So, this way we are able to find the fuzzy distance
between 2 discrete fuzzy sets.

(Refer Slide Time: 17:51)

So, here we have just to help you we have included the MATLAB code for finding the
distance between to discrete fuzzy sets A and B. So, you can make use of this MATLAB
code to find the distance between any 2 discrete fuzzy sets.

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(Refer Slide Time: 18:12)

Now, let us find the distance between 𝐵 and 𝐴 and this way we will try to verify whether
distance between 2 fuzzy sets 𝐴 and 𝐵 whether this is equal to the distance between 2
fuzzy sets 𝐵 and 𝐴. So, this is very interesting example which will show us that these 2
distances these 2 the distance either we find the distance between 𝐴 and 𝐵 are 𝐵 and 𝐴 and
the where these 𝐴 and 𝐵 are two fuzzy sets these two are always equal. So, let us verify
this. So, we have already we have fuzzy set 𝐴 and 𝐵. So, let’s now find the distance
between 𝐵 and 𝐴.

(Refer Slide Time: 19:11)

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So, let us now for the same example that we have already taken let us find the distance
between fuzzy set 𝐵 and 𝐴. Please note that here we will take 𝐵 first and then 𝐴 and let us
now enter all these values in the table which we have already discussed as to how we can
fill up these values. So, here also we’ll have because we are taking the same sets same
discrete fuzzy sets 𝐴 and 𝐵. So, obviously, here we are going to have the delta the
difference as 0, 1, 2 and 3.

(Refer Slide Time: 20:02)

So, for all these delta values, let us now enter these values in this table. So, for 0 for 𝛿 =
0, again we have 2 pairs here we have 2, 2 and 3, 3 for which we are going to get delta is
equal to 0. So, let us now enter this value, this value of 𝛿 = 0 and for this 0 for this value
of 𝛿 = 0, we are now entering corresponding values of 𝑥𝐵 and 𝑥𝐴 .

So, when we do that we are going to get 2, 2 ,0.4, 1 and then if we take min of these 2 we
are going to get 0.4. Similarly, for 3, 3 we are going to get 0.3. Now, as we have already
seen in the previous example 𝜇𝑑(𝐵,𝐴) (𝛿) we are going to take the max of these two values.
So, when we take max of these two values we are going to get 0.4. Now, here we have
completed all the entries for 𝛿 = 0. Now, let us do the same for 𝛿 = 1.

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(Refer Slide Time: 21:36)

So, when we take 𝛿 = 1, we have 4 entries as we have seen in the previous example also,
here also we have 4 entries 4 pairs. So, let us now put all these in the table. So, here we
will write 𝛿 = 1 and the first pair we write 2 1 and then 0.4, 0.5 and then we take minimum
of these two, we are going to get 0.4 and this way we are able to complete the entries for
the first pair for which 𝛿 = 1. Now the next pair we have 3, 2.

Then the next pair the third pair we have 2, 3 and then the last pair which is the 4th a pair
we have 4, 3 and then we at this is stage we know how are we getting these values these
corresponding values of membership and then we have taken the min of the corresponding
membership values of A and B. And then when we take max of these values we are going
to get here 0.4. So, this is 0.4.

So, if we look at this, if you look at this we have completed all the entries corresponding
to 𝛿 = 1. Now, let us move ahead and fill all the entries corresponding to 𝛿 = 2.

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(Refer Slide Time: 23:45)

So, when we apply the same logic, we have two pairs for here for 𝛿 = 2; one is the 3, 1
and the second one is the 4, 2. So, here we will write all the entries in the table and when
we again take the maximum of this, we are going to get 1 which is mentioned over here.
Now, next when we move to 𝛿 = 3, so, we see that we have only one pair only one pair
for which we are getting 𝛿 = 3.

(Refer Slide Time: 24:27)

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So, we fill the table with 𝛿 = 3 and its corresponding values. So, here since we have only
one value if we take max we get only 0.5. Now, please note that we do not have any pair
for which we get 𝛿 = 4.

So, we will stop here. And then again as we have done in the previous example, we take
the first column which is delta column and the second column which is the𝜇𝑑(𝐵,𝐴) (𝛿). So,
here we will take these 2 columns, we will make use of these 2 entries and we will write
the fuzzy set which is which will be representing the distance between 2 fuzzy sets B and
A.

(Refer Slide Time: 25:50)

So, when we make use of these entries here we are going to write 𝑑(𝐵, 𝐴) this is nothing
but the distance between the fuzzy set discrete fuzzy set 𝐴, 𝐵 and 𝐴. In earlier case we had
the distance between the fuzzy set A and B. So, the difference here is that we have change
the sequence. So, here we are taking B first the fuzzy set B first and then A. So, 𝑑(𝐵, 𝐴)
represents the distance between fuzzy set B and A and which is nothing but the 0.4/1.

So, 0.4 is the 𝜇 corresponding to 𝛿 and this 𝛿 = 0, this delta here is 0. So, 4 delta is equal
to 0 which is which we can see in the very first column we have all the delta values
mentioned. We have delta is equal to 0, 1, 2, 3 and then we when we take the corresponding
membership values which is which are mentioned in the last column, so, we have for 𝛿 =
1, we are getting 0.4.

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So, this 0.4 is from here and then for 𝛿 = 1 we are getting the corresponding membership
value from here and then similarly for 𝛿 = 2 we are getting the corresponding membership
value here, similarly for 𝛿 = 3 we are getting the corresponding membership. So, this way
we have got a fuzzy set which is nothing, but the distance between discrete fuzzy set B
and discrete fuzzy set A.

(Refer Slide Time: 28:18)

And when we plot these fuzzy set we are going to get here a fuzzy set which is exactly
same as the fuzzy set as the distance between fuzzy set A and B we have got in the previous
example. So, we can clearly see here that whether we calculate the distance between A and
B fuzzy sets or B and A both are same. So, we can clearly see here if you plot or even the
values also we see that 𝑑(𝐴, 𝐵) is same as 𝑑(𝐵, 𝐴).

So, this way, we can say that we have verified that 𝑑(𝐴, 𝐵) = 𝑑(𝐵, 𝐴). So, this means that
the distance between fuzzy sets A and B is exactly same as the distance between fuzzy sets
B and A. So, by now we have understood very clearly as to how we can find the distance
between any two fuzzy sets. So, if we can manage to find these entries here first the delta
value, so difference.

So, we will first point all the possible values of the delta from the given fuzzy set like we
have seen in the, this example and then we find all such pairs for which we are getting the
different delta values. So, these pairs will be used to find their corresponding membership
values and this these values then we will fill in the table as 𝑥𝐵 , 𝑥𝐴 ,𝜇𝐵 (𝑥𝐵 ), 𝜇𝐴 (𝑥𝐴 ).

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And then we take the minimum of these values these membership values and then when
we have done for all such pairs all the minimums are all the minimum values are found
out then we take max of these values to get the final value which is 𝜇𝑑(𝐵,𝐴) (𝛿). So, this
way we can manage to get all the entries all the values of delta and its corresponding
membership values and these 2 values delta and mu of delta will help us in constructing
the fuzzy set the discrete fuzzy set here in this case because we have taken discrete fuzzy
set only.

And so, the distance between the two discrete fuzzy set is of course, of obviously, going
to return us the discrete fuzzy set as the distance between the two discrete fuzzy sets.

(Refer Slide Time: 31:37)

So, with this I would like to stop here and in the next lecture I will try to cover a few more
examples on distance between fuzzy sets.

Thank you.

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 Fuzzy Sets, Logic and Systems and Applications
Prof. Nishchal K. Verma
Department of Electrical Engineering
Indian Institute of Technology, Kanpur

Lecture - 19
Distance between fuzzy sets

Welcome to lecture number 19 of Fuzzy Sets, Logic and Systems and Applications. And
here in this lecture we will have examples on a distance between two fuzzy sets. And this
lecture is in continuation to lecture number 18 where we have already discussed the
formulation for finding the membership values, corresponding to the generic variable
value that was lambda for the distance between two fuzzy sets.

(Refer Slide Time: 00:48)

So, here before I move to the example, I would quickly go through the formulation. So,
here as I mentioned earlier also that when we find the distance between two fuzzy sets let
us say A and B. So, this distance is the fuzzy set this distance value is fuzzy and this is
represented by a fuzzy set. So, this fuzzy set of course, will have the generic variable value
delta as well as it is membership values. So, and of course here the delta is the difference
between corresponding generic variable values up to fuzzy sets for a generic variable.

So, as it is written here that the distance between two fuzzy sets A and B that is represented
by 𝑑(𝐴, 𝐵), which is equal to the summation of all the elements containing it’s membership
values over the universe of discourse. So, summation is used here for the discrete fuzzy

327
set case and on the same lines we have another formulation, where we used integration
sign integration symbol in place of a summation which is here for the continuous case.

And the membership value can be calculated by this formulation and these membership
values are nothing but for respective delta values. So, since we have already explained
enough in my previous lecture. So, let us now directly go to the example.

(Refer Slide Time: 03:30)

In this example we have three discrete sets A B and C and as I mentioned these are all
discrete fuzzy sets. So, since these are the discrete fuzzy sets, we will apply the summation
sign and we can you know accordingly minus to get all the values of the table that I already
mentioned in the last lecture. So, we have here the fuzzy A this fuzzy which is given here
is represented by a fuzzy set plot.

So, we have here a fuzzy set A and then we have fuzzy set B which is represented by the
fuzzy set as shown here and then we have the third fuzzy set which is here. So, please note
that these the generic variable for these fuzzy sets are nothing but the same the 𝑋. So,
generic variable is 𝑋 here in all the cases.

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(Refer Slide Time: 04:54)

So, here let us now start with finding the distance between two discrete fuzzy sets, here
also we have 𝑋 here also we have 𝑋 as generic variable in both the cases.

(Refer Slide Time: 05:13)

And if we see here as I mentioned in the last lecture that we have a table and this table will
have the delta value which is here. And then we have 𝑥𝐴 so 𝑥𝐴 is nothing but the generic
variable value which is present in the discrete set A similarly here is the generic variable
value present in fuzzy set B. And here 𝜇 𝐴 𝑥𝐴 and 𝜇 𝐵 𝑥𝐵 both are the respective membership
values and then we take min of these two 𝜇 𝐴 𝑥𝐴 and 𝜇 𝐵 𝑥𝐵 .

329
So, that is how we fill these entries. So, let us now quickly go through and please note here
that we have delta which is here this is always positive real number. So, that has to be
noted because we are using here the mod of x A minus x B, so obviously delta is going to
be the real positive value. So, let us quickly go through completing the this exercise of
calculating the difference between two the generic variable value or in other words all the
possible differences right from 0 and above.

So, when we find all combinations we get these values of 𝛿. So, this 𝛿 will have 1 0, 1, 2,
1, 0, 3, 2, 1, 4, 3 ,2. So, like that when we rearrange this we will get 𝛿(𝐴) set which is 0,
1, 2, 3, 4. So, let us now start with delta is equal to 0.

(Refer Slide Time: 07:25)

And here we will see that we have these two entries these two pairs for which we get 𝛿 =
0. So, it is like this and these values are filled over here see here these values will go here
and here so 𝑥𝐴 , 𝑥𝐵 and of course this is for 𝛿 = 0. Similarly, we will find from the discrete
fuzzy set given the respective membership values and then when we take min we are going
to get 0.4. Now let us enter the values for the second pair.

330
(Refer Slide Time: 08:26)

So, second pair is 3, 3 and we can see that these values are entered and the minimum we
are getting here is 0.3. Now we have to take max of these two. So, this is here is used to
compute the max, max is going to be 0.4 max of these two. So, as I already explained in
my last lecture, lecture number 18 and 17 both that we first arrange all the values of lambda
and then we find all the entries fill in the table and then we take the max values in the last
column.

And then we use first column and the last column for constructing the fuzzy set and this
fuzzy set is nothing but the distance between these two discrete fuzzy sets here. So now,
let us move ahead and take 𝛿 = 1.

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(Refer Slide Time: 09:41)

So when we take 𝛿 = 1. So, we see that these are the combinations these are the pairs
these four pairs are there and these four pairs will be entered in the table. You can see here
I am quickly moving ahead and since you already know as to how we fill these entries, so
we can quickly fill these entries here. Now for 𝛿 = 1. So, we get the min values as listed
here and when we take the max of these we are going to get 0.9, so this is written over
here.

(Refer Slide Time: 10:32)

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Similarly, let us go and have all these entries for 𝛿 = 2. So, when we take 𝛿 = 2 we get
these three pairs and as I have already explained these pairs are coming from both the
fuzzy sets that are given. The both the discrete fuzzy sets A and B, now let us fill these
entries also here in the table.

So, when we fill these entries we are going to get here all the entries filled and we see here
that we are getting the max of all these as 1. So, this way we are getting 𝜇𝑑(𝐴,𝐵) (𝛿) = 1.
So now, let us move ahead and find all the entries for 𝛿 = 3.

(Refer Slide Time: 11:40)

So, when we move ahead we see that we have two pairs which are resulting 𝛿 = 3. So, let
us fill these entries as well. So, when we fill these entries we find that the min is coming
as 0.2 and then for the second pair when we see we are getting 0.4 as the min of these two.
And as we have done in previous entries we take the max here finally and when we take
max we are going to get 0.4, so that is how we have entered here 0.4. Now for the last
value that is a 𝛿 = 4

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(Refer Slide Time: 12:43)

So, when we take 𝛿 = 4 we are getting here a 5 and 1 only one pair for which we are
getting the difference that is delta 4 and this we are entering here and of course since we
have only one row here. So, max is going to be the same that is 0.2. So, when we do that
we quickly get all the delta values which are mentioned in the first column and the
corresponding membership values that are mentioned in the last column.

(Refer Slide Time: 13:31)

So, now may make use of these two columns like for 𝛿 = 0, the corresponding
membership value is 0.4 and then for 𝛿 = 1 corresponding membership value is 0.9. And

334
then for 𝛿 = 2 corresponding membership value is 1 here and then for 𝛿 = 3 the
corresponding membership value is 4.

Similarly, for 𝛿 = 4 we are getting corresponding membership value 0.2. Now let us make
use of this to construct the fuzzy set which is the resultant fuzzy set which is nothing but
the distance. So, here we have the distance between a two fuzzy sets A and B.

So, since we have written here we have made the use of the entries of column number one
and the first column and last column and then we are getting here all these entries. And
when we construct a fuzzy set this will look like this, this is nothing but the distance
between two fuzzy sets A and B and this way we are able to get the distance between two
fuzzy sets. Here in this case we have two discrete fuzzy sets and that is how we are able to
calculate the distance and which is again coming out to be a discrete fuzzy set.

(Refer Slide Time: 15:09)

Now, we have a MATLAB code here. So, if you are interested you can make use of these
MATALB codes, you can generate you can compute the distance between two fuzzy sets
A and B and then you can generate the resultant fuzzy set. So, this is the same code that
was used to generate all these figures plots and entries as well.

335
(Refer Slide Time: 15:42)

So, let us now take another example here, where we have to compute the distance between
two discrete fuzzy sets A and C. In earlier case we computed the distance between fuzzy
A and fuzzy B here we are computing the distance between fuzzy set A and fuzzy set C.

So, since these sets are already given A has been given like this and C fuzzy set has been
given like this, both of these fuzzy sets are discrete fuzzy sets again I am writing this
discrete fuzzy sets. So of course, as we have seen in the previous examples that the
resultant fuzzy set the distance is also going to be the discrete fuzzy set. So, here also we
see that we have the generic variable x in both the cases.

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(Refer Slide Time: 16:49)

And when we take these two fuzzy sets again on the same lines as we have done in the
previous example, if we try to find the delta values. So, as I had already explained and I
already suggested that we should start with delta is equal to 0.

(Refer Slide Time: 17:05)

So, for 𝛿 = 0 here we get these pairs from 𝑥𝐴 and 𝑥𝐶 , the generic variables of both the
fuzzy sets and in this case 𝛿 = 0 we are getting only one pair that is 5, 5. So, let us quickly
enter this value in the table. So, as we have already seen in the previous examples we have
a delta column which is the first column. So, let us enter the delta value and this is nothing

337
but 0. So, we enter here in the first row we are entering here 𝛿 = 0. Now let us quickly
take these values of 𝑥𝐴 and 𝑥𝐶 which are like this 5 and 5 for which delta is coming out to
be 0.

So, we can fill these entries and now we will go ahead and find the corresponding
membership values. So, let us now see what are the corresponding membership values. So,
here if you see 5 is from here so corresponding membership value here is 1 and then in the
fuzzy set C we see that we have corresponding to 5 we have membership value here as
0.3. So, when we take minimum of these two we are getting 0.3 and then similarly see here
we have a 𝛿 = 0 we have only one row.

So, the max of 0.3 since we have only one element so we can straight away right 0.3. So,
this way we are able to enter the first row values for 𝛿 = 0.

(Refer Slide Time: 19:00)

Now, let us quickly go ahead and take delta is equal to 1. So, we have delta is equal to 1
here and let us find all these pairs which are responsible for creating delta is equal to 1.
So, we have two pairs here first pair and the second pair first pair is 4, 5, second pair is 5,
6. So, let us now enter these values. So, first pair here is a for delta is equal to 1, we have
𝑥𝐴 4 and 𝑥𝐶 5 and here we have 5 the other pair values 6 like this and then we’ll find the
corresponding membership values. So, corresponding membership value here will be 0.9
corresponding to 4 so 0.9 and then corresponding to 5 we have 0.3.

338
Similarly for 5, 6 we have here corresponding to 5 we have membership value 1 and then
we have here corresponding to 6 we have 0.8. So now, here if you take minimum of these
two we are getting 0.3 and minimum of these two we are getting 0.8. When we take
maximum of these two we are getting 0.8. So, this is how we are able to enter the values
corresponding to 𝛿 = 1.

(Refer Slide Time: 21:03)

Now, let us enter the values of 𝛿 = 2. So, when we take 𝛿 = 2. So, let us see how many
pairs we get which are giving us the 𝛿 = 2 the difference 2. So, we get three pairs so first
pair, second pair and third pair all these pairs are responsible for generating the difference
of two.

339
(Refer Slide Time: 21:33)

So, let us now quickly enter these values here and we know as to how we enter these values
and then when we take maximum of these we are going to get 1. So, this is how we are
getting the corresponding to 𝛿 = 2, we are getting it’s corresponding membership value
as 1.

(Refer Slide Time: 22:00)

So, now let us find the number of pairs for 𝛿 = 3. So, here we see that there are 4 pairs
which are coming from fuzzy set A and C, as 𝑥𝐴 and 𝑥𝐶 values and the generic variable

340
values. So, these four pairs let us enter quickly here in the table. So, here we have entered
all these values and then when we take maximum we are getting here as 0.9.

(Refer Slide Time: 22:40)

Similarly, now let us go ahead and find all those pairs which are responsible for generating
the difference that is 𝛿 = 4. So, 𝛿 = 4 let us see how many pairs are responsible for
generating. So, there are three pairs so first pair here 2, 6 and then 3, 7 and 4, 8 these three
pairs are responsible for generating 𝛿 = 4. So, 𝛿 = 4 as I already mentioned is nothing
but the difference between the 𝑥𝐴 and 𝑥𝐶 and these values are basically the generic variable
values and these values are coming from the fuzzy sets that are given to us.

Here in this case these fuzzy sets are A and C and these both the fuzzy sets are discrete
fuzzy sets. So, this way we are able to quickly manage to enter all the values in the table
corresponding to 𝛿 = 4 and the maximum value that we are getting here is the max of all
these entries we are getting here as 0.7.

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(Refer Slide Time: 23:56)

Now, let us do the same exercise for 𝛿 = 5 and when we see we are getting two pairs 2, 7
and 3, 8 and both are responsible for 𝛿 = 5. So, let us enter these values here and here we
are getting 0.7 as the maximum of 0.7 and 0.3 so this way this is also done.

(Refer Slide Time: 24:34)

And now let us go ahead and try to find all the entries corresponding to 𝛿 = 6. So, when
we want to have the difference of 6, let’s see how many pairs that we are having which are
responsible for creating the difference of 6. So, this is pair of 2 and 8 which is responsible

342
for creating 𝛿 = 6 that is the difference. And of course as I already mentioned that this is
coming from the A set and this is coming from the 8 is coming from C fuzzy set.

(Refer Slide Time: 25:24)

So, are in the other words we can say this is the generic variable value from A set and this
is the generic variable value from C set. So, here we have only one pair and this pair is
responsible for creating the distance of 6, now let us enter this quickly. So, we will right
here at 6 and then we will right here two we will right here 8 and then let us now look for
the corresponding membership values, we will directly get these values from the fuzzy set
that has been given to us.

So, corresponding to 2 we are getting it’s membership value 0.7 this is coming from here
from fuzzy set A. And now corresponding to 8 which is here now let us quickly find and
enter it’s membership value which is nothing but 0.5. When we take min of these we have
getting 0.5 and since we have only one row, so when it comes to taking max so max will
remain the same.

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(Refer Slide Time: 26:38)

So, this way we got all the entries that are needed to compute the distance between two
fuzzy sets. And now if we see here at the table we have the first row as the delta row that
is all the possible values or delta are there. And I would like to mention one thing here that
see delta is equal to 6 we have computed we have found all the values we have entries in
the table. So, can we have any entry for 𝛿 = 7 also. So, the answer is no because we cannot
find any pair here which are responsible for creating the difference of that is 𝛿 = 7. So, no
pair is responsible for creating 𝛿 = 7.

So, that is how or that is why we are not including the 𝛿 = 7, because no pair here is
present which can create a difference 𝛿 = 7. So, that is why we stop here and in the table
we are having only entries of 𝛿 = 6.

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(Refer Slide Time: 28:04)

So, now we are using these delta values and it is corresponding membership values which
are in the last column and we see that we have corresponding to 𝛿 = 0. So, for 0 we are
getting 0.3 as the membership value. So, we have written over here and then we have 1 for
delta value 1 we have 0.8. So, this is also written and similarly for all the values of delta
up to 6 we have written all the elements all the membership values and this 𝑑(𝐴, 𝐶) this
𝑑(𝐴, 𝐶), means the distance between two fuzzy sets A and C we are getting A fuzzy set
which is nothing but the distance between two discrete fuzzy sets A and B.

So, this way we are able to find the distance in between two fuzzy sets and it is needless
to say that here both the fuzzy sets are discrete and hence they are getting the distance also
a discrete fuzzy set and this can be plotted here like this. So, I in the last lecture I mentioned
as to how we can plot the resultant fuzzy set which is here you can see.

345
(Refer Slide Time: 29:36)

So, here also for the distance between two fuzzy sets A and B we have the MATALB code
and you can use these MATALB code for computing the distance between two fuzzy sets
and this MATALB code can also be used for plotting the resultant fuzzy sets.

(Refer Slide Time: 30:04)

Now, let us find the distance between fuzzy sets B and C. Where we have B as this discrete
fuzzy set which is given and C here is another fuzzy set which is given by this expression.
So, we have again all these two fuzzy sets B and C both are on the same generic variables.

346
(Refer Slide Time: 30:42)

So, let us quickly now go ahead and find out the distance between these two fuzzy sets.
So, we remember that we first need to find a set of delta values and this delta values will
start from 0. So, we will quickly see that first we will try to find the difference of 0 and for
this difference of 0, we try to use the generic variable values of both the sets.

So, we see that we do not find any pair any such pair which creates 𝛿 = 0. Similarly, we
do not find any such pair which create the difference of one that is 𝛿 = 1. So, here a delta
is equal to 0 and 1 we are not including in the set and then when we move ahead we find
that for 𝛿 = 2, means the difference of two can easily be created and from the entries that
are given in the fuzzy sets B and C. So, 2, 3, 4, 5, 6, 7 all these delta values are possible,
we will see as to how we are getting these differences.

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(Refer Slide Time: 32:10)

So, let us take first the 𝛿 = 2. So, 𝛿 = 2 we have one pair these entries we have. So, 3 and
5 so this is one pair and this pair is responsible for creating the difference of 2 that is 𝛿 =
2. So, let us quickly enter all these entries in the table and that is how we get the max here
also 0.3 in the last column. And now let us quickly go ahead and find all these entries for
𝛿 = 3.

(Refer Slide Time: 32:49)

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So, for 𝛿 = 3 we see all these entries and you can follow the same procedure for finding
all these entries. And you know when you take max of these min values here then you get
0.8 for 𝛿 = 3 and let us now see what we are getting for delta is to 4.

(Refer Slide Time: 33:17)

So, for 𝛿 = 4 we are getting three pairs, here first pair second pair and then we have third
pair. So, this way we get three pairs and all these three pairs we are entering in three rows
and then when we take the max of these min entries. So, we are getting 1.0 means 1. So,
we quickly get the corresponding to 4 means 𝛿 = 4 we are getting it’s membership value
1.

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(Refer Slide Time: 33:57)

Now, let us take 𝛿 = 5 and again we have a three pairs of 𝑥𝐵 and 𝑥𝐶 . So, let us now enter
these values on the same manners, you see here so we get for 𝛿 = 5, we get it is
membership value as 0.5.

(Refer Slide Time: 34:24)

And when we take 𝛿 = 6 we get all these entries filled and since we have two pairs only
which are responsible for creating the difference of a 6, that is delta is equal to 6 we have
two rows here and when we take max of this we are getting 0.4. So, we are having it’s
membership value 0.4.

350
(Refer Slide Time: 34:58)

Next is a 𝛿 = 7. So, for 𝛿 = 7 we are having only one pair and this pair here for delta is
to 7. So, we enter the values of 𝑥𝐵 and 𝑥𝐶 which are responsible for creating the difference
between these as 7. So, we just enter these values here so 𝑥𝐵 here is 1 and 𝑥𝐶 here is 8 and
this way we are getting it’s corresponding membership values from the fuzzy sets B and
C and that is how we fill these entries.

(Refer Slide Time: 35:45)

So, we see here that when we pick these membership values and the corresponding delta
values from the last column and first column respectively. We see that we are able to get

351
a discrete fuzzy set and this is nothing but the distance between two fuzzy sets B and C
over the generic variable x. So, when we are interested in plotting this we are getting this
plot, so this plot represents the distance between B and C.

(Refer Slide Time: 36:30)

And here we have the MATALB code for finding the distance between two fuzzy sets B
and C. So, if you are interested this MATALB codes you can use for finding the distance
between two fuzzy sets and generating the plot for the resultant discrete fuzzy set. So, this
way we have seen that the distance between two fuzzy set can be found.

So, if the fuzzy sets are discrete fuzzy sets, the resultant distance between these two fuzzy
sets will be a discrete fuzzy set and if these fuzzy sets are continuous fuzzy sets, ofcourse,
the distance between these two fuzzy sets are going to be the continuous fuzzy sets. So,
this way we understood as to how we can manage to find the distance between two fuzzy
sets. So, here we will stop and then in the next lecture we will study the fuzzy arithmetic.

Thank you very much.

352
 Fuzzy Sets, Logic and Systems and Applications
Prof. Nishchal K. Verma
Department of Electrical Engineering
Indian Institute of Technology, Kanpur

Lecture – 20
Arithmetic Operations on Fuzzy Numbers

Welcome to lecture number 20 of Fuzzy Sets Logic and Systems and Applications. So, in
this lecture we will learn the Arithmetic Operations on Fuzzy Numbers. So, let us first
understand what is a fuzzy number.

(Refer Slide Time: 00:41)

In one of the lectures we had already discussed what is a fuzzy number. So, a fuzzy number
is a fuzzy set basically that holds the condition of normality and convexity. So, what does
this mean? This means that any fuzzy set which satisfies the property of normality and
convexity is qualified to be fuzzy number. So, all the fuzzy numbers are the most basic
types of fuzzy sets.

So, here we have a fuzzy set the right side of this diagram. So, we have a plot of fuzzy
number, we call this as fuzzy number 5. So, if we see a fuzzy number 5, we see that we
have a fuzzy set which has its highest membership value that is 1 at 5, but at around 5 also
we have some membership values and that’s how this is a fuzzy number.

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So, when we talk of fuzzy number; obviously, let us also understand the crisp number. A
fuzzy number is a fuzzy set, now if we see what is a crisp number. So, crisp number can
also be regarded as a fuzzy set, but this fuzzy set is a fuzzy singleton. So, this is a special
kind of fuzzy set that is singleton fuzzy set 5. And, it is because if we look at this diagram
plot of fuzzy singleton, we see that at the generic variable 𝑥 = 5 we see that we have only
one membership value corresponding to 5 that is the highest; that means, the core.

So, we have only one core at 5. So, it is because this number is crisp number and we have
at 5 its belongingness or it is membership value 1. So, no other membership values are
possible around 5. So, that is why it is crisp. So, either we call a crisp number or in fuzzy
systems we call this crisp numbers or we represent these crisp numbers by fuzzy singleton.
So, we see here the fuzzy singleton 5 which is a crisp number. So, this is a crisp number,
right side we have the fuzzy number. Fuzzy number is always a fuzzy set, but this fuzzy
set must satisfy the condition of normality and convexity.

(Refer Slide Time: 04:11)

In this coming lectures, this lecture and the coming lectures we will learn the arithmetic
operations on fuzzy number. So, by now we know what is a fuzzy number. So, similarly
when we talk about fuzzy numbers when we say fuzzy numbers, it means we are we will
be dealing with fuzzy sets. And, these fuzzy sets will be satisfying the criteria of normality
and the convexity. And, when we talk of arithmetic operations here; so, arithmetic
operations will be here arithmetic operations are addition of fuzzy numbers and then we

354
have subtraction of fuzzy numbers, multiplication of fuzzy numbers, division of fuzzy
numbers.

(Refer Slide Time: 05:04)

Now, let us see how we add two fuzzy numbers. So, when we say two fuzzy numbers;
obviously, these fuzzy numbers will be a fuzzy sets and it is needless here in this lecture
to say that these fuzzy sets again these should be with these conditions of convexity and
normality satisfied. So, let us take two fuzzy numbers A and B and let us right here C
which is the sum of these two fuzzy numbers A and B.

So, we have the resultant fuzzy set which you C here and this C let us see how do we
obtain ok. So, when we say C is a resultant fuzzy set obviously, C will have representation
as we have already discussed. So, if it is a discreet fuzzy set if this resultant is a discrete
fuzzy set, we will represent this fuzzy set C by

𝐶 = ∑ 𝜇𝐶 (𝑥 𝐶 )/𝑥 𝐶
𝑋

where this 𝑥𝐶 is the generic variable values that will be coming from the fuzzy set the
resultant fuzzy set C.

We can also say C as a fuzzy number because here when we are adding two fuzzy numbers
A and B we will be getting another fuzzy number C. When the resultant fuzzy set here is

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a continuous fuzzy set C, we will replace the summation by the integration sign which is
here.

𝐶 = ∫ 𝜇𝐶 (𝑥 𝐶 )/ 𝑥 𝐶
𝑋

So, as we already know that in continuous in representation of the continuous fuzzy set,
we represent the fuzzy set like this. Our job is to find the corresponding membership values
of C when we add two fuzzy sets A and B.

So, we clearly see here the max criteria which gives us the addition of two fuzzy sets and
this is with respect to the addition of two fuzzy sets A and B. So, 𝜇𝐶 (𝑥 𝐶 ) = 𝜇𝐴+𝐵 (𝑥 𝐶 ) =
max [𝜇𝐴 (𝑥 𝐴 ) ∧ 𝜇𝐵 (𝑥 𝐵 )]. So, it means what? It means that we will take min of all the
𝑥 𝐴 ,𝑥 𝐵

corresponding membership values and then whatever values that we will be getting for all
𝑥𝐴 𝑥𝐵 we will take max. And please note that the C is nothing, but A plus B.

So, we will see as to how we are going to get the complete fuzzy set we will take the
example and then we will understand as to how we get the addition of two fuzzy numbers.

(Refer Slide Time: 08:46)

So, let us take this example here we have a fuzzy number A and then we have another
fuzzy number B. If we see this is represented by these two discrete fuzzy expressions,
these are the expressions. So, I will write here the discreet fuzzy sets because these fuzzy

356
number, but finally, this is a fuzzy set discrete fuzzy set. So, these two are discrete fuzzy
sets.

(Refer Slide Time: 09:24)

Let us know quickly go ahead and see how to add these two fuzzy numbers. Alright, so
we have A fuzzy number here and B fuzzy number here. As we have seen the first most
operation is to get the minimum of x A 𝜇𝐴 (𝑥𝐴 ) and 𝜇𝐵 (𝑥𝐵 ). So, let us see as to how we
are going to get this if we take the elements from A and B. So, the first element of A is
0.3/1 and the first element of B is 0.5/10. So, let us get this min operation and then we
right by oblique and then 𝑥𝐴 + 𝑥𝐵 .

So, what do we do here is, we see that 𝜇𝐴 (𝑥𝐴 ) here is 0.3 which is here and 𝜇𝐵 (𝑥𝐵 ) is 0.5
which is here. When we take the min of these two we will get 0.3; as I mentioned this by
oblique 1 plus 10. So, this is generic variable value which is from the first fuzzy set, fuzzy
number A. So, this is from fuzzy number A and I can write it here this as 𝑥𝐴 𝑥1𝐴 .

Similarly this value is from fuzzy set or fuzzy number B, I can write here this as 𝑥1𝐵 . So,
this way we have written here the min of 0.3 and 0.5 as 0.3 and then oblique 11 because 1
plus 10 here is 11. So, finally, we are getting 0.3/11. So, this way we have got the first
term and similarly we’ll go through all the combinations of A and B elements.

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Let us now go ahead and see what we are getting. And, then when we have got these
combinations since we have the summation, so, we will sum these elements to get the final
expression for fuzzy set C which is the resultant of fuzzy number A and B.

(Refer Slide Time: 12:39)

So, let us quickly go through this and here we have written the final expression, but I will
explain as to how we are going to get these values as listed here. This we have already
seen that how we are getting 0.3/11. And, I can repeat here that this is the outcome when
we have taken the first elements from both the sets, both the numbers of the elements from
both the fuzzy numbers, next is the next combination.

So next combination will be we see here that, when we take 0.3/1 and then from the fuzzy
number A and then when we take the second element from fuzzy number B, we obtain
0.3/12 and we know how are we getting this generic variable 12. This is because we have
here 1 is 1 I can write again here 𝑥1𝐴 and here I can write here as 𝑥2𝐵 . So, this 12 is the
outcome of 𝑥1𝐴 + 𝑥2𝐵 . So, this way the generic variable values are computed and now when
we go ahead for other combinations let us say we are taking 0.3/1 and 0.5/12.

So, 0.3/1 is from fuzzy number A and 0.5/12 is from fuzzy number B. So, we are going
to obtain here out of these two when we use these two, we are getting this as the. Similarly,
when we go with other combinations you see the min of 0.6 and 0.5 we are getting 0.5 and
then the generic variable values when we add, we add the addition we will get 2 + 10 and
this is nothing, but 12. So, 0.5/12 we are getting and then here if we take this combination

358
that is 0.6/2 and 1/ 11 we are going to get the min value as 0.6 and the generic variable
value here is 13.

So, finally, this element is here this element is 0.6/13. I will quickly go ahead with other
combinations here. So, with this combination we are going to get 0.5/14, with this
combination we are going to get 0.5/ 13 and then with this combination we are going to
get 1/ 14. With this combination we are going to get 0.5/15, with this combination we
are going to get 0.5/14 with this combination here we get 0.7/15.

Similarly, we get here with this combination 0.5/ 16, with this combination we see here
0.2/15, here with this combination we get 0.2/ 16. And finally, with this combination
0.2/5 and 0.5/12, we get 0.2/17, 0.2 because here the min of 0.2 and 0.5 will be 0.2 and
then the generic variable values will be straight away added to give us 17.

(Refer Slide Time: 17:09)

So, that is how we get all these values, now the next step is to rearrange these values that
we have got these elements that we have got. So, let us now rearrange these values, these
findings and when we rearrange we see that for the generic variable value 11, we have
only one term that is with membership value 0.3. And, when we see for generic variable
value 12 we have two terms here one is 0.3/12 and the other one is 0.5/12.

So, we keep writing all such terms for which we have the same generic variable values.
So, for 12 for generic variable value 12 we are writing these two terms together, similarly

359
for generic variable 13 we have three terms and we are writing these together for generic
variable value 14 we have 1, 2, 3 three terms and here this also is being written together.
Similarly, generic variable value 15, we have three terms and it is also written together
and here also for generic variable value 16 we have two terms.

And finally, here as the last term we have for generic variable value 17 only one term that
is with 0.2 as the membership value. So, this will be rearrange these terms and now what
we do here is that, we take the max of these terms for which we have generic variable
values same.

(Refer Slide Time: 19:29)

So, let us do that and when we do that here when we take the max as we see here. So, since
we have 0.3/11 as this is only one term. So, we just write this as it is, but here we if we
have more number of terms for the same generic variable values. So, like for 12 we have
two terms and then for these two terms we take the max of its corresponding membership
values. So, max when we take max of 0.3 and 0.5 we get 0.5 here. So, we write 0.5/12.

Similarly for generic variable value 13 when we take max we get only one membership
value which is 0.6. So, we have this term as 0.6/ 13. So, for generic variable value 14 we
have 1 as the membership value. So, 1/14 will remain here and then similarly for 15 we
have 0.7, for 16 we have 0.5, for 17 we have 0.2. So, all these are the corresponding max
values with respect to the generic variable values. So, this way we have the expression
here of the fuzzy set the resultant fuzzy C and which is nothing but a fuzzy number here

360
and this is, this C as I have already mentioned the C is the addition of two fuzzy numbers
A and B.

(Refer Slide Time: 21:23)

So, this is how we obtain an addition of two fuzzy numbers A and B. So, let’s now see as
to what was given to us. So, A was given to us as the discrete fuzzy set, B was also given
to us as the discrete fuzzy set and then what we are obtaining here is this. So, when we add
these two fuzzy numbers we have C fuzzy number. So, this is the resultant fuzzy number,
let us know represent these three fuzzy numbers A, B and C and let us see how these three
look like.

So, if we see here fuzzy number A, fuzzy number B and then when we compute the
addition, when we find the addition of these two fuzzy numbers we get another fuzzy
number which is C which is outcome of the addition. So, we see that the spread is increased
the spread of the resultant fuzzy number is increased. It is probably because of the
uncertainty level that is getting increased in the addition of two fuzzy sets A and B.

So, the uncertainty level here in fuzzy number A was some uncertainty and then the
uncertain level in fuzzy number B, but whatever uncertain levels in the both fuzzy numbers
were there we see that the uncertainty level in fuzzy number C is more than the uncertainty
level of fuzzy number A and fuzzy number B. So, this way we see that we how can we
add fuzzy number A and fuzzy number B to get another fuzzy number C.

361
(Refer Slide Time: 23:22)

On the same lines now we can find we can compute the addition of the fuzzy number B
and A. So, here we are just changing the order, we are just taking B first and then A. So,
let us see what are we getting.

(Refer Slide Time: 23:44)

So, when we take B first and we do the same exercise.

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(Refer Slide Time: 23:53)

We see that here we are getting when we take this

∑(𝜇𝐵 (𝑥 𝐵 ) ∧ 𝜇𝐴 (𝑥 𝐴 ))/(𝑥 𝐵 + 𝑥 𝐴 )
𝑋

So, we see that for this combination we are getting 0.3/11, 0.5/12, 0.5/13, 0.5/14, 0.2/
15, 0.3/12, 0.6/13. So, 1.0/14 and then here we are getting 0.7/15, 0.2/16, 0.3/ 13, 0.5/
14, 0.5/15, 0.5/16, 0.2/17.

Now, when we rearrange these terms as I mentioned earlier, so we see that we are getting
the expression like this and when we take the maximum or the max of all the membership
values corresponding to the same generic variable value, we are going to get the final
expression like this.

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(Refer Slide Time: 25:20)

So, this is after taking the max. So, this is coming out to be 0.3/11 + 0.5/12 + 0.6/13 +
1.0/14 + 0.7/15 + 0.5/16 + 0.2/17. So, this is how we are getting the result the
resultant fuzzy set.

(Refer Slide Time: 25:53)

So, if we see here that we are obtain fuzzy set C which is the outcome of we obtain fuzzy
number C, which is the outcome of fuzzy number A fuzzy number B and fuzzy number A.

364
(Refer Slide Time: 26:10)

So, we see the same fuzzy number the same fuzzy set that we have obtained here and this
way we can say that either we take A first or B first both the outcomes remain same. So,
we can say the addition is commutative here the fuzzy addition is commutative.

So, this means the 𝐴 + 𝐵 = 𝐵 + 𝐴 for fuzzy numbers. So, this way we have been able to
know as to how we can add two fuzzy numbers and any two fuzzy numbers can be added
this way. And, please note once again that these fuzzy numbers when we when we say
fuzzy numbers means these fuzzy numbers are the fuzzy sets that satisfy the criteria of
normality and convexity. It means these fuzzy numbers must be a normal fuzzy set first
and then these fuzzy numbers should be satisfying the criteria of convexity.

365
(Refer Slide Time: 27:40)

So, this way we understood as to how we can add any two fuzzy numbers. And, in the next
lecture, we will study the other operations other arithmetic operations like subtraction,
multiplication, divisions of fuzzy numbers.

Thank you.

366
 Fuzzy Sets, Logic and Systems and Applications
Prof. Nishchal K. Verma
Department of Electrical Engineering
Indian Institute of Technology, Kanpur

Lecture – 21
Arithmetic Operations on Fuzzy Numbers

So, welcome to lecture number 21 of Fuzzy Sets, Logic and Systems and Applications.
So, this lecture is in continuation to our discussions on Arithmetic Operations and this
arithmetic operations are on Fuzzy Numbers. We have already covered the arithmetic
operation that is the addition on fuzzy numbers in our last lecture.

(Refer Slide Time: 00:58)

Today we will be discussing the subtraction of fuzzy numbers. So, when we have two
fuzzy numbers 𝐴𝐴 and 𝐵𝐵 and these two fuzzy numbers are within some universe of discourse
capital 𝑋𝑋. And if we perform the subtraction it results in a new fuzzy number let’s say C
and 𝐶𝐶 this can be defined as 𝐶𝐶 = 𝐴𝐴 − 𝐵𝐵.

So, please note here, that 𝐴𝐴 and 𝐵𝐵 both are fuzzy numbers. So, what does this mean here
is if we say 𝐴𝐴 is a fuzzy number means 𝐴𝐴 is a fuzzy set which satisfies the criteria of
normality as well as the convexity. So, we can say 𝐴𝐴 is a normal fuzzy set and 𝐴𝐴 is a convex
fuzzy set. Similarly, 𝐵𝐵 also is a normal fuzzy set and convex fuzzy set.

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Now, when we subtract 𝐵𝐵 from 𝐴𝐴 fuzzy numbers we have a 𝐶𝐶 fuzzy number which is again
convex and normal fuzzy set. So, for discrete fuzzy sets the subtraction will govern by this
equation here. So, 𝐶𝐶 is nothing but 𝐶𝐶 will be equal to

𝐶𝐶 = � 𝜇𝜇𝑐𝑐 (𝑥𝑥 𝐶𝐶 )/𝑥𝑥 𝐶𝐶

𝑋𝑋

So, this is for discrete fuzzy sets and for continuous fuzzy set this summation will be
replaced by the integration sign. Please note that this summation is a symbolic
representation here. Similarly the integration is also a symbolic representation, we do not
have to either sum or integrate.

So, these have to be there these values these terms in summation we have to just separate
by plus signs and in integration we do not have to integrate these the whatever is the
outcome because of course, here there is no dx or something like that.

So, the resultant fuzzy set that is coming out of 𝐴𝐴 − 𝐵𝐵 at 𝐶𝐶, so 𝐶𝐶 will have a membership
function and its generic variable. So, 𝜇𝜇𝐶𝐶 here can be found by 𝜇𝜇𝐴𝐴−𝐵𝐵 (𝑥𝑥 𝐶𝐶 ) where 𝐶𝐶 is nothing
but the resultant fuzzy number and this is equal to max
𝐴𝐴 𝐵𝐵
(min�𝜇𝜇𝐴𝐴 (𝑥𝑥 𝐴𝐴 ), 𝜇𝜇𝐵𝐵 (𝑥𝑥 𝐵𝐵 )�).
𝑥𝑥 ,𝑥𝑥

So, it is clearly understood that 𝑥𝑥 𝐶𝐶 = 𝑥𝑥 𝐴𝐴 − 𝑥𝑥 𝐵𝐵 for subtraction purpose and for all
𝑥𝑥 𝐴𝐴 , 𝑥𝑥 𝐵𝐵 , 𝑥𝑥 𝐶𝐶 they are belonging all values are belonging to the universe of discourse. So, we
can write here this 𝐶𝐶 is nothing but 𝐶𝐶 is represented by this expression.

368
(Refer Slide Time: 05:09)

So, let us now take an example of subtraction of fuzzy numbers A and B and understand
the operation. So, if we have two fuzzy sets 𝐴𝐴 and 𝐵𝐵, 𝐴𝐴 = 0.3/1 + 0.6/2 + 1/3 +
0.7/4 + 0.2/5. 𝐵𝐵 is represented as 𝐵𝐵 = 0.5/10 + 1/11 + 0.5/12 and both these 𝐴𝐴 and 𝐵𝐵
sets are discrete fuzzy sets. And these are represented here as the fuzzy numbers 𝐴𝐴 and 𝐵𝐵.

(Refer Slide Time: 06:11)

So, let us now apply the formula that we have just seen what is resulting. So, 𝐶𝐶 = 𝐴𝐴 − 𝐵𝐵,
𝐶𝐶 is resulting fuzzy number here and this is equal to the fuzzy number 𝐴𝐴 − 𝐵𝐵. And when

369
we apply this formula let’s see what is happening. So, first let us compute this value for
one combination of the elements from 𝐴𝐴 and 𝐵𝐵.

So, let us take min of these two, this term and this term. So, 0.3/1 and 0.5/10 we have
two elements and let us take min of these. So, 𝑚𝑚𝑚𝑚𝑚𝑚(0.3, 0.5) we are going to get 0.3. And
then here for generic variable value since we are subtracting, so we’ll simply subtract so,
the generic variable value will be the 𝑥𝑥 𝐶𝐶 here for this case and 𝑥𝑥 𝐶𝐶 will be here nothing but
𝑥𝑥 𝐴𝐴 − 𝑥𝑥 𝐵𝐵 ; 𝑥𝑥 𝐶𝐶 = 𝑥𝑥 𝐴𝐴 − 𝑥𝑥 𝐵𝐵 and this is here in this case 𝑥𝑥 𝐴𝐴 = 1 and 𝑥𝑥 𝐵𝐵 = 10. So, what we
are going to get here is −9.

So, that is how we have computed here and the value of this term here the when we have
taken the one combination and this combination; the first combination is giving us 0.3/−9
for 𝐴𝐴 − 𝐵𝐵. So, similarly for all the combinations of elements of fuzzy number 𝐴𝐴 and fuzzy
number 𝐵𝐵. So, we will follow the same procedure for finding the other elements for
corresponding to the other combinations.

(Refer Slide Time: 08:39)

So, let’s quickly go through this and we find the other elements here as 0.3/−10. So, this
we will get out of this, so when we have these two pairs we clearly see that the 𝑚𝑚𝑚𝑚𝑚𝑚(0.3, 1)
we are going to get 0.3. So, that’s for how this 0.3 is coming. And then this is the outcome
of the generic variable valued subtraction, so 1 − 11. So, this is going to give us −10
which is here.

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So, on the same lines here we go ahead and find all the combinations. So, this way we take
all these combinations of pairs we get these values coming you can see here this is shown
by the outcome of each and every pair is coming and shown in the red color you see here.

(Refer Slide Time: 10:03)

So, that’s how we are getting the subtraction of all these pairs. So, that’s how we get the
complete total terms the total elements, the total outcomes are listed.

Now, what we have to do is, we rearrange all these terms means we form a group of all
these elements which are having same generic variable value. So, for generic variable
value −11 we have only one term, so that is how it is written here. Now, for generic
variable value minus 10 we have two values, the first one is this and the second one is this.

So, we see that we have two terms for generic variable value −10. Similarly we go ahead
with all the generic variable values that are listed here and we form a group of same generic
variable values. And then we apply the max criteria, if we take the max of membership
values of the same generic variable values.

371
(Refer Slide Time: 11:17)

So, let us now go ahead and see what is happening here and when we do that we see, so
here we are now applying the max and when we apply the max we see here this is the term
which is having two membership values. First membership is 0.5 and the second
membership here is 0.3 for the same generic variable value −10.

So, for −10 we will have to have only one membership value which is 0.5 when we take
𝑚𝑚𝑚𝑚𝑚𝑚(0.5, 0.3). So, that is how we are getting here 0.5. So, we here we can say here that
we are applying the max in between or 𝑚𝑚𝑚𝑚𝑚𝑚(0.5, 0.3). So, that is how we are getting 0.5
and this 0.5 is coming over here.

And on the same lines we are obtaining all these terms. So, far corresponding to −9 we
are getting 0.6 − 8 we are getting 1 − 7 we are getting 0.7 − 6 we are getting 0.5 − 5 we
are getting 0.2. So, all these terms we have got here.

And now we can clearly see that there is no duplication or in other words we can say there
is no duplication in the sense that we do not have here more than two values of generic
variables.

Means we have only the generic variable −10, we do not have other places −10. So, it
means we have the generic variable value −11 and then −10, −9, −8, −7, −6, −5 like
that. So, this way we have obtained the expression for 𝐶𝐶 and we can clearly see this is a
fuzzy number.

372
(Refer Slide Time: 13:33)

And now let us plot this fuzzy number 𝐶𝐶 which is the outcome of 𝐴𝐴 − 𝐵𝐵 here. So, fuzzy
number 𝐴𝐴 minus fuzzy number 𝐵𝐵 we have fuzzy number 𝐶𝐶 and 𝐶𝐶 is you see here also in
this case comparatively we have more spread than 𝐴𝐴 and 𝐵𝐵. So, this is because the
uncertainty level in the fuzzy number 𝐶𝐶 is increased.

So, when we have a fuzzy number 𝐴𝐴 which has some uncertainty level and again we have
fuzzy number 𝐵𝐵 which also has some uncertainty level. And when we take 𝐴𝐴 − 𝐵𝐵 or 𝐴𝐴 +
𝐵𝐵. So, the outcome fuzzy set which is a fuzzy number in both the cases will have
comparatively more spread. So, it means the uncertainty level gets increased. And please
understand that this uncertainty that I am talking of is the uncertainty due to ambiguity,
imprecision and vagueness. So, in this category these uncertainty is normally accounted in
the fuzzy sets.

373
(Refer Slide Time: 14:52)

Now, let us find the subtraction 𝐵𝐵 − 𝐴𝐴. So, let us now subtract fuzzy number 𝐴𝐴 from fuzzy
number 𝐵𝐵 and see what is happening.

(Refer Slide Time: 15:06)

So, on the same lines as we have discussed 𝐴𝐴 − 𝐵𝐵, so we have seen that we have first
computed the min of the membership values of corresponding terms, corresponding pairs.
And then when we have all these pairs we have found then we moved ahead and we took
max of these and then we moved ahead. So, we can quickly go through this and see what

374
is happening here when we subtract fuzzy number 𝐴𝐴 from fuzzy number 𝐵𝐵. So, we can
directly here go through this.

And here also when we subtract fuzzy number 𝐴𝐴 from fuzzy number 𝐵𝐵 we see that we
have (𝜇𝜇𝐵𝐵 (𝑥𝑥 𝐵𝐵 ) ∧ 𝜇𝜇𝐴𝐴 (𝑥𝑥 𝐴𝐴 ))/𝑥𝑥 𝐵𝐵 − 𝑥𝑥 𝐴𝐴 is because we are subtracting fuzzy number 𝐴𝐴 from
fuzzy number 𝐵𝐵. So, here when we take the min of the corresponding membership values.
We see here that we get 0.3 and then when we subtract here 1 from 10 we are getting 9.
So, 0.3/ 9 we are getting as one of the terms of fuzzy number 𝐶𝐶 which is the outcome of
𝐵𝐵 − 𝐴𝐴.

So, similarly for all the combinations of elements of fuzzy number 𝐵𝐵 and fuzzy number 𝐴𝐴
we will follow the same procedure.

(Refer Slide Time: 17:09)

So, we see here that we are getting 𝐶𝐶 as if we see here we are getting here result of 𝐵𝐵 − 𝐴𝐴
fuzzy numbers we get 0.2/5 + 0.5/6 + 0.7/7 + 1.0/8 + 0.6/9 + 0.5/10 + 0.3/11.

375
(Refer Slide Time: 17:42)

Now, our 𝐵𝐵 − 𝐴𝐴 is coming here as fuzzy number 𝐶𝐶 which is plotted here and when we
clearly see that this is very similar to the outcome which we got in the earlier case of 𝐵𝐵 −
𝐴𝐴.

(Refer Slide Time: 18:06)

So, when we bring these two outcomes 𝐴𝐴 − 𝐵𝐵 and 𝐵𝐵 − 𝐴𝐴 together we see that these two
are symmetric. So, or in other words we can say that 𝐴𝐴 − 𝐵𝐵 and 𝐵𝐵 − 𝐴𝐴 both are the mirror
images of each other. Of course, here we see when we take 𝐴𝐴 − 𝐵𝐵 and 𝐵𝐵 − 𝐴𝐴. We see that

376
these two are not equal to each other. So, it means commutativity property doesn’t hold
good with respect to the subtraction.

(Refer Slide Time: 18:47)

Now, let us discuss the multiplication of fuzzy numbers. So, multiplication can be defined
here as 𝐶𝐶 = 𝐴𝐴 ∗ 𝐵𝐵. So, star here signifies the multiplication. So, and 𝐴𝐴, 𝐵𝐵, 𝐶𝐶 are fuzzy sets
where 𝐴𝐴 and 𝐵𝐵 both are the fuzzy numbers. So, once again I would like to mention that
when we say fuzzy number it means it is fuzzy set which is satisfying both the properties,
the convexity as well as the normality.

So, when we say here 𝐴𝐴 and 𝐵𝐵 both are fuzzy numbers it means both the sets 𝐴𝐴 and 𝐵𝐵 are
the normal fuzzy sets as well as convex fuzzy sets. So, here we use this formula for finding
out the fuzzy set the multiplication of 𝐴𝐴 and 𝐵𝐵 and which is resulting let us say C fuzzy
set is the 𝐴𝐴 ∗ 𝐵𝐵. So, of course, when we multiply any two fuzzy number we are going to
get a fuzzy set out of it.

This fuzzy set here may not be fuzzy number because this may not satisfy the criteria of
the normality as well as the convexity. So, we will see ahead what is happening when we
multiply two fuzzy numbers.

So, let us quickly go ahead and see how do we multiply two fuzzy numbers. So, formula
remains the same if we see the only difference that we see here is this 𝑥𝑥𝐶𝐶 comes out to be
𝑥𝑥𝐴𝐴 multiplied by 𝑥𝑥𝐵𝐵 which is here as well which is mentioned. So, other than this there is

377
no difference at all. So, all min and max remains intact, so we need to be careful while
finding the multiplication of two fuzzy numbers 𝐴𝐴 and 𝐵𝐵. So, in addition this 𝑥𝑥𝐶𝐶 = 𝑥𝑥𝐴𝐴 +
𝑥𝑥𝐵𝐵 , in subtraction the 𝑥𝑥𝐶𝐶 = 𝑥𝑥𝐴𝐴 −𝑥𝑥𝐵𝐵 . So, in multiplication here we have 𝑥𝑥𝐶𝐶 = 𝑥𝑥𝐴𝐴 ∗ 𝑥𝑥𝐵𝐵 .

(Refer Slide Time: 21:42)

So, let us quickly go through an example here which will make us understand little better
the multiplication of two fuzzy numbers. So, here we have the fuzzy numbers 𝐴𝐴 and 𝐵𝐵 and
this fuzzy numbers here are within the universe of discourse −15 to 15. So, here we have
to consider the universe of discourse as well.

Now, let us find the multiplication of these two fuzzy numbers. So, when we plot the fuzzy
number 𝐴𝐴 this will look like this, when we plot the fuzzy number 𝐵𝐵 this will look like this
here it is shown by the green color and fuzzy number 𝐴𝐴 is shown by the blue color.

378
(Refer Slide Time: 22:44)

So, let us now apply the formulation that we have just discussed and when we apply we
first try to find this term which is (𝜇𝜇𝐴𝐴 (𝑥𝑥 𝐴𝐴 ) ∧ 𝜇𝜇𝐵𝐵 (𝑥𝑥 𝐵𝐵 ))/(𝑥𝑥 𝐴𝐴 ∗ 𝑥𝑥 𝐵𝐵 ). So, let us quickly find
out these values 𝜇𝜇𝐴𝐴 (𝑥𝑥 𝐴𝐴 ), 𝜇𝜇𝐵𝐵 (𝑥𝑥 𝐵𝐵 ) from this pair the first combination. So, we have first
combination here and from here we get all these values 𝜇𝜇𝐴𝐴 (𝑥𝑥 𝐴𝐴 ), 𝜇𝜇𝐵𝐵 (𝑥𝑥 𝐵𝐵 ) and also we get
𝑥𝑥 𝐴𝐴 and 𝑥𝑥 𝐵𝐵 as generic variable values, 0.3 is 𝜇𝜇𝐴𝐴 (𝑥𝑥 𝐴𝐴 ), 0.5 is 𝜇𝜇𝐵𝐵 (𝑥𝑥 𝐵𝐵 ).

So, when we substitute here and take the min of these we find the 𝑚𝑚𝑚𝑚𝑚𝑚 as 0.3 and 1 ∗ 10
which are the generic variable values of 𝐴𝐴 and 𝐵𝐵 respectively as 𝑥𝑥 𝐴𝐴 and 𝑥𝑥 𝐵𝐵 , so when we
multiply this we are going to get 10. So, this is going to result 0.3/10. So, similarly for all
the combinations of elements of fuzzy number 𝐴𝐴 and fuzzy number 𝐵𝐵 we will follow the
same procedure.

379
(Refer Slide Time: 24:23)

And that is how we are going to get 0.3 all the elements as 0.3/10 ,0.3/11, 0.3/12, 0.5/
20, 0.6/22, 0.5/24, 0.5/30, 1.0/33, 0.5/36, 0.5/40, 0.7/44, 0.5/48, 0.2/50, 0.2/
55, 0.2/60.

So, all these elements we have got after computing minimum of 𝜇𝜇𝐴𝐴 (𝑥𝑥 𝐴𝐴 ) and 𝜇𝜇𝐵𝐵 (𝑥𝑥 𝐵𝐵 )
oblique 𝑥𝑥 𝐴𝐴 into 𝑥𝑥 𝐵𝐵 from all the combinations that are possible from fuzzy number 𝐴𝐴 and
𝐵𝐵. So, when we have done that that will result here these terms. So, when we rearrange;
when we write all these terms we see that we have all these terms are; some of the terms
that are there which are not within the universe of discourse. So, let’s see what we need to
do further.

380
(Refer Slide Time: 25:59)

Since the universe of discourse that has been given to us is -15 to 15, it means whatever
elements that we will be having or that will be getting here should be within the universe
of discourse.

So, this means what? This means we have to consider only those terms for which we have
the generic variable values within this universe of discourse. So, this way if we will look
here in this expression. So, we see that we have generic variable value 10 which is within
the universe of discourse, 11 which is within the universe of discourse, 12 which is within
the universe of discourse. So, like that these are within the universe of discourse these
values these generic variable values. Now, what about this 20? So, 20 is not within the
universe of discourse.

Similarly, 22 also is not within the universe of discourse that has been given to us.
Similarly, 24 which is also not within the universe of discourse, similarly 30 is also not
within the universe of discourse, 33 is also not within the universe of discourse, 36 is also
not within the universe of discourse, 40 is also not within the universe of discourse, 44 is
also not within the universe of discourse, 48 is also not within the universe of discourse,
50 is also not within the universe of discourse, 55 is also not within the universe of
discourse, 60 is also not within the universe of discourse.

Only 10 is within the universe of discourse this 10 as the generic variable value. So, this
generic variable value is within the universe of discourse. Then 11 is within the universe

381
of discourse, 12 is also within the universe of discourse. So, when we have now known
this thing that only 10, 11, 12 are the generic variable values present here are the part of
the universe of discourse.

So, we will take up only those terms which are having the generic variable values within
the universe of discourse. So, it means we are only considering these 3 terms and these 3
terms we see here that 0.3/10, 0.3/11, 0.3/12 these 3 are lying within the universe of
discourse which is −15 to 15.

(Refer Slide Time: 29:03)

And when we plot this outcome here as fuzzy set 𝐶𝐶 this is not a fuzzy number because you
see this doesn’t qualify to be a fuzzy number because this is not a normal set. So, this is a
fuzzy set 𝐶𝐶 this is not a fuzzy number please note that.

So, this way when we have 𝐴𝐴 fuzzy number 𝐴𝐴 and fuzzy number 𝐵𝐵 when we multiply this
as 𝐴𝐴 ∗ 𝐵𝐵 or in other words we can say when 𝐴𝐴 and 𝐵𝐵 both are fuzzy numbers and they are
multiplied we are going to get a fuzzy set and this is not going to be the fuzzy number in
our case here. So, in a more appropriate way I would say that the multiplication of two
fuzzy numbers may not be a fuzzy number.

382
(Refer Slide Time: 30:09)

So, now let us find the 𝐵𝐵 ∗ 𝐴𝐴 means now let us take 𝐵𝐵 first and then 𝐴𝐴.

(Refer Slide Time: 30:21)

So, let’s now see what is happening. So, when we do that and we apply the same procedure
of computing 𝐵𝐵 multiplication 𝐴𝐴.

383
(Refer Slide Time: 30:33)

We see that we are getting here after doing all the computations and intermediate steps.
So, we are landing up here.

(Refer Slide Time: 30:40)

Again, we are landing up with only 3 elements means the same elements.

384
(Refer Slide Time: 30:49)

So, when we plot this 𝐵𝐵 ∗ 𝐴𝐴 means when we multiply 𝐵𝐵 and 𝐴𝐴 we get the same set. So, we
can say the 𝐴𝐴 multiplication 𝐵𝐵 is equal to 𝐵𝐵 multiplication 𝐴𝐴 means 𝐴𝐴 and 𝐵𝐵 the
multiplication of the fuzzy numbers 𝐴𝐴 and 𝐵𝐵 are commutative with respect to
multiplication.

(Refer Slide Time: 31:20)

So, this way we see that here both the fuzzy sets are same and as I mentioned earlier that
these are not the fuzzy numbers the result is not a fuzzy number here. Please understand
that since we have limited the universe of discourse, we have shown only a portion of the

385
actual multiplication. We could have seen some more terms which would have been spread
probably the in wider universe of discourse, but here since the universe of discourse is
very limited.

So, that is why only few terms are coming as are few terms are included within the universe
of discourse few terms are included here and in fuzzy set 𝐶𝐶. So, that is why you can see
here how these two look like within the universe of discourse -15, 15

And as I already mentioned 𝐴𝐴 multiplication 𝐵𝐵 is equal to 𝐵𝐵 multiplication 𝐴𝐴. So, that’s

how we have seen as to how we can multiply two fuzzy numbers 𝐴𝐴 and 𝐵𝐵. So, with this
we will stop here.

(Refer Slide Time: 32:43)

And in the next lecture we will study the division of fuzzy numbers and apart from these
we will do some more examples on the addition, subtraction, multiplication and division
of fuzzy numbers.

Thank you.

386
 Fuzzy Sets, Logic and Systems and Applications
Prof. Nishchal K. Verma
Department of Electrical Engineering
Indian Institute of Technology, Kanpur

Lecture – 22
Arithmetic Operations on Fuzzy Numbers

(Refer Slide Time: 00:18)

So, welcome to lecture number 22 of Fuzzy Sets Logic and Systems and Applications. So,
this lecture is in continuation to our previous lectures, where we discussed Arithmetic
Operations on Fuzzy Numbers. So, before we move ahead, we need to know that a fuzzy
number is a fuzzy set which satisfies the condition of normality as well as the condition of
convexity.

387
(Refer Slide Time: 00:50)

So, before this arithmetic operation of division, we had already discussed the other
arithmetic operations like addition of fuzzy numbers, subtraction of fuzzy numbers,
multiplication of fuzzy numbers. So, now, today here, we will be discussing the division
of fuzzy numbers.

So, if we have two fuzzy numbers 𝐴𝐴 and 𝐵𝐵 within the universe of discourse capital X and
if we are supposed to divide the fuzzy number 𝐴𝐴 by fuzzy number 𝐵𝐵, let us see what
happens. So, this is expressed by this formula here and this formula is for discrete fuzzy
numbers, means when we have, when we are dealing with discrete fuzzy numbers, we use
the formula for division here as

𝐶𝐶 = ∑𝑋𝑋 𝜇𝜇𝐶𝐶 (𝑥𝑥 𝐶𝐶 )/𝑥𝑥 𝐶𝐶

Similarly, if we are dealing with the continuous fuzzy numbers 𝐴𝐴 and 𝐵𝐵 instead of using
the summation sign, we are using the integration sign.

And this is needless to say here that, the summation and integration both the signs are just
symbolic representation; we are not supposed to add any of the elements or integrate the
membership function here, so this needs to be understood. Now next is when we are
dividing a fuzzy number 𝐴𝐴 by another fuzzy number 𝐵𝐵, we are going to get a resultant
which is again a fuzzy set. I am not saying here a fuzzy number, because the resultant is
not going to be a fuzzy number because resultant fuzzy set is not going to be normal always

388
and the convex. And since this is a fuzzy set, so we’ll have its membership values and the
corresponding generic variable values of the resultant fuzzy set.

So, let us first find the membership values and, this is represented by the 𝜇𝜇𝐶𝐶 because C
here will be the resultant of 𝐴𝐴 ÷ 𝐵𝐵. So, 𝐶𝐶 has been designated here as the fuzzy set. So, the
membership value of this fuzzy set corresponding to 𝑥𝑥 𝐶𝐶 is designated as 𝜇𝜇𝐶𝐶 (𝑥𝑥 𝐶𝐶 ) =
𝜇𝜇𝐴𝐴÷𝐵𝐵 (𝑥𝑥 𝐶𝐶 ) = max
𝐴𝐴 𝐵𝐵
[𝜇𝜇𝐴𝐴 (𝑥𝑥 𝐴𝐴 ) ∧ 𝜇𝜇𝐵𝐵 (𝑥𝑥 𝐵𝐵 )] where this 𝑥𝑥 𝐶𝐶 is nothing but it is 𝑥𝑥 𝐴𝐴 ÷ 𝑥𝑥 𝐵𝐵 . What are
𝑥𝑥 ,𝑥𝑥

these 𝑥𝑥 𝐴𝐴 , 𝑥𝑥 𝐵𝐵 and 𝑥𝑥 𝐶𝐶 ? These 𝑥𝑥 𝐴𝐴 , 𝑥𝑥 𝐵𝐵 , 𝑥𝑥 𝐶𝐶 ’s are the generic variable values coming from the
fuzzy number 𝐴𝐴, 𝐵𝐵 and fuzzy set 𝐶𝐶 respectively and these 𝑥𝑥 𝐴𝐴 , 𝑥𝑥 𝐵𝐵 , 𝑥𝑥 𝐶𝐶 will be part of the
universe of discourse.

So, we can finally write here as shown here in this equation in this expression, if we write
together the membership value of fuzzy set 𝐶𝐶 and the corresponding generic variable
values. So, when we take discrete fuzzy numbers, we get the discreet fuzzy set here and
𝐴𝐴, 𝐵𝐵 are the two discrete fuzzy numbers fuzzy numbers, whereas the 𝜇𝜇𝐴𝐴 , 𝜇𝜇𝐵𝐵 are the
corresponding membership values. So, this expression

𝐶𝐶 = � 𝜇𝜇𝐶𝐶 (𝑥𝑥 𝐶𝐶 ) /𝑥𝑥 𝐶𝐶 = � 𝜇𝜇𝐴𝐴÷𝐵𝐵 (𝑥𝑥 𝐶𝐶 )/𝑥𝑥 𝐶𝐶 = � max

𝐴𝐴 𝐵𝐵
[𝜇𝜇𝐴𝐴 (𝑥𝑥 𝐴𝐴 ) ∧ 𝜇𝜇𝐵𝐵 (𝑥𝑥 𝐵𝐵 )]
𝑥𝑥 ,𝑥𝑥
𝑋𝑋 𝑋𝑋 𝑋𝑋

So, this expression is for discreet or we can say division of discreet fuzzy numbers. We
will replace this summation when we are taking the continuous fuzzy numbers instead of
discrete fuzzy numbers, we will replace the summation sign by the integration sign. So, I
hope now this is understood clearly.

389
(Refer Slide Time: 08:07)

Now, let us take an example and understand the division of fuzzy numbers. So, here we
are having two discrete fuzzy sets and of course these two fuzzy sets are qualified to be
fuzzy numbers. So, we can say these two fuzzy numbers 𝐴𝐴 and 𝐵𝐵 are there and we are
supposed to divide 𝐴𝐴 fuzzy number by 𝐵𝐵 fuzzy number. And it is given here that, 𝐴𝐴 and 𝐵𝐵
are within the universe of discourse 𝑋𝑋 belonging to the natural number, is given.

So, let us first plot the fuzzy number 𝐴𝐴 here and the fuzzy number 𝐵𝐵 on the same generic
variable value 𝑋𝑋 here as the x axis and the membership grades here as the y axis. So, I can
write here 𝜇𝜇𝐴𝐴 (𝑥𝑥), this is 𝜇𝜇𝐴𝐴 (𝑥𝑥); here we have 𝜇𝜇𝐵𝐵 (𝑥𝑥). So, these two fuzzy numbers are
shown and these two fuzzy numbers are discrete fuzzy numbers.

390
(Refer Slide Time: 09:38)

Let us see how we proceed for the division of 𝐴𝐴 by 𝐵𝐵 and this is going to result as another
fuzzy set which is 𝐶𝐶. So, for this we need to take the first element from 𝐴𝐴 and the first
element from 𝐵𝐵 fuzzy numbers, and let us take their corresponding membership values
and the generic variable values and let us find the min of the membership values of the
first combination that is the combination is here. So, this is the first combination.

So, likewise we will have so many combinations, we will have all the combination; means
we will take all the elements of 𝐴𝐴 and 𝐵𝐵, we will be combine them together and then we
will process this, as we will see here in this couple of slides. So, when we substitute these
values here, we find the min(0.3, 0.5)/2 ÷ 1; this comes here 2 divided by 1, because we
have 𝑥𝑥 𝐴𝐴 ÷ 𝑥𝑥 𝐵𝐵 . And 𝑥𝑥 𝐴𝐴 , 𝑥𝑥 𝐵𝐵 are nothing, but the generic variable values from the fuzzy
numbers 𝐴𝐴 and 𝐵𝐵 respectively. When we are taking the min of this, we are getting 0.3 and
then when we divide 𝑥𝑥 𝐴𝐴 by 𝑥𝑥 𝐵𝐵 , we are getting 2.

So, this way we are getting the one element of C fuzzy set as 0.3/ 2. Please note that, we
are not supposed to divide here 0.3/ 2, so this is just the representation, we I have to keep
this element as it is, where 0.3 of this element represents the membership value
corresponding to the generic variable value of 2 for the resulting fuzzy set 𝐶𝐶. So, similarly
let us go ahead and take up all the combinations here.

391
(Refer Slide Time: 12:29)

We see all the combinations because summation is here, so we are taking all the
combinations separated by the plus signs, we see here all these plus signs are there, again
we need not add these values together this is just for the representation purpose. So, these
values have been shown here and now let us rearrange these values and we find that we
have some elements, we have same generic variable values, like for these two elements,
we have the same generic variable, means the 𝑥𝑥 𝐶𝐶 here is one for both the terms.

So, for such cases what we do here is, we avoid the conflict and by taking the max of the
membership values. So, we will take max in the next step, so but what we need to
understand that all such elements need to be written together.

392
(Refer Slide Time: 13:54)

So, we have written here and then we see that here, we see that; when we apply the max
criteria, we are getting 𝐶𝐶 = 0.3/0.5 + 0.5/1 + 0.5/1.5 + 0.6/2 + 1.0/3 + 0.5/4 +
0.5/6. So, here we see that, we have as a result of fuzzy number 𝐴𝐴 divided by a fuzzy
number 𝐵𝐵, we are getting another fuzzy set 𝐶𝐶. So, here I am saying a fuzzy set 𝐶𝐶, because
this fuzzy set need not be a normal fuzzy set and a convex fuzzy set always.

This can be possible, but it may not be necessary that you are always getting a fuzzy
number as a result of the division of one fuzzy number by the another fuzzy number.

(Refer Slide Time: 15:16)

393
So, here when we see the fuzzy set which has been plotted. So, we see that we have a fuzzy
number here A and B that was given to us as a result of fuzzy number A divided by fuzzy
number B, we got another fuzzy set C and we can check if this satisfies the criteria of
normality and convexity. And if it this satisfies, we can say we are getting a fuzzy number
C; but I as I said as I mentioned that, this fuzzy set C which is a result of the division of
fuzzy A and B, we do not always get the fuzzy number. So, that is why we need to check
before we say that C is a fuzzy number.

So, here in this case it looks like it is a fuzzy number. So, C is a fuzzy number, because
you see this is a normal fuzzy set and then we can check with convexity and if it this
follows the convexity, we can say C fuzzy set is a fuzzy number.

(Refer Slide Time: 16:33)

So, now let us find the 𝐵𝐵 ÷ 𝐴𝐴. So, let us see whether we are getting the same result as we
have gotten for the fuzzy number A divided by fuzzy number; of course we will not,
because in crisp number also when we do like this, we are not getting the same. So, here
also let us now divide fuzzy number B by another fuzzy number A and see what we are
getting.

394
(Refer Slide Time: 17:06)

So, if we do the same exercise here, we get the first element of the resultant fuzzy set as
0.3/0.5. And we know how are we getting this; we are taking the first element of the fuzzy
number B and we take first element of fuzzy number A and with this combination, we get
all 𝜇𝜇𝐵𝐵 (𝑥𝑥 𝐵𝐵 ) and 𝜇𝜇𝐴𝐴 (𝑥𝑥 𝐴𝐴 ) and their corresponding generic variable values. And this way we
are getting 0.3/0.5 as the first element or I would say one of the elements of the resultant
fuzzy set C.

(Refer Slide Time: 18:16)

395
(Refer Slide Time: 18:20)

(Refer Slide Time: 18:24)

So, likewise we find all such elements with the same exercise and we see that, finally we
are getting after applying the max criteria to avoid the conflicts for the same generic
variable values. So, we see that we are getting 0.5/0.16667.

Similarly, 0.5/0.25, then we get 1.0/0.333 here and then we get 0.6/0.5, then we get here
0.5/0.6667, here we get another term which is 0.5/1 and then we get 0.3/2. So, we have
got here this out of the division of fuzzy number B by fuzzy number A.

396
(Refer Slide Time: 19:26)

(Refer Slide Time: 19:27)

So, now let us quickly go ahead and plot these fuzzy numbers. So, we had taken the fuzzy
number B first and then fuzzy number A and then when we divide it the fuzzy number C
which is the result of the fuzzy number B divided by fuzzy number A, so we are getting
this as a plot.

397
(Refer Slide Time: 19:59)

But you see why are we getting this plot here? Here we had so many values as we have
seen here as I mentioned. So, please now let us look at the universe of discourse. So, the
universe of discourse that is given to us here in the, this example is the natural number.
So, we see that we have only out of these elements, we have only two natural numbers
here as the generic variable values, rest others do not qualify. So, that is how we discard
other elements and only we are keeping here 0.5/1 and 0.3/2 as the elements of the
resultant fuzzy set

So, let us now quickly see that, here we have only two elements of C, we are dealing with
the universe of discourse which is natural number here. So, that’s how other elements do
not qualify and then we are only left with only two elements. And when we have plotted
this we see that, we have getting this as a result which is not a normal discrete fuzzy set.
So, we can say that, the C is not a fuzzy number here. So, this is not a fuzzy number,
because the normality condition is not satisfied.

398
(Refer Slide Time: 21:45)

So, this way we have been able to know that, the fuzzy set that is we see that is we are
going to get out of the division is not always a fuzzy number.

And another point that I would like to make here is, if you recall what we have gotten
when we divided fuzzy number A by fuzzy number B, we have got a fuzzy number C
which is shown here. So, this is fuzzy number C, this is a full C fuzzy set and here in this
case when we have divided fuzzy number B by fuzzy number A, we have gotten C here
like this. So, we can clearly say that, 𝐴𝐴 ÷ 𝐵𝐵 ≠ 𝐵𝐵 ÷ 𝐴𝐴 when we have A and B as fuzzy
numbers and the same is true for the crisp sets as well.

399
(Refer Slide Time: 23:11)

Now, by now we have finished all the arithmetic operations, the addition of two fuzzy
numbers, subtraction of two fuzzy numbers, multiplication of two fuzzy numbers, division
of two fuzzy numbers. So, I would like to take up this example here, because by now we
have understood as to how we undertake the arithmetic operations on fuzzy numbers.

So, let us take this example, where we have (𝐴𝐴 + 𝐵𝐵)2 ≠ 𝐴𝐴2 + 𝐵𝐵2 + 2𝐴𝐴𝐴𝐴, as we have this
true for crisp numbers. So, when we deal with crisp numbers, (𝐴𝐴 + 𝐵𝐵)2 = 𝐴𝐴2 + 𝐵𝐵2 +
2𝐴𝐴𝐴𝐴, but here we will see, when we take fuzzy numbers, this is not going to be the true.

(Refer Slide Time: 24:39)

400
So, let us now first try to get (𝐴𝐴 + 𝐵𝐵)2 . So, here we have taken discrete fuzzy set here. So,
we have two discrete fuzzy sets A and B, A and B and these fuzzy sets are fuzzy numbers
as you can see. So, these two are the fuzzy numbers and let us now find the addition of
these two fuzzy numbers.

(Refer Slide Time: 25:12)

So, first what we are going to get here as 𝐴𝐴 + 𝐵𝐵 is here, I am not going to explain each and
every step of this addition, because we have already done this in detail. And here I am
quickly going to show you the 𝐴𝐴 + 𝐵𝐵 which is here, you can see. So, when we do the
addition; when we add 𝐴𝐴 + 𝐵𝐵, we are going to get this as the result. So, this is our 𝐴𝐴 + 𝐵𝐵.
So, when we have 𝐴𝐴 + 𝐵𝐵, now we need to get the (𝐴𝐴 + 𝐵𝐵)2 . So, let us now multiply 𝐴𝐴 + 𝐵𝐵
by 𝐴𝐴 + 𝐵𝐵 and see what is happening. So, when we do that here finally we are getting as a
result 0.7/(−1) + 0.5/0 + 1.0/1 + 0.7/2.

So, this way you see here that, we are able to find (𝐴𝐴 + 𝐵𝐵)2 , when A and B are the fuzzy
numbers.

401
(Refer Slide Time: 26:56)

Now, let us go to the right hand side of the expression. So, earlier was the LHS, the left
hand side of the expression. So, this was the LHS, I will write here LHS.

(Refer Slide Time: 27:24)

So, let us work for the RHS, the right hand side of the expression. So, RHS is here, in RHS
we have the first term as 𝐴𝐴2 . So, 𝐴𝐴2 is here, A multiplied by A. Once again, I would like
to tell you that this A is a fuzzy number.

402
So, A multiplied by A is resulting as 0.5/(−1) + 1.0/0 + 0.7/1. So, this is how we are
getting these three terms as a square.

(Refer Slide Time: 28:09)

Similarly, we can find here the 𝐵𝐵2 and as 𝐵𝐵2 we are getting here, 0.7/(−2) + 0.5/(−1) +
1.0/1 + 0.5/2 + 0.7/4; but understand here we need to know here that the given universe
of discourse here is −1,2, means all the points that we will be dealing with should be right
from −1 to up to 2.

So, this is an important point that we should always check the universe of discourse and
all the results, all the generic variable values that are that we are using must be lying within
the universe of discourse. So, when we see 𝐴𝐴2 and 𝐵𝐵2 , so in 𝐴𝐴2 we are fine, but in 𝐵𝐵2 we
see that, there are some elements, there are some terms that are not within the universe of
discourse, means there are some terms whose generic variable values are not lying within
the universe of discourse. So, now, what we do here is that, such elements we will be
discarding.

403
(Refer Slide Time: 29:55)

So, let us now rewrite the resulting fuzzy set 𝐵𝐵2 and here we have only 0.5/(−1) + 1/1 +
0.5/2.

Now, let us go ahead and find the AB? AB is nothing, but the fuzzy number A multiplied
by fuzzy number B. So, when we do that here, we find the resulting fuzzy set as this and
again when we apply the criteria of universe of discourse that all the generic variable
values must be from the or must be lying within the universe of discourse.

(Refer Slide Time: 30:47)

404
So, when we apply this, we see that we are getting the A multiplied by B has 0.5/(−1) +
1/0 + 0.7/1 + 0.5 /2. Now we have to find the twice of this thing. So, for twice of this
thing, we will add these two fuzzy sets that we have just gotten as A into B. So, A into B
plus A into B see here.

(Refer Slide Time: 31:50)

And when we apply the addition in between the AB and AB will become 2 AB. So, this 2
AB is going to give us a new fuzzy set which is here; this has four elements, so 0.5/−1 +
1/0 + 0.7/1 + 0.7/2. So, this way we have you see result of A square plus B square plus
2 AB as, 0.7/(−1) + 0.7/0 + 0.7/1 + 0.7/2.

405
(Refer Slide Time: 32:39)

So, this is what is the expression that we are getting as a result of 𝐴𝐴2 + 𝐵𝐵2 + 2𝐴𝐴𝐴𝐴, which
is nothing, but the RHS. So, we are writing here, this was our LHS and this was our RHS.
And when we compare these two, we can clearly see that these two are not equal, means
the RHS or LHS is not equal to RHS.

So, this we can better judge when we plot these two, the LHS and RHS and compare. So,
here we have plotted the resultant fuzzy set, which is LHS which is (𝐴𝐴 + 𝐵𝐵)2 , which is
coming like this. So, here x axis is the genetic variable and y axis the membership values
are grades, we can write it like this and this is nothing but here A plus B whole square.

So, we can clearly see that these two, the LHS and RHS are not equal. So, by plotting these
we can very clearly visualize. And this way we can say that, although (𝐴𝐴 + 𝐵𝐵)2 = 𝐴𝐴2 +
𝐵𝐵2 + 2𝐴𝐴𝐴𝐴, but when we take fuzzy numbers (𝐴𝐴 + 𝐵𝐵)2 ≠ 𝐴𝐴2 + 𝐵𝐵2 + 2𝐴𝐴𝐴𝐴. So, you see the
difference that, this does not hold good for fuzzy numbers and the same holds good for the
crisp numbers.

406
(Refer Slide Time: 35:07)

So, we will stop here in this lecture, and in the next lecture we will study the fuzzy
compliments.

Thank you.

407
 Fuzzy Sets, Logic and Systems and Applications
Prof. Nishchal K. Verma
Department of Electrical Engineering
Indian Institute of Technology, Kanpur

Lecture - 23
Complement of Fuzzy Sets

Welcome to lecture number 23 of Fuzzy Sets, Logic and Systems and Applications. So, in
this lecture, we will discuss the Complement of Fuzzy Sets. Although, we have already
discussed in one of our previous lectures, the basic type of complement of fuzzy sets, but
here, we will discuss the complement of fuzzy sets in more detail. So, let us now take up
a fuzzy set, whose complement we say is A bar. So, the complement A bar is going to be
obtained by operating a complement function.

(Refer Slide Time: 01:06)

So, here the c, you can see here. The 𝑐𝑐 is such that this is going to give us when it is
operated on any membership value; we are going to get all those values in between or right
from 0 to 1. So, what does this mean? This means that, if we operate the 𝑐𝑐 which is a
complement which is, 𝑐𝑐 is a complement operator and if this is operated on any
membership value and this membership value will be anywhere from 0 to 1 and this is
going to give us again the value which is from 0 to 1 somewhere in between. So, we can
say let 𝑐𝑐 which is a complement operator is such that interval 0 to 1 is returning us the

408
value in between 0 to 1. So, this be a mapping function that transforms the membership
function of a fuzzy set A into the membership function of a complement of fuzzy set.

So, here we have written membership function. So, this is applicable when we are dealing
with a continuous fuzzy set. But if we are dealing with the discrete fuzzy set, then we will
use the membership values or membership grades. So, we can see here that if we apply the
complement operator 𝑐𝑐. So, let me right here c is the complement operator. So, 𝑐𝑐 is a
complement operator, when this c is applied to the membership value or membership
function in case of continuous fuzzy set, it is returning us 𝜇𝜇𝐴𝐴̅ (𝑥𝑥). This means that this going
to give us the membership value corresponding to the complement of fuzzy set and this
complement of fuzzy set is denoted as 𝐴𝐴̅.

So, 𝐴𝐴̅ is nothing but I will I will write it here. 𝐴𝐴̅ is nothing but it is the complement of fuzzy
set A. And what we have studied so far for the complement is the basic complement of a
fuzzy set is here like if we have membership function or membership value and when we
want to have the complement for finding the complement of a fuzzy set, we are going to
simply subtract all these membership values, membership function from 1 to get the
membership values or functions of the complement of fuzzy set A.

So, this is simply denoted by I will just make a box here so that, this can be clearly
understood as to what is the basic fuzzy complement. So, but apart from basic complement
we have so many other complement operators. So, in order for the function c that is the
fuzzy complement to be qualified as fuzzy complement, it should satisfy certain criteria,
certain axioms. So, here we have two axioms. Axiom number 1, Axiom number 2, these
are designated by 𝑐𝑐1 and 𝑐𝑐2. So, let us now look at 𝑐𝑐1.

So, c 1 says that the axiom c 1 says that if we apply the complement operators c on 0, we
are going to get 1. We should be getting one and if we apply this complement operator to
1, where we are we should be getting 0 and this is also called as the Boundary condition.
So, this is what is the axiom 𝑐𝑐1. Now, let us now look at the 𝑐𝑐2, axiom 𝑐𝑐2. So, axiom 𝑐𝑐2
says that if we have membership values 𝜇𝜇𝑥𝑥1 , 𝜇𝜇𝑥𝑥2 and of course, these 𝑥𝑥1 and 𝑥𝑥2 are from
the universe of this course and 𝜇𝜇𝑥𝑥1 , 𝜇𝜇𝑥𝑥2 should belong to the range 0 to 1. So, if 𝜇𝜇𝑥𝑥1 ≤ 𝜇𝜇𝑥𝑥2 ,
then the 𝑐𝑐 of 𝜇𝜇𝑥𝑥1 should be greater than or equal to 𝑐𝑐 of 𝜇𝜇𝑥𝑥2 . So, c here is the complement
operator.

409
So, what does this mean? This means that a complement operator or I would say the c is
some operator. So, if this c is following these two axioms 𝑐𝑐1, 𝑐𝑐2 then, we can say the c is
a complement operator. So, c for c to be called as the complement operator, the c should
satisfy axiom 𝑐𝑐1 and axiom 𝑐𝑐2. So, if these two axioms are satisfied, we can say the
operator that we are taking as c here is a complement operator. So, in other words we can
also say as its mentioned here that any function c, either we say function or operator both
conveys the same meaning.

So, any function 𝑐𝑐 such that range 0 to 1; this for the membership value. So, if it is applied
to some membership function or membership value which is right from the 0 up to 1 is
going to be again the resulting membership value for the complement fuzzy set right from
0 to 1. And of course, the function c should satisfy the axiom 𝑐𝑐1 and 𝑐𝑐2. So, this needs to
be clearly understood that if we take any complement, if we take any function or any
operator say that this is a complement operator, then we need to first check the axiom 𝑐𝑐1
and 𝑐𝑐2. And if these 2 axioms are satisfied, then this function or this operator will be
regarded as the complement operator. And this is going to give us the membership values
or membership function for the corresponding complement of the fuzzy set that has been
taken.

(Refer Slide Time: 09:40)

Then, we have some other complements. We have a Sugeno’s class of complement here
and which is written here which is mentioned as the 𝑐𝑐𝜆𝜆 (𝜇𝜇(𝑥𝑥)), which is nothing but is

410
 1−𝜇𝜇(𝑥𝑥)
equal to , where, 𝑐𝑐𝜆𝜆 here is the Sugeno’s complement. So, 𝑐𝑐𝜆𝜆 is Sugeno’s
1+𝜆𝜆𝜆𝜆(𝑥𝑥)
1−𝜇𝜇(𝑥𝑥)
complement. So, Sugeno’s class of complement is 𝑐𝑐𝜆𝜆 �𝜇𝜇(𝑥𝑥)� = , where 𝜆𝜆 values
1+𝜆𝜆𝜆𝜆(𝑥𝑥)

will be in the range −1 𝑡𝑡𝑡𝑡 ∞. So, this has to be understood. And please understand here
that since this is a complement, so this Sugeno’s class of complement is also satisfying the
axioms 𝑐𝑐1 and 𝑐𝑐2 and that is how it is regarded as it is called as the Complement operator.

Now, if we take 𝜆𝜆 = 0. So, let us see what do we get. So, when we take 𝜆𝜆 = 0, the value
of lambda 0, then the Sugeno’s class of complement becomes the basic fuzzy complement.
That is another interesting thing to note. So, here the lambda is equal to 0 means 𝑐𝑐0 . So,
you can see 0 here, this is for lambda is equal to 0. This is this becomes c is 𝑐𝑐0 𝜇𝜇(𝑥𝑥) = 1 −
𝜇𝜇(𝑥𝑥). So, when lambda is 0, the Sugeno’s class of complement gives us the basic
complement, the basic fuzzy complement that we have studied earlier.

(Refer Slide Time: 12:25)

Let us take an example here to understand better the Sugeno’s class of complement. So,
here we have an example, where we have a fuzzy set, discrete fuzzy set 𝐴𝐴 as 0.7/1 +
0.5/2 + 0.1/3 + 0.6/4. So, we have this fuzzy set and we need to find Sugeno’s class of
complement of this fuzzy set for the values which are listed here. So, values of lambda are
minus 0.8, 0, 1 and 2. So, let us first write the Sugeno’s class of complement here which
1−𝜇𝜇𝐴𝐴 (𝑥𝑥)
is nothing but 𝑐𝑐𝜆𝜆 �𝜇𝜇𝐴𝐴 (𝑥𝑥)� = which is here.
1+𝜆𝜆𝜇𝜇𝐴𝐴 (𝑥𝑥)

411
(Refer Slide Time: 13:32)

And let us now quickly take the value of lambda is equal to minus 0.8, which is here and
substitute this in the expression. The expression of the 𝑐𝑐𝜆𝜆 = 1 − 𝜇𝜇(𝑥𝑥). If it is A then, we
write here A and then, the we have here 1 + 𝜆𝜆𝜇𝜇𝐴𝐴 (𝑥𝑥). So, when we use this, we see that,
we are getting for the value of lambda is equal to −0.8, we are getting A complement as
0.6818/1 + 0.833/2 + 0.9783/3 + 0.7692/4.

So, let us now see what we getting? We have taken this fuzzy set A and here, we have
plotted all these values means we have shown this discrete fuzzy set A and let us see what
we are getting as the complement of this fuzzy set A for lambda is equal to minus 0.8. So,
we are getting this fuzzy set here which is shown here as the complement of fuzzy set A
that is A bar and this is for lambda is equal to −0.8.

412
(Refer Slide Time: 15:31)

Now, let us take the another value of lambda, which is 𝜆𝜆 is equal to 0 and let us see what
we are getting. So, here we have the fuzzy set 𝐴𝐴 which was given to us and then here the
fuzzy set A and then, we have 𝐴𝐴̅ here, the complement of 𝐴𝐴 here for 𝜆𝜆 is equal to 0 and we
see that when we plot, we see that this is basic complement, we can clearly see that each
and every membership value here of the complement fuzzy set, we are getting from 𝐴𝐴 by
subtracting with the respective membership values from 1.

(Refer Slide Time: 16:20)

413
So, similarly when we take the 𝜆𝜆 is equal to 1, let us see what we are getting here. So, this
is also shown here that 𝐴𝐴̅ is the complement of fuzzy set 𝐴𝐴 for 𝜆𝜆 is equal to 1. So, this is
𝑐𝑐𝜆𝜆 is the complement operator here. The Sugeno’s complement for 𝜆𝜆 is equal to 1 in other
way we can say and then here for 𝜆𝜆 is equal to 2. So, we are getting the complement
Sugeno’s complement for 𝜆𝜆 is equal to 2.

(Refer Slide Time: 16:44)

(Refer Slide Time: 16:54)

So, this way we are getting four different fuzzy sets as complements Sugeno’s complement
for different values of 𝜆𝜆. So, let us now plot all these outcomes, all these complement fuzzy

414
sets or the complement of fuzzy set A. And when we plot membership values here, when
we plot the membership values as to how they are changing with the various values of 𝜆𝜆
is equal to minus 0.8, 𝜆𝜆 is equal to 0, 𝜆𝜆 is equal to 1, 𝜆𝜆 is equal to 2.

So, when we plot these membership values and its complement membership values we see
that here we have this as 𝜆𝜆 is equal to −0.8 is and this is for 𝜆𝜆 is equal to 0 and this curve
we are getting as the 𝜆𝜆 is equal to 1 and then, we are getting this as the 𝜆𝜆 is equal to 2. So,
this means what? What we are trying to relate? So, we are trying to relate here like if we
increase the value of lambda, when we increase the value of lambda, we are going to get
the changes like this. We can clearly see here that the relationship is changing here the in
between the membership values and its corresponding complement of membership values.

(Refer Slide Time: 18:51)

Now, let us take another example here. Earlier we took the example of discrete fuzzy set,
now we take the example of continuous fuzzy set which is a triangular fuzzy set here whose
vertices are at 1 and 2 and then 3; this is 3. So, now let us find the Sugeno’s complement
of this fuzzy set for lambda values and lambda values again are from this set here. And
this means that lambda will take the value minus 0.8 and lambda is equal to can be again
the 𝜆𝜆 is equal to 0, 𝜆𝜆 is equal to 1, 𝜆𝜆 is equal to 2. So, like that. Let us now quickly go
ahead and see what we are getting when we take the 𝜆𝜆 is equal to −0.8 and what we are
getting here as the complement of this fuzzy set Sugeno’s complement of this fuzzy set.

415
(Refer Slide Time: 19:57)

So, again, we have this expression for computing the Sugeno’s complement of Sugeno’s
complement of membership function, membership values, because these membership
values will be needing for expressing the complement of the fuzzy set A. So, that is how
we use this expression, we use this formula for computing the membership values,
membership function for the complement of the fuzzy set A.

(Refer Slide Time: 20:36)

So, let us now quickly see as to how we are getting it. So, we have a fuzzy set A as given.
This fuzzy set is a triangular fuzzy set and when we take 𝜆𝜆 is equal to −0.8, we are getting

416
the Sugeno’s complement you can see. And then, when we take the lambda is equal to 0,
the complement of this fuzzy set which is expressed by the inverted triangular membership
function.

So, here for lambda is equal to 0, we see that we are taking the basic complement. We are
applying the basic complement, which is nothing but 1 − 𝜇𝜇(𝑥𝑥) means the all the
corresponding membership values we get by just subtracting from highest membership
value that is 1. So, all the corresponding membership values of the complement of fuzzy
set, we get by just subtracting the values of the fuzzy set we are taking from 1 which we
can see here.

(Refer Slide Time: 21:50)

Now, similarly we can go ahead and we can use lambda is equal to 1 and we can get the
membership values accordingly and then, we can plot the A complement by taking all the
respective membership values or membership function here in case of a continuous
because this is continuous. So, we can we can use the term membership function.
Similarly, for lambda is equal to 2, we are getting A complement like this. So, we see that
these the complement of the fuzzy set that we are getting for different values of lambda,
we verify two axiom 𝑐𝑐1 and 𝑐𝑐2 for complement are satisfied.

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(Refer Slide Time: 22:38)

Now, here we have another class of complement which is Yager’s class of complement.
So, Yager’s class of complement is defined by the first of all Yager’s class of complement
a complement operator here is 𝑐𝑐𝑤𝑤 . So, this is called Yager’s complement. Yager’s
complement and this is defined as 𝑐𝑐𝑤𝑤 𝜇𝜇𝐴𝐴 (𝑥𝑥) = (1 − 𝜇𝜇𝐴𝐴 (𝑥𝑥)𝑤𝑤 )1/𝑤𝑤 . Here, the 𝑤𝑤 values can
be anywhere in between 0 𝑡𝑡𝑡𝑡 ∞. So, or in other words, we can say the 𝑤𝑤 can take any
value from 0 𝑡𝑡𝑡𝑡 ∞.

So, from for each value of the parameter w, we obtain the complement of fuzzy set and it
is easier to check that Yager’s class of complement satisfies the axioms 𝑐𝑐1 and 𝑐𝑐2 that we
have just discussed. And here also if we take 𝑤𝑤 is equal to 1, we are landing up in the basic
fuzzy complement. So, we can see here if we take 𝑤𝑤 is equal to 1, what we are getting here
is the 𝑐𝑐𝑤𝑤 𝜇𝜇𝐴𝐴 (𝑥𝑥) = 1 − 𝜇𝜇𝐴𝐴 (𝑥𝑥). So, 𝑐𝑐𝑤𝑤 becomes here, 𝑐𝑐𝑤𝑤 becomes the basic fuzzy
complement operator. So, when it becomes basic complement operator, when you have 𝑤𝑤
is equal to 1. So, for 𝑤𝑤 is equal to 1. So, for 𝑤𝑤 is equal to 1, 𝑐𝑐𝑤𝑤 becomes basic fuzzy
complement operator.

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(Refer Slide Time: 25:15)

Now, let us understand Yager’s class of complement better and let us take an example and
see what is happening, how these complement of this fuzzy set A look like by going
through different values of w’s. So, we have the fuzzy set here. The discrete fuzzy set here
again, so we have taken a discrete fuzzy set and we need to find its complement. So, as we
are interested in Yager’s complement, we have to use this, we have to use this formula
here.

(Refer Slide Time: 26:13)

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So, let us now apply this formula for 𝑤𝑤 is equal to 0.5 which is here and when we apply
this we get 𝑐𝑐0.5 = (1 − 𝜇𝜇𝐴𝐴 (𝑥𝑥)0.5 )2 which is here and this way when we compute this
finally, we are going to get this value which is mentioned here that is 0.0267/1. Similarly,
other values other corresponding values of membership for 𝑤𝑤 is equal to 0.5, we get here
and then when we plot, we see that the complement of 𝐴𝐴 for 𝑤𝑤 is equal to 0.5 looks like
this.

(Refer Slide Time: 27:21)

Next, when we take up the 𝑤𝑤 is equal to 1. So, for 𝑤𝑤 is equal to 1, we apply the same
formula and the complement here complement of the fuzzy set A, we are getting as this.
And since, we have taken here 𝑤𝑤 is equal to 1, this is returning us the basic complement.
So, this is after basic this basic complement. So, we can clearly see that these are the basic
complement.

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(Refer Slide Time: 28:07)

Now, when we take 𝑤𝑤 is equal to 2 we are getting here A complement. Then when we
plot, we are getting the 𝐴𝐴̅ for 𝑤𝑤 is equal to 2.

(Refer Slide Time: 28:23)

Similarly, for 𝑤𝑤 is equal to 3, we are getting here 𝐴𝐴 as the complement of fuzzy set A. So,
𝐴𝐴̅; 𝐴𝐴̅ is changed here.

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(Refer Slide Time: 28:40)

And again, when we plot when we try to plot the membership values of the set that we
have taken and then, the membership values of the complement set, complement of the set
𝐴𝐴 which is here, this is complement of the set A and yes, this has to be noted that this
complement is Yager’s class of complement or Yager’s complement. So, when we do that
we are when we plot, we are going to get this kind of relationship. So, this is for 𝑤𝑤 is equal
to this is for 𝑤𝑤 is equal to 0.5 and then this for 𝑤𝑤 is equal to 1 and this is for 𝑤𝑤 is equal to
1, then this is for 𝑤𝑤 is equal to 2 and this is for 𝑤𝑤 is equal to 3.

So, what does this mean? This means that, when we are increasing the value values of w’s,
we are moving towards this and when we are this is increasing and I can say here that this
is decreasing. So, this is how we are getting this kind of relationship in between the
membership values, membership functions and its complement membership values and
membership functions.

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(Refer Slide Time: 30:37)

Let us take another example which is a continuous fuzzy set. Here we would like to find
the Yager’s class of complement on the continuous fuzzy set A. And we have here the 𝑤𝑤,
set of w’s that will be taking and let us see what we are getting, what we are getting as the
output of the Yager’s complement of the fuzzy set A.

(Refer Slide Time: 31:15)

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(Refer Slide Time: 31:22)

So, let us now apply Yager’s complement and see what we are getting. So, when we have
applied we first took the fuzzy set A’s which is a continuous fuzzy set A, which is a
triangular fuzzy set A whose vertices are at 1, 2 and 3. And then when we apply 𝑤𝑤 is equal
to 0.5, we are getting a complement fuzzy set of A, which is like this. Similarly, when we
when we take 𝑤𝑤 is equal to 1, we get completely inverted fuzzy set, inverted of the original
fuzzy set and this is as I have already discussed that 𝑤𝑤 is equal to 1 for Yager’s class of
complement is going to give us the basic complement. So, this is a basic complement basic
complement of a fuzzy set that we have taken as A.

(Refer Slide Time: 32:24)

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Next is for 𝑤𝑤 is equal to 2, we are getting a bar like this the complement of fuzzy set like
this, the Yager’s complement of fuzzy like this. Here also we are getting Yager’s
complement of the fuzzy set A for 𝑤𝑤 is equal to 3 and this way, we have seen that as to
how we can manage to compute the Yager’s class of the complement of the fuzzy set, any
fuzzy set either continuous or discrete very easily.

So, this way we have seen Sugeno’s class of complement and Yager’s class of complement
and we have also seen in Sugeno’s class of complement for 𝜆𝜆 is equal to 0, we are getting
this operators change to the basic fuzzy complement. And similarly, in Yager’s class of
complement, we have seen that for 𝑤𝑤 is equal to 1, the Yager’s complement changes to
the basic fuzzy complement.

So now, with this we will stop here and in the next lecture, we will discuss t-norm operators
on fuzzy sets.

Thank you.

425
 Fuzzy Sets, Logic and Systems and Applications
Prof. Nishchal K. Verma
Department of Electrical Engineering
Indian Institute of Technology, Kanpur

Lecture – 24
T-norm Operators

So, welcome to lecture number 24 of Fuzzy Sets, Logic and Systems and Applications.
So, in this lecture, we will discuss T-norm Operators. This is also called S-conorm
operator.

(Refer Slide Time: 00:30)

Now, understand what is a T-norm or S-conorm operator. So, let this be defined by capital
T and this is nothing but mapping function that transforms the membership function of any
fuzzy set. So, here we have taken let’s say two fuzzy sets A and B into the membership
function of the T-norm of fuzzy sets A and B with the universe of discourse capital X.

So, this can be defined by this expression here, and please read this as the operator T of
𝜇𝜇𝐴𝐴 (𝑥𝑥), that is and 𝜇𝜇𝐵𝐵 (𝑥𝑥). So, this means that if we apply T-norm or S-conorm here on
pairs of membership values 𝜇𝜇𝐴𝐴 (𝑥𝑥) and 𝜇𝜇𝐵𝐵 (𝑥𝑥) this is going to be 𝜇𝜇𝐴𝐴∩𝐵𝐵 (𝑥𝑥), for every 𝑋𝑋
belongs to the universe of discourse. And it is very clear here that the 𝜇𝜇𝐴𝐴 (𝑥𝑥) you can see
here, and 𝜇𝜇𝐵𝐵 (𝑥𝑥) these two as I already mentioned they denote the membership function
values for fuzzy sets A and B respectively.

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Please also understand that T-norm can be represented by a triangle sign, which is open
triangle like this here, this a T-norm or S-conorm. So, wherever we want to write a T-norm
we can either use capital T or we can simply use this open triangle sign or symbol for this
thing.

So, we can say open triangle symbol is the T-norm operator which is mentioned over here
and T-norm is also known as S-conorm which I have already mentioned. So, let us
understand few more things related to T-norm or S-conorm.

(Refer Slide Time: 02:55)

So, we have 4 axioms and these axioms are namely axiom T1, axiom T2, axiom T3, axiom
T4. So, the axiom T1 is boundary condition, this means that if we take T-norm of 0 and 0,
as I have already mentioned both the 0s are the membership values, so this is going to give
us 0. So, if we apply the T-norm on two membership values, here these two membership
values are 0, 0, means the lowest values and this is going to return as the 0 which is again
a membership value. I can use the symbol open triangle symbol for T-norm or S-conorm.

So, I can write the same thing like this or this can also be written as the 0 ∧ 0 which is
going to give us 0. Now, if we apply the T-norm or we can say in other words, if we take
T-norm of any membership value which is represented by 𝜇𝜇𝐴𝐴 (𝑥𝑥) here and then we have 1.
So, if we take these two membership values, first one is 𝜇𝜇𝐴𝐴 (𝑥𝑥) and the second one is 1 and
please note that this 1 is the highest value which a membership value can attain.

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So, if we have these two together and if we apply the T-norm or we take T-norm, this is
going to give us the value which is a 𝜇𝜇𝐴𝐴 (𝑥𝑥) which is here and this can also be written by
𝑇𝑇[1, 𝜇𝜇𝐴𝐴 (𝑥𝑥)]. So, this means that whatever membership value that 𝜇𝜇𝐴𝐴 (𝑥𝑥) can attain right,
this does not matter if we have 1. So, whatever value that 𝜇𝜇𝐴𝐴 (𝑥𝑥) has will be returned if we
take 𝑇𝑇[𝜇𝜇𝐴𝐴 (𝑥𝑥)] and 1. And please note, 1 is the highest value of any membership value
which can be there in the in case of fuzzy sets, alright.

So, we see here that axiom T1 in boundary condition we have the lowest value and we
have the highest value. The lowest value here we have taken 0s in case of highest value
we have taken 1. So, this boundary condition needs to be clearly understood.

Now, let us understand axiom T2. This is commutativity property of the T-norm. So, when
we have two membership values, T of 𝜇𝜇𝐴𝐴 (𝑥𝑥) and 𝜇𝜇𝐵𝐵 (𝑥𝑥) this can also be written as T of
𝜇𝜇𝐵𝐵 (𝑥𝑥) and 𝜇𝜇𝐴𝐴 (𝑥𝑥). Means, the order can be interchanged, ok.

So, next is the axiom T3, the third axiom. So, which is non decreasing property. We see
here, if we have 𝜇𝜇𝐴𝐴 (𝑥𝑥) ≤ 𝜇𝜇𝐵𝐵 (𝑥𝑥); means that the value of 𝜇𝜇𝐴𝐴 (𝑥𝑥) ≤ 𝜇𝜇𝐵𝐵 (𝑥𝑥) 𝜇𝜇𝐴𝐴 (𝑥𝑥) and
𝜇𝜇𝐶𝐶 (𝑥𝑥) ≤ 𝜇𝜇𝐷𝐷 (𝑥𝑥), then we have triangular norm or the T-norm of 𝜇𝜇𝐴𝐴 (𝑥𝑥) and 𝜇𝜇𝐶𝐶 (𝑥𝑥) will be
less than or equal to triangular norm of.

So, I am saying triangular norm, so either T-norm or triangle, this both these terms are
used interchangeably. So, we can either use T-norm or we can use triangular norm or we
can use S-conorm. So, please understand all these 3 terms we will be using
interchangeably.

So, this way I am repeating that if this is the condition which is satisfied, means if we have
𝜇𝜇𝐴𝐴 (𝑥𝑥) ≤ 𝜇𝜇𝐵𝐵 (𝑥𝑥) and 𝜇𝜇𝐶𝐶 (𝑥𝑥) ≤ 𝜇𝜇𝐷𝐷 (𝑥𝑥) then the triangular norm of 𝜇𝜇𝐴𝐴 (𝑥𝑥) and 𝜇𝜇𝐶𝐶 (𝑥𝑥) will be
less than or equal to triangular norm of 𝜇𝜇𝐵𝐵 (𝑥𝑥) and 𝜇𝜇𝐷𝐷 (𝑥𝑥). So, this is called the non-
decreasing property.

Now, the next axiom is the T 4 which is for associatively property. So, when we have 3
membership values say 𝜇𝜇𝐴𝐴 (𝑥𝑥), 𝜇𝜇𝐵𝐵 (𝑥𝑥), 𝜇𝜇𝐶𝐶 (𝑥𝑥), these 3 membership values if we have, so
we can apply the triangular norm like this that the triangular norm of, the triangular norm
of 𝜇𝜇𝐴𝐴 (𝑥𝑥) and 𝜇𝜇𝐵𝐵 (𝑥𝑥) and 𝜇𝜇𝐶𝐶 (𝑥𝑥) you can see here clearly, this is going to be equal to the
triangular norm of 𝜇𝜇𝐴𝐴 (𝑥𝑥) and the triangular norm of 𝜇𝜇𝐵𝐵 (𝑥𝑥) and 𝜇𝜇𝐶𝐶 (𝑥𝑥). Please understand

428
that this 𝜇𝜇𝐴𝐴 (𝑥𝑥), 𝜇𝜇𝐵𝐵 (𝑥𝑥), 𝜇𝜇𝐶𝐶 (𝑥𝑥), these 3 are the membership values of the respective fuzzy
sets A, B and C, so which is mentioned here.

And we have one more membership value which is 𝜇𝜇𝐷𝐷 (𝑥𝑥). So, D here signifies that we
have the membership value, which is with respect to the D fuzzy set. And this is needless
to say that all of these 𝑥𝑥 the generic variable that has been included in all the fuzzy sets
that we have just seen, the respective membership values that we have used the 𝑥𝑥 here the
generic variable here is belonging into the universe of discourse. So, now we can clearly
say that any function 𝑇𝑇: [0,1] × [0,1] → [0,1].

If it is a triangular norm this is going to satisfy all the axioms of the T-norm. So, we can
say axioms T 1 to T; we can say that this operator is qualifying to be called as the T-norm
or S-conorm. Once again we can also say this as the triangular norm. So, this way we
understand that how these axioms needs to be satisfied before we call T as the triangular
norm.

(Refer Slide Time: 11:44)

Now, there are 4 commonly used T-norm operators that we will be seeing here. So, we
have the minimum and this is defined as the T min of 𝜇𝜇𝐴𝐴 (𝑥𝑥) and 𝜇𝜇𝐵𝐵 (𝑥𝑥) or can be written
as min of 𝜇𝜇𝐴𝐴 (𝑥𝑥) and 𝜇𝜇𝐵𝐵 (𝑥𝑥). So, what we are seeing here is that, we have replaced T min
by min. So, this is when we are dealing with the T-norm operator as the minimum case.
So, this is there are 4 cases here, there are 4 commonly used T-norm operators. So,

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minimum is a one of the operators, if we are interested in a minimum T-norm operator
then we will simply replace T min by min.

So, let us say if we are interested in finding the minimum T-norm operator. The minimum
T norm operator 𝑇𝑇𝑚𝑚𝑚𝑚𝑚𝑚 �𝜇𝜇𝐴𝐴 (𝑥𝑥), 𝜇𝜇𝐵𝐵 (𝑥𝑥)� is going to be min�𝜇𝜇𝐴𝐴 (𝑥𝑥), 𝜇𝜇𝐵𝐵 (𝑥𝑥)� = 𝜇𝜇𝐴𝐴 (𝑥𝑥) ∧ 𝜇𝜇𝐵𝐵 (𝑥𝑥).
And please understand that we can write this min by using the ∧ symbol here. So, we can
see here that this open triangle symbol for T-norm as min operator.

Alright so, next is we have the T-norm as the algebraic product this is denoted by 𝑇𝑇𝑎𝑎𝑎𝑎 ,
small a, small p is here for algebraic product. So, 𝑇𝑇𝑎𝑎𝑎𝑎 (𝜇𝜇𝐴𝐴 (𝑥𝑥), 𝜇𝜇𝐵𝐵 (𝑥𝑥)) is going to give us
the membership values that are 𝜇𝜇𝐴𝐴 (𝑥𝑥) and 𝜇𝜇𝐵𝐵 (𝑥𝑥). So, this is simply here when we are
interested in T-norm as algebraic product, so we will simply multiply the membership
values.

Next is the bounded product. This represented by 𝑇𝑇𝑏𝑏𝑏𝑏 . So, T subscript bp, small b, small
p, here we have used the triangular norm, but this is inverted triangle, ∨. So, here we can
use this inverted triangle here which is used normally for max, like in the first case we use
the open triangle symbol here, this is the inverted open triangle. So, we see that we take
the (0 ∨ (𝜇𝜇𝐴𝐴 (𝑥𝑥) + 𝜇𝜇𝐵𝐵 (𝑥𝑥) − 1)). So, this way we can find the bounded product.

The fourth one is here the drastic product when we use this we write this as
𝑇𝑇𝑑𝑑𝑑𝑑 (𝜇𝜇𝐴𝐴 (𝑥𝑥), 𝜇𝜇𝐵𝐵 (𝑥𝑥)), so this is going to be like this is equal to 𝜇𝜇𝐴𝐴 (𝑥𝑥) when we have 𝜇𝜇𝐵𝐵 (𝑥𝑥)
equal to 1, and this is going to give us 𝜇𝜇𝐵𝐵 (𝑥𝑥) when 𝜇𝜇𝐴𝐴 (𝑥𝑥) is equal to 1. And this is going
to give us 0, when we have 𝜇𝜇𝐴𝐴 (𝑥𝑥) and 𝜇𝜇𝐵𝐵 (𝑥𝑥) less than 1. So, this way we understand how
these 4 commonly used T-norm operators are being defined.

Let us take an example here to understand T-norm operator better.

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(Refer Slide Time: 16:02)

So, we have taken here fuzzy set A here and another fuzzy set B here. Both the fuzzy sets
A B are triangular, continuous fuzzy sets. So, we see that these are defined by the fuzzy
set A and B mathematically defined. And what we are interested here in is that we are
interested in finding the intersection of fuzzy sets A and B using T-norm operator. So,
basically when we say T-norm operator so if nothing has been set so we use min operator
as T-norm.

(Refer Slide Time: 16:53)

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So, here we see that we have used minimum here because nothing has been set as the T-
norm operator and which we can find here as the for corresponding every value of each
and every value of 𝜇𝜇𝐴𝐴 (𝑥𝑥) and 𝜇𝜇𝐵𝐵 (𝑥𝑥) like this. So, we’ll take the minimum of each and
every corresponding values of 𝜇𝜇𝐴𝐴 (𝑥𝑥) and 𝜇𝜇𝐵𝐵 (𝑥𝑥) and you can see how it is found.

So, this way let us now proceed, and one more thing I would like to mention here that
when we say the minimum T-norm operator or T-norm operator as the minimum operator.
So, when we say this, this is also called a basic intersection operator. Whenever we are
using the minimum T-norm operator, we normally get the intersection of the fuzzy sets.

(Refer Slide Time: 18:09)

So, when we apply this condition of T-norm which is once again I would like to mention
that the minimum class.

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(Refer Slide Time: 18:44)

So, when we take the minimum T-norm operator we after applying the condition on fuzzy
set A and fuzzy set B, and when we superimpose these two fuzzy sets here together we see
that this is A, and this is B. So, when we apply this criteria we use the minimum criteria,
minimum condition for each and every values of fuzzy membership. So, we see that we
are getting this as the minimum, means the lower envelop we are getting as the outcome
of 𝑇𝑇𝑚𝑚𝑚𝑚𝑚𝑚 , which can be seen here that we are getting here as the output of intersection of
fuzzy set A and fuzzy set B.

We are saying intersection, but since we are studying the fuzzy T-norm in this lecture we
will use the term the 𝑇𝑇𝑚𝑚𝑚𝑚𝑚𝑚 . When we use 𝑇𝑇𝑚𝑚𝑚𝑚𝑚𝑚 it means we are getting the intersection of
the two fuzzy sets.

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(Refer Slide Time: 19:43)

Now, on the same fuzzy sets let us see what happens when we use algebraic product. So,
we have just seen that the 𝑇𝑇𝑎𝑎𝑎𝑎 we get by multiplying each and every membership values
of fuzzy set A and fuzzy set B. So, there is how written here. And when we do that and we
will multiply we are going to get this membership function I would say because here A
and B are continuous fuzzy sets after multiplying the corresponding membership values
of fuzzy set A and B we are getting a continuous membership function.

(Refer Slide Time: 20:09)

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So, it has been shown by the red color and we can see here that this is we are getting as
𝑇𝑇𝑎𝑎𝑎𝑎 (𝜇𝜇𝐴𝐴 (𝑥𝑥), 𝜇𝜇𝐵𝐵 (𝑥𝑥)).

(Refer Slide Time: 20:46)

Now, let us quickly understand the bounded product as the T-norm operator. So, keeping
these fuzzy sets same A and B, when we apply the bounded product let us see what we are
getting.

(Refer Slide Time: 21:06)

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So, we are getting here, a continuous curve which is the membership curve, which is the
resultant of the bounded product of membership values of A and B. So, I would say here
that each and every values of membership for A and B fuzzy sets. So, when we write this
separately, we see that we are getting here bounded product 𝑇𝑇𝑏𝑏𝑏𝑏 (𝜇𝜇𝐴𝐴 (𝑥𝑥), 𝜇𝜇𝐵𝐵 (𝑥𝑥)).

(Refer Slide Time: 21:55)

Similarly, when we see that drastic product, if we are interested and we apply the drastic
product conditions, the drastic product formula that we have just covered, so we see that
we are getting here this as the outcome of the drastic product of the membership function
of fuzzy set A and the membership function of fuzzy set B.

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(Refer Slide Time: 22:11)

And please understand, here we are getting only two values. So, here we are getting the
discreet values or I would say you are getting only two membership values as the outcome
of the drastic product of the continuous fuzzy sets after applying this condition. So, here
we can clearly see that we are getting drastic product as the mu A the value of 𝜇𝜇𝐴𝐴 (𝑥𝑥) is
equal to 1, if 𝜇𝜇𝐵𝐵 (𝑥𝑥) is equal to 1. So, we can see here this is our membership function for
fuzzy set A and this is the membership function for the fuzzy set B.

So, we can see here that we are going to get only this value here as the outcome at this we
have 𝜇𝜇𝐴𝐴 (𝑥𝑥) which is equal to 1, and at this condition here we have 𝜇𝜇𝐵𝐵 (𝑥𝑥) is equal to 1. So,
at both the points we have here we have 𝜇𝜇𝐴𝐴 (𝑥𝑥) is equal to 1. So, at this point we will have
𝜇𝜇𝐵𝐵 (𝑥𝑥) whatever value of 𝜇𝜇𝐵𝐵 (𝑥𝑥) is and then here we will get at a 𝜇𝜇𝐵𝐵 (𝑥𝑥) is equal to 1 and
for all other values we are going to get 0. So, that is why we are getting here only two
membership values out of the drastic product of the membership functions of continuous
fuzzy set A and continues fuzzy set B respectively.

437
(Refer Slide Time: 24:08)

Let us take another example where we are taking the discreet fuzzy sets A and B, and these
are represented by here the discrete points. So, we can see here fuzzy set A and fuzzy B.

(Refer Slide Time: 24:30)

And when we apply the 𝑇𝑇𝑚𝑚𝑚𝑚𝑚𝑚 criteria, like when we are interested in T-norm of the 𝜇𝜇𝐴𝐴 (𝑥𝑥)
and 𝜇𝜇𝐵𝐵 (𝑥𝑥) as the minimum. So, then we apply this criteria, similarly for 𝑇𝑇𝑎𝑎𝑎𝑎 we applied
this criteria and 𝑇𝑇𝑏𝑏𝑏𝑏 we apply this criteria, 𝑇𝑇𝑑𝑑𝑑𝑑 we apply this criteria.

So, when we apply this for 𝑇𝑇𝑚𝑚𝑚𝑚𝑚𝑚 when we apply we are going to get this as the outcome.

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(Refer Slide Time: 25:00)

You can see here for each and every values of membership for the fuzzy set A and B we
are taking min. So, you can clearly see here we are using min here. And when we take this,
this is the outcome that we are getting. And when we show it here as the fuzzy set you can
see that these values are plotted here and we see here that we get the 𝑇𝑇𝑚𝑚𝑚𝑚𝑚𝑚 of these two
discrete fuzzy set has shown here.

(Refer Slide Time: 25:51)

So, next is the 𝑇𝑇𝑎𝑎𝑎𝑎 of the same discreet fuzzy sets A and B. So, as you have already seen
these two fuzzy sets A and B, now when we take the algebraic product when we apply the

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this criteria, the multiplication and when we multiply the respective membership values
for the A fuzzy set and B fuzzy set we are getting this as the outcome which has come like
this and when we plot this we are getting.

So, now, the next is for the same fuzzy set we have bounded product and when we apply
this criteria, this formula for𝑇𝑇𝑏𝑏𝑏𝑏 , we are going to get here this as the outcome 0/1 +
0.3/2 + 0/3 + 0/4.

(Refer Slide Time: 26:29)

So, we can clearly see here that we are getting only one membership value out of many
membership values here, where we have in fuzzy set we have 1, 2, 3, 4, 4 membership
values and in fuzzy set we have two membership values, but as the outcome we are getting
only one membership value. So, when we are interested in the bounded product of these
two fuzzy sets, I would say that 𝑇𝑇𝑏𝑏𝑏𝑏 you are going to get this as the outcome.

440
(Refer Slide Time: 27:41)

Similarly, when we find the 𝑇𝑇𝑑𝑑𝑑𝑑 of the same fuzzy set, same discreet fuzzy sets A and B
we are getting here null fuzzy set. So, means, we are getting for the same fuzzy sets, we
are not going to get any output. You can clearly see here that we have all the elements of
this fuzzy sets discrete fuzzy set as the outcome, so each and every element has its
membership value 0, which we never account as the element of the fuzzy set. We only
account only those elements, which has the membership value more than 0. So, here we
can say we are getting a null fuzzy set out of these two when we are taking the 𝑇𝑇𝑑𝑑𝑑𝑑 of these
two discrete fuzzy sets.

(Refer Slide Time: 28:42)

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In the next lecture, we will study the S-norm which is also known as T-conorm.

Thank you very much.

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 Fuzzy Sets, Logic and Systems and Applications
Prof. Nishchal K. Verma
Department of Electrical Engineering
Indian Institute of Technology, Kanpur

Lecture – 25
S-norm Operators

So, welcome to lecture number 25 of Fuzzy Sets, Logic and Systems and Applications. In
this lecture we will study the S-norm Operators this also known as T-conorm operators.

(Refer Slide Time: 00:44)

So, S-norm or T-conorm operators are nothing but they are the mapping function that
transforms the membership function of fuzzy sets A and B into the membership function
of the S-norm of fuzzy set A and B with the universe of discourse capital X. Can be defined
as you see here the formula,

𝑆𝑆[𝜇𝜇𝐴𝐴 (𝑥𝑥), 𝜇𝜇𝐵𝐵 (𝑥𝑥)] = 𝜇𝜇𝐴𝐴∪𝐵𝐵 (𝑥𝑥), ∀𝑥𝑥 ∈ 𝑋𝑋

so S is just the operator and which is called the S-norm it is also called as T-conorm as I
just mentioned.

So, the S of the two membership values 𝑆𝑆[𝜇𝜇𝐴𝐴 (𝑥𝑥), 𝜇𝜇𝐵𝐵 (𝑥𝑥)] = 𝜇𝜇𝐴𝐴∪𝐵𝐵 (𝑥𝑥). So here when we are
taking the S-norm of two membership values that are coming from the two fuzzy sets A
and B this is returning us the membership value which is termed as the 𝜇𝜇𝐴𝐴∪𝐵𝐵 (𝑥𝑥). Why?

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Because this we are getting by taking the union of the two fuzzy sets as far as the
membership values are concerned.

So, here let us also understand that as in case of T-norm we have seen we were using open
triangle symbol. So, here in for S-norm we will use this triangle inverted open triangle
which is here. So, either S or inverted open triangle is used for the S-norm operator. S-
norm is also known as T-conorm as we have already discussed.

(Refer Slide Time: 02:43)

Now, on the same lines as we have discussed the T-norms in last couple of lectures. So,
we have here 4 axioms. So, any operator if it satisfies the all 4 axioms S1, S2, S3, S4, this
will qualify to be called as the S-norm or T-conorm. So, for any function S to be qualified
as a fuzzy union I am saying here fuzzy union because S-norm normally is giving us the
fuzzy union for its basic norms. Like we have seen in the basic T-norm is the intersection.

So, here the basic S-norm is the union. So, here we can say there is a function S to be
qualified as a fuzzy union it must satisfy at least the following four requirements these are
the four requirements and these requirements are the axioms, axioms S1, S2, S3, S4.

So, let me just quickly go through all these 4 axioms one by one. First one is the boundary
condition. So, here we have 1, 1. So, these 1,1 are the highest values of the membership
function of particular fuzzy set or I would say membership values of the fuzzy set. So, let

444
this be the membership value of a fuzzy set A and this is another fuzzy set let us say B.
So, both the membership values are coming from two different fuzzy sets.

So, if we take the S-norm of the these two fuzzy sets that is going to give us 1 as a result.
So, please understand that this m S-norm is applied on membership values. So, here S of
1 and 1 is going to give us 1 and then we have here another condition the 𝑆𝑆[𝜇𝜇𝐴𝐴 (𝑥𝑥), 0]
means if we apply the S-norms on two values two membership values one membership
value is 𝜇𝜇𝐴𝐴 (𝑥𝑥) and the other one is 0.

So, if this kind of condition happens like any membership value like 𝜇𝜇𝐴𝐴 (𝑥𝑥) you have and
it is with 0 lowest possible membership value. So, if this is with 0 what is going to be
returned here is the 𝜇𝜇𝐴𝐴 (𝑥𝑥). So, it means that if we have any membership value which is
with 0, if we take the S-norm of these two values we are going to get the value which is
with 0.

We are not getting 0 we are getting 𝜇𝜇𝐴𝐴 (𝑥𝑥) which is here this can also be written as S of or
S-norm of 0 comma 𝜇𝜇𝐴𝐴 (𝑥𝑥). And then we have the commutatively property if we take the
𝑆𝑆[𝜇𝜇𝐴𝐴 (𝑥𝑥), 𝜇𝜇𝐵𝐵 (𝑥𝑥)] = 𝑆𝑆[𝜇𝜇𝐵𝐵 (𝑥𝑥), 𝜇𝜇𝐴𝐴 (𝑥𝑥)]. So, this is the second axiom S2.

Now the third one is non decreasing, if we have 𝜇𝜇𝐴𝐴 (𝑥𝑥) ≤ 𝜇𝜇𝐵𝐵 (𝑥𝑥) and 𝜇𝜇𝐶𝐶 (𝑥𝑥) ≤ 𝜇𝜇𝐷𝐷 (𝑥𝑥) then
S-norm of 𝜇𝜇𝐴𝐴 (𝑥𝑥) and 𝜇𝜇𝐶𝐶 (𝑥𝑥) will be less than or equal to the S-norm of 𝜇𝜇𝐵𝐵 (𝑥𝑥) and 𝜇𝜇𝐷𝐷 (𝑥𝑥).
So, this condition is the non decreasing condition.

Now the 4th axiom S4 is the associativity condition. So, here we have
the𝑆𝑆[𝑆𝑆[𝜇𝜇𝐴𝐴 (𝑥𝑥), 𝜇𝜇𝐵𝐵 (𝑥𝑥)]𝜇𝜇𝐶𝐶 (𝑥𝑥)] = 𝑆𝑆[𝜇𝜇𝐴𝐴 (𝑥𝑥), 𝑆𝑆[𝜇𝜇𝐵𝐵 (𝑥𝑥), 𝜇𝜇𝐶𝐶 (𝑥𝑥)]]. So, this needs to be clearly
understood for three different membership values and these are coming from three
different fuzzy sets A B and C. And this is needless to say here that the generic variable
here is coming from the universe of discourse. So, that way now we can say any function
S if it satisfies all 4 axioms will be called as S-norm.

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(Refer Slide Time: 08:13)

So, let us now see as to how many commonly S-norm operators exist. So, we have four
commonly used S-norm operators and these are maximum, algebraic sum, bounded sum,
drastic sum. So, 𝑆𝑆 − 𝑛𝑛𝑛𝑛𝑛𝑛𝑛𝑛 or I would say the S-norm as maximum is represented by
𝑆𝑆𝑚𝑚𝑚𝑚𝑚𝑚 . So, 𝑆𝑆𝑚𝑚𝑚𝑚𝑚𝑚 of the two membership values 𝑆𝑆𝑚𝑚𝑚𝑚𝑚𝑚 �𝜇𝜇𝐴𝐴 (𝑥𝑥), 𝜇𝜇𝐵𝐵 (𝑥𝑥)� =
max�𝜇𝜇𝐴𝐴 (𝑥𝑥), 𝜇𝜇𝐵𝐵 (𝑥𝑥)� = 𝜇𝜇𝐴𝐴 (𝑥𝑥) ∨ 𝜇𝜇𝐵𝐵 (𝑥𝑥).

So, we simply take the maximum of these two and get the 𝑆𝑆𝑚𝑚𝑚𝑚𝑚𝑚 . So, this way we get the
S-norm of these two membership values. And here in this case we can also use the inverted
triangle as the symbol this is ∨ symbol. So, which is used here you can see. So, either we
write max�𝜇𝜇𝐴𝐴 (𝑥𝑥), 𝜇𝜇𝐵𝐵 (𝑥𝑥)� or we write 𝜇𝜇𝐴𝐴 (𝑥𝑥) ∨ 𝜇𝜇𝐵𝐵 (𝑥𝑥).

So, this is how the maximum S-norm is computed. This is also called the basic S-norm.
So, now in the next you see algebraic sum defined by 𝑆𝑆𝑎𝑎𝑎𝑎 �𝜇𝜇𝐴𝐴 (𝑥𝑥), 𝜇𝜇𝐵𝐵 (𝑥𝑥)� = 𝜇𝜇𝐴𝐴 (𝑥𝑥) +
𝜇𝜇𝐵𝐵 (𝑥𝑥) − (𝜇𝜇𝐴𝐴 (𝑥𝑥) × 𝜇𝜇𝐵𝐵 (𝑥𝑥)). So, this way we can compute the algebraic sum and the
representation here the symbol here is 𝑆𝑆𝑎𝑎𝑎𝑎 .

S is capital as is both are is small and these are in subscript of S. Then the third one comes
as the bounded sum S operator is presented by capital S, small b, small s as subscript. We
operate this on two membership values 𝑆𝑆𝑏𝑏𝑏𝑏 �𝜇𝜇𝐴𝐴 (𝑥𝑥), 𝜇𝜇𝐵𝐵 (𝑥𝑥)� = 1 ∧ (𝜇𝜇𝐴𝐴 (𝑥𝑥) + 𝜇𝜇𝐵𝐵 (𝑥𝑥)).

The fourth one and the last one here is the drastic sum as the S-norm, this is represented
by capital S and small d small s as the subscript. So, here it is 𝑆𝑆𝑑𝑑𝑑𝑑 .

446
 𝜇𝜇𝐴𝐴 (𝑥𝑥) 𝑖𝑖𝑖𝑖 𝜇𝜇𝐵𝐵 (𝑥𝑥) = 0
𝑆𝑆𝑑𝑑𝑑𝑑 �𝜇𝜇𝐴𝐴 (𝑥𝑥), 𝜇𝜇𝐵𝐵 (𝑥𝑥)� = �𝜇𝜇𝐵𝐵 (𝑥𝑥) 𝑖𝑖𝑖𝑖 𝜇𝜇𝐴𝐴 (𝑥𝑥) = 0
1 𝑖𝑖𝑖𝑖 𝜇𝜇𝐴𝐴 (𝑥𝑥), 𝜇𝜇𝐵𝐵 (𝑥𝑥) > 0

So, this way we can compute the drastic sum. So, we see here that we have these four S-
norm operators and depending upon the need we use these criterias to find the four S-norm
operators minimum, algebraic sum, bounded sum, drastic sum. And as I mentioned the
first one is the maximum which is the basic S-norm operator when nothing is said we
normally use the maximum S-norm operators we take the max of the two values.

(Refer Slide Time: 12:54)

So, let us take a simple example here and let us find the S-norm of these membership
values which are coming from the fuzzy set A and B. So, since these membership values
are coming from the continuous membership functions of the respective fuzzy sets A and
B.

So, here this fuzzy set A and membership function here is a continuous membership
function this is continuous I can write here this is a continuous membership function. So,
we can find any membership value with respect to its generic variable value for both the
cases here. So, these are the two membership functions that is coming out from a fuzzy set
A and fuzzy set B and there is needless to say that both the fuzzy sets here are continuous
fuzzy sets and these are defined by this two expressions.

447
(Refer Slide Time: 14:14)

So, let us now find the four S-norms of the respective membership values from the
continuous fuzzy sets A and B. So, let us first find the S-norm of these two fuzzy set. So,
as per the definition S max is nothing, but S max of any two membership value is going to
give us the max of these two membership values. So, let us see what is happening here in
this case. So, since we have here the continuous membership functions. So, you can get
any membership value for the corresponding generic variable value.

(Refer Slide Time: 15:06)

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So, here let us now super impose these two fuzzy sets because we are taking the maximum
of the two membership values corresponding to the generic variable value throughout and
within the universe of discourse. Here we are interested in max because we are computing
the max S-norm operator.

So, here we will take the maximum of the corresponding membership values within the
universe of capital X for the generic variable. So, when we are interested in max we super
impose these two membership functions. So, let us see what we are getting. So, when we
take the max of the two membership values throughout we are getting the outer envelope
here. So, this is represented by the red color.

So, when we take the max the 𝑆𝑆𝑚𝑚𝑚𝑚𝑚𝑚 is coming out to be like this. So, this is the max of the
two. So, since we have multiple membership values these two fuzzy sets are continuous
fuzzy sets. So, we have the continuous membership function as the outcome. So, this way
we see that we have computed the 𝑆𝑆𝑚𝑚𝑚𝑚𝑚𝑚 .

(Refer Slide Time: 16:38)

Now, let us look at the 𝑆𝑆𝑎𝑎𝑎𝑎 . So, when we apply this 𝑆𝑆𝑎𝑎𝑎𝑎 �𝜇𝜇𝐴𝐴 (𝑥𝑥), 𝜇𝜇𝐵𝐵 (𝑥𝑥)� = 𝜇𝜇𝐴𝐴 (𝑥𝑥) +
𝜇𝜇𝐵𝐵 (𝑥𝑥) − (𝜇𝜇𝐴𝐴 (𝑥𝑥) × 𝜇𝜇𝐵𝐵 (𝑥𝑥)) for the algebraic sum. So, what we are doing here is when we
are interested in finding the algebraic sum of the membership values 𝜇𝜇𝐴𝐴 (𝑥𝑥) and 𝜇𝜇𝐵𝐵 (𝑥𝑥). So,
we first add them together and then subtract the multiplication of these two. So, here when
we applied this formula for each and every pair of the membership values let us see what
we are getting.

449
(Refer Slide Time: 17:19)

So, when we do that we have you see here this is the first fuzzy set A, this the second fuzzy
set B and this is also characterized by corresponding membership functions that we can
call as 𝜇𝜇𝐴𝐴 (𝑥𝑥) and 𝜇𝜇𝐵𝐵 (𝑥𝑥).

(Refer Slide Time: 17:39)

So, when we apply this formula here means for all the pairs of the membership values
corresponding to the generic variable values when we take the 𝑆𝑆𝑎𝑎𝑎𝑎 apply this formula we
are going to get the membership function as a result which is represented by the red color,

450
you can see here. So, this is algebraic sum this is the outcome of the algebraic sum. So,
here we see that the algebraic sum is different from the max S-norm operator.

(Refer Slide Time: 18:27)

Now, the 𝑆𝑆𝑏𝑏𝑏𝑏 when we are taking the bounded sum of the corresponding membership
values what we are getting here after applying this formula 𝑆𝑆𝑏𝑏𝑏𝑏 �𝜇𝜇𝐴𝐴 (𝑥𝑥), 𝜇𝜇𝐵𝐵 (𝑥𝑥)� = 1 ∧
(𝜇𝜇𝐴𝐴 (𝑥𝑥) + 𝜇𝜇𝐵𝐵 (𝑥𝑥)), you see here this is nothing but the you have to take the minimum of 1
and the sum of the two membership values 𝜇𝜇𝐴𝐴 (𝑥𝑥) and 𝜇𝜇𝐵𝐵 (𝑥𝑥).

(Refer Slide Time: 19:09)

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So, when we compute this for each and every pair of membership values corresponding to
its generic variable values for fuzzy set A and fuzzy set B we are going to get this thing
this is plotted by the red color this is the outcome of the membership values. So, since the
fuzzy set that we have taken is continuous fuzzy sets. So, the corresponding membership
values are also coming from the respective membership functions 𝜇𝜇𝐴𝐴 (𝑥𝑥) 𝜇𝜇𝐵𝐵 (𝑥𝑥).

So, when we are finding the bounded sum, as the S-norm the result is also going to be
returned as the continuous membership function and this is computed by taking the
bounded sum as S norm. You can see here very clearly and this is how it can be
represented. So, this is actually nothing, but the fuzzy set as a whole and this is
characterized by the membership function which has been shown by the red color.

(Refer Slide Time: 20:09)

Now, we have the drastic sum, now when we apply the 𝑆𝑆𝑑𝑑𝑑𝑑 let us see what we are getting
here.

452
(Refer Slide Time: 20:26)

When we apply this formula we see it is returning us 𝜇𝜇𝐴𝐴 (𝑥𝑥) when we have 𝜇𝜇𝐵𝐵 (𝑥𝑥) = 0. So,
similarly when we apply this criteria 𝜇𝜇𝐵𝐵 (𝑥𝑥) when we have 𝜇𝜇𝐴𝐴 (𝑥𝑥) = 0. So, this way we are
getting this portion and we get 1 when we have 𝜇𝜇𝐴𝐴 (𝑥𝑥) and 𝜇𝜇𝐵𝐵 (𝑥𝑥) > 0. So, we see here this
portion, this portion we are getting for the intermediate values. So, let us now make it little
better and see that we are getting here this as the outcome of the drastic sum of two
membership functions.

(Refer Slide Time: 21:33)

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(Refer Slide Time: 22:57)

So, this way we have seen how the different S operators are giving us different results
when we apply the different S-norm. So, first S-norm that we have discussed was here as
the maximum, you can see here I am quickly going back here S-norm and then algebraic
sum on the same set of fuzzy sets, and then you see here the bounded sum here, and this
is what is being returned as the outcome when we apply bounded sum and then we have
the drastic sum.

So, see how the outputs are different from each other. Now let us take one more example
here earlier we took two continuous fuzzy sets A and B. Now, let us take two discrete
fuzzy sets A and B and see what is the result what is the outcome has we changed the
different kinds of S-norm.

454
(Refer Slide Time: 22:52)

So, when we apply the maximum S-norm we see that we have here for the same example
here we have the fuzzy set the discreet fuzzy set A and B like this, the A is 0.7 /1 +
0.5 /2 + 0.1 /3 + 0.6 /4 here. And the B is another discrete fuzzy set which is 0 by 1 plus
normally we do not write 0, 0 is written here just for the computation purpose, but
otherwise when we write the fuzzy set discrete fuzzy set B here no element will be written
for 0 membership value.

So, this will not be included, but here for competition purposes we have written. So, B is
equal to 0/ 1 + 0.8 /2 + 0.3 /3 + 0 /4. So, if we take these examples here. So we see
that when we are taking the maximum. So, here you see the max of all the pairs
corresponding pairs are taken you will see here for all the elements. So, since we have 4
elements here in first discrete fuzzy set A and then second fuzzy set that is B which is also
discreet fuzzy set it has only two non-zero membership value elements.

So, two more elements are added with the 0 membership value just for the computation
purpose here because we had to take the max of the two membership values corresponding
to the same generic variable value. So, for one when we take the max the membership
value the max membership value is coming out to be 0.7 and then for 2 we are getting 0.8
for 3 we are getting 0.3 for 4 we are getting 0.6.

So, this way we are getting four elements of the resulting fuzzy set or fuzzy membership
function. So, this is you see here are the discreet fuzzy set or discreet membership function

455
when we take the max S-norm of the two fuzzy set we are getting the outcome like this.
You can see here that we are getting the maximum of each pairs of the corresponding
generic variable value.

(Refer Slide Time: 25:29)

So, now let us use the same example here and take the S-norm algebraic sum. So, when
we take algebraic sum when we apply the formula here we see that we are getting 4
elements after using this formula. And we can plot here we can just represent here all these
four membership values for the corresponding pairs of the membership values that we
have taken and again this is corresponding to the same generic variable values. So, we see
that the algebraic sum S-norm when we do we are getting something different than what
we have gotten for the max as the S-norm operator.

456
(Refer Slide Time: 26:22)

Now, let us go to the third one that is bounded sum and the bounded sum S-norm when we
apply this formula which we have already discussed when we apply this we are again going
to get 4 elements of the resulting fuzzy set. So, when we plot these membership values we
see that we have the result again this is different from the previous one that is you see here
the algebraic sum. So, all these are very similar, but little different.

(Refer Slide Time: 27:04)

Now, the fourth one is the drastic sum S-norm when we apply this formula which we have
already discussed we are getting 4 elements out of this drastic sum of discrete fuzzy set A

457
and discrete fuzzy set B. So, we see that all these four membership values of the resulting
fuzzy set is plotted here as you shown in red color. So, this way we can see that how the
different S-norms are giving us the different results for the same fuzzy sets. By now we
have understood that we have S-norm operators or T-conorm operators as the max first.

So, first one is a max I would say the 𝑆𝑆𝑚𝑚𝑚𝑚𝑚𝑚 and then we have𝑆𝑆𝑎𝑎𝑎𝑎 , and then we have the
𝑆𝑆𝑏𝑏𝑏𝑏 , and then lastly here the 𝑆𝑆𝑑𝑑𝑑𝑑 .

(Refer Slide Time: 28:16)

So, in today’s lecture we have seen S-norm operators, in the next lecture we will study
parameterized T-norm operators and parameterized S-norm operators.

Thank you.

458
 Fuzzy Sets, Logic and Systems and Applications
Prof. Nishchal K. Verma
Department of Electrical Engineering
Indian Institute of Technology, Kanpur

Lecture – 26
Parameterized T-norm Operators

So, welcome to lecture number 26 of a Fuzzy Sets Logic and Systems and Applications.
So, in this lecture we will discuss parameterized T-norm as we all know parameterized T-
norm is also known as parameterized S-conorm.

(Refer Slide Time: 00:42)

So, in parameterized T-norm operator, we have certain classes of T-norm like Dombi’s
class of T-norm, Dubois Prade’s class of T-norm and Yager’s class of T-norm.

459
(Refer Slide Time: 00:56)

So, let us first discuss Dombi’s class of T-norm here if we have 2 fuzzy sets 𝐴𝐴 and 𝐵𝐵 with
the universe of discourse capital 𝑋𝑋. So, the Dombi’s class of T-norm is defined as T
lambda of 𝜇𝜇𝐴𝐴 (𝑥𝑥) and 𝜇𝜇𝐵𝐵 (𝑥𝑥). Here, T lambda is for Dombi’s class and this lambda here is
a parameter. So, this will appear in the right hand side. So, 𝑇𝑇𝜆𝜆 (𝜇𝜇𝐴𝐴 (𝑥𝑥), 𝜇𝜇𝐵𝐵 (𝑥𝑥)) =
1
, where this lambda is in between 0 𝑎𝑎𝑎𝑎𝑎𝑎 ∞.
𝜆𝜆 𝜆𝜆 1/𝜆𝜆
1 1
1+�� −1� +� −1� �
𝜇𝜇𝐴𝐴 (𝑥𝑥) 𝜇𝜇𝐵𝐵 (𝑥𝑥)

So, the lambda values can take any value in between 0 𝑎𝑎𝑎𝑎𝑎𝑎 ∞. This 𝜇𝜇𝐴𝐴 (𝑥𝑥) and 𝜇𝜇𝐵𝐵 (𝑥𝑥)
denote the membership function values for fuzzy sets A and B respectively, you can see
here. So, it is very clear from this expression of Dombi’s class of T-norm that this T lambda
applies to any two values of membership functions.

460
(Refer Slide Time: 02:57)

So, this can be very well understood by this example here. So, if we take an example here
where we have two fuzzy sets 𝐴𝐴 and 𝐵𝐵 in this case we have fuzzy sets 𝐴𝐴 and 𝐵𝐵 are discrete
fuzzy sets, then let us use Dombi’s class of T-norm for the intersection of 𝐴𝐴 and 𝐵𝐵 under
the universe of discourse 𝑋𝑋 = {1,2}.

So, let us now find out this the Dombi’s class of T-norm for the 𝐴𝐴 ∩ 𝐵𝐵. So, you can see
here that we have one fuzzy set here which is a discrete fuzzy set as I mentioned and this
is represented by you see here by this figure and similarly we have another fuzzy set B
which is you can see here represented by this figure. So, we have two discrete fuzzy sets
𝐴𝐴 and 𝐵𝐵 and you see here these are defined by these two expressions.

461
(Refer Slide Time: 04:18)

Now, let us apply the Dombi’s class of T-norm for finding the intersection of 𝐴𝐴 and 𝐵𝐵. So,
it’s very simple, we will have to take the membership values of the fuzzy sets 𝐴𝐴 and 𝐵𝐵
respectively and then we’ll apply the Dombi’s class of T-norm formula and this where we
get the result as you see here when we apply. So, we see that, we get two terms as a result
of Dombi’s class of T-norm for 𝜆𝜆 = 1. And when we simply these two terms, we are going
to get these two terms as the element of the resultant fuzzy set which is nothing but the
intersection of fuzzy set 𝐴𝐴 and fuzzy set 𝐵𝐵 based on the Dombi’s class of T-norm.

So, this is coming out to be 0.096/1 + 0.44/2. So, we have two elements of the resultant
fuzzy set. So, we can say when we take the intersection of these two fuzzy set using the
Dombi’s class of T-norm, we are going to get this thing and you can see here this is plotted
and please understand here we have taken the value of the parameter 𝜆𝜆 = 1. So, here we
can change the value of a 𝜆𝜆, let us take some other values of 𝜆𝜆 and see how the results
vary.

462
(Refer Slide Time: 06:10)

So, when we take 𝜆𝜆 = 2, we see that we are getting the intersection of A and B as a fuzzy
set whose elements are 0.10 / 1 + 0.49 /2. So, this way here, the intersection of fuzzy set
𝐴𝐴 and fuzzy set 𝐵𝐵, obviously both the fuzzy sets 𝐴𝐴 and 𝐵𝐵 are the discrete fuzzy sets. And
when we take the intersection of these two based on the Dombi’s class of T-norm, we get
this as the output which has been plotted here. So, similarly we can now increase the value
of 𝜆𝜆 further and see what is happening.

(Refer Slide Time: 07:11)

463
So, we see that on applying the Dombi’s class of T-norm for 𝜆𝜆 = 3, we are getting here
the elements of the resultant fuzzy set and this resultant fuzzy set is the intersection of the
fuzzy set 𝐴𝐴 and fuzzy set 𝐵𝐵 which is giving us this the fuzzy set. So, here also when we
plot the fuzzy set because it is the discrete fuzzy sets, so we are getting the plots for the
discrete points and we see this as the result of the intersection of the two discrete fuzzy
sets 𝐴𝐴 and 𝐵𝐵.

So this way we have understood that the Dombi’s class of T-norm can be found by
1
applying this expression 𝑇𝑇𝜆𝜆 (𝜇𝜇𝐴𝐴 (𝑥𝑥), 𝜇𝜇𝐵𝐵 (𝑥𝑥)) = . And this way by
𝜆𝜆 𝜆𝜆 1/𝜆𝜆
1 1
1+�� −1� +�𝜇𝜇 (𝑥𝑥)−1� �
𝜇𝜇𝐴𝐴 (𝑥𝑥) 𝐵𝐵

substituting 𝜆𝜆 = 3, we are getting the expression of the resultant fuzzy set and please
understand this expression is applied for the membership values of the corresponding
fuzzy set for which we are interested in taking the intersection.

So, since this is the discrete fuzzy set, we first take all the elements and then we take
corresponding numbers of values of the fuzzy sets 𝐴𝐴 and fuzzy set 𝐵𝐵 and then apply the
T-norm of the Dombi’s class. So, if we look at all the results we see here for 𝜆𝜆 = 1, 𝜆𝜆 =
2, 𝜆𝜆 = 3, we see that the membership values of the resultant fuzzy set increases as we
increase the values of 𝜆𝜆. So, when we increase the value of 𝜆𝜆, the resultant membership
value also increase.

(Refer Slide Time: 09:53)

464
Now, let us go to Dubois Prade’s class of T norm. So, if we take 𝐴𝐴 and 𝐵𝐵 fuzzy sets again
and with the universe of a discourse 𝑋𝑋. The Dubois Prade’s class of T-norm is defined by
this expression you can see here and here we have another parameter that is 𝛼𝛼. So, we
write this Dubois Prade’s class of T-norm as 𝑇𝑇𝛼𝛼 and 𝛼𝛼 value can be from 0 to 1. So, Dubois
𝜇𝜇𝐴𝐴 (𝑥𝑥)×𝜇𝜇𝐵𝐵 (𝑥𝑥)
Prade’s class of T-norm is defined by the 𝑇𝑇𝛼𝛼 �𝜇𝜇𝐴𝐴 (𝑥𝑥), 𝜇𝜇𝐵𝐵 (𝑥𝑥)� = .
max(𝜇𝜇𝐴𝐴 (𝑥𝑥),𝜇𝜇𝐵𝐵 (𝑥𝑥),𝛼𝛼)

So this is very simple expression here and Dubois Prade’s class of T-norm is found by
using this expression. Again the Dubois Prade’s class of T-norm is used for taking the
intersection of any two fuzzy sets, here we have two fuzzy sets 𝐴𝐴 and 𝐵𝐵 whose intersection
is found by using Dubois Prade’s class of T-norm. So, when we use this formula here we
have as I mentioned 𝑇𝑇𝛼𝛼 , 𝛼𝛼 here is a parameter and 𝛼𝛼 this parameter value can be any value
from 0 up to 1, as I mentioned. And when we apply this, we get the resultant membership
value or I would say the membership value of the fuzzy set which is the intersection of
fuzzy set 𝐴𝐴 and fuzzy set 𝐵𝐵.

(Refer Slide Time: 12:00)

So, this can also be understood by an example here. So, we have two discrete fuzzy sets
fuzzy set 𝐴𝐴 and fuzzy set 𝐵𝐵, here you can see. So, fuzzy set 𝐴𝐴 = 0.7 / 1 + 0.5 / 2 and
then 𝐵𝐵 = 0.1/1 + 0.8 / 2. So, these two fuzzy set we have and we are here interested in
finding out the intersection of these two fuzzy sets for different values of 𝛼𝛼 and these
values are 𝛼𝛼 are 0.1, 0.2, 0.3 and this is again through the intersection is found through the
Dubois Prade’s class of T-norm.

465
So, if we take two fuzzy sets as shown here in these two plots these two fuzzy sets are the
discrete fuzzy sets 𝐴𝐴 and fuzzy set 𝐵𝐵. So, 𝐴𝐴 is this fuzzy set and 𝐵𝐵 is this fuzzy set.

(Refer Slide Time: 13:19)

And when we use the formula that I have already discussed. Just now. the formula here is
𝜇𝜇𝐴𝐴 (𝑥𝑥)×𝜇𝜇𝐵𝐵 (𝑥𝑥)
𝑇𝑇𝛼𝛼 �𝜇𝜇𝐴𝐴 (𝑥𝑥), 𝜇𝜇𝐵𝐵 (𝑥𝑥)�, this is giving us the . So, let us now take 𝛼𝛼 = 0.1 and
max(𝜇𝜇𝐴𝐴 (𝑥𝑥),𝜇𝜇𝐵𝐵 (𝑥𝑥),𝛼𝛼)

see what is the membership value of the corresponding resultant fuzzy set and this again
this resultant fuzzy set is there a intersection of fuzzy set 𝐴𝐴 and fuzzy set 𝐵𝐵. So, when we
apply this, we find here the resultant fuzzy set which is the intersection of fuzzy set 𝐴𝐴 and
fuzzy set 𝐵𝐵 and this resultant fuzzy set has two elements. So, we have here 0.1 / 1 +
0.5 / 2 as two elements of the resultant fuzzy set.

So, when we plot this the fuzzy set looks like this. So, here I can write that we are taking
the intersection of fuzzy set 𝐴𝐴 and fuzzy set 𝐵𝐵 and this is giving us the resultant fuzzy set
here and this fuzzy set has again two elements and this looks like as it is shown here in
this slide as a result.

466
(Refer Slide Time: 15:03)

Now, when we take some other value of alpha, so here we are now increasing the value of
alpha. In earlier example we have 𝛼𝛼 = 0.1, in this example we have 𝛼𝛼 = 0.2. And let see
what is happening how the result is varying, how the elements are changing. I mean these
elements are from the intersection of fuzzy set 𝐴𝐴 and fuzzy set 𝐵𝐵.

So, again I can write here that this is the intersection of fuzzy set 𝐴𝐴 and fuzzy set 𝐵𝐵 and
here we are getting the resultant fuzzy set which again is having two elements as 0.1 / 1 +
0.5 / 2 and when it is plotted it looks like as it shown here.

(Refer Slide Time: 16:12)

467
Now, again we change the value of alpha and we increase the value of alpha here. Now
for 𝛼𝛼 = 0.3, when we substitute in the expression that is given for Dubois Prade’s class of
T-norm. We see that the resultant fuzzy set again has two terms two elements and this
these elements are 0.1 / 1 + 0.5 / 2.

So, when we take intersection based on the Dubois Prade’s class of T-norm, we are getting
here the resultant fuzzy set which is shown here like this. So, what we have done in this
Dubois Prade’s class of T-norm, we apply Dubois Prade’s class of T-norm for taking the
intersection of fuzzy set 𝐴𝐴 and fuzzy set 𝐵𝐵 and then the resultant fuzzy set 𝐵𝐵 we have
obtained based on the Dubois class of T-norm. And we see that when we increase the value
of 𝛼𝛼 like in the first case we took 𝛼𝛼 = 0.1, the second case we took 𝛼𝛼 = 0.2 and then we
took 𝛼𝛼 = 0.3.

And we see that the values of the membership we can see here the values of the
membership. So, let us now quickly go to the first case where we took 𝛼𝛼 = 0.1. So, you
see here 0.1 and then when we change the value of 𝛼𝛼 to 0.2, we see that there is no change
in the output, there is no change in the membership value even when we change the value
of 𝛼𝛼.

So, this way we see that for the values of alpha, here the change is not very significant in
the values of 𝛼𝛼. So, because of that the membership value of the resultant fuzzy set is also
not changing significantly.

(Refer Slide Time: 18:26)

468
Now, let us talk about the Yager’s class of T-norm which is the another class. So, in
Yager’s class we see that if we have two fuzzy sets 𝐴𝐴 and 𝐵𝐵 and within the universe of
discourse capital 𝑋𝑋. So, Yager’s class is defined by the expression which is given here, the
Yager’s class operator is represented by 𝑇𝑇𝑤𝑤 . So, 𝑇𝑇𝑤𝑤 is operated on a pair of membership
values, here we have this pairs as 𝜇𝜇𝐴𝐴 (𝑥𝑥) and 𝜇𝜇𝐵𝐵 (𝑥𝑥).

So, we can say that 𝑇𝑇𝑤𝑤 (𝜇𝜇𝐴𝐴 (𝑥𝑥), 𝜇𝜇𝐵𝐵 (𝑥𝑥)) = 1 − 𝑚𝑚𝑖𝑖𝑛𝑛�1, ((1 − 𝜇𝜇𝐴𝐴 (𝑥𝑥))𝑤𝑤 + (1 −
𝜇𝜇𝐵𝐵 (𝑥𝑥))𝑤𝑤 )1/𝑤𝑤 �, you can see here. So, we will take min of this and this. And here the value
of 𝑤𝑤 lies in between 0 and ∞. So, when we are interested in finding the intersection by
using the Yager’s class of T-norm, so let us see how it works.

(Refer Slide Time: 20:13)

We have this example here where we have taken two discrete fuzzy sets 𝐴𝐴 and 𝐵𝐵, the same
fuzzy sets same discrete fuzzy sets as we have taken in the previous example. So, here
when we apply the Yager’s class of T-norm on these two fuzzy sets 𝐴𝐴 and 𝐵𝐵.

469
(Refer Slide Time: 20:43)

When we apply the expression for the Yager’s class of T-norm on the membership values
of fuzzy set 𝐴𝐴 and 𝐵𝐵. So, we get this expression finally, you can see all these steps are
shown very nicely for understanding. So, we get again two terms where we have a taken
𝑤𝑤 = 1 and for 𝑤𝑤 = 1, when we apply the Yager’s class of T-norm for finding the
intersection of fuzzy set 𝐴𝐴 and fuzzy set 𝐵𝐵, we are getting here two elements. In fact, we
are getting only 1 element, because the two elements that are mentioned here, one has the
0 membership value. So, 0 membership value is normally not included when we represent
a fuzzy set.

So, you see that when we take the Yager’s class of T-norm, we get the resultant fuzzy set
whose elements are 0 / 1 + 0.3 / 2. So, this way as I already mentioned that one of these
elements has 0 membership value. So, it is shown here, but it need not be included while
writing the discrete fuzzy sets of the resultant of fuzzy set and fuzzy set 𝐵𝐵. So, it is shown
here the resultant fuzzy set shown here which you can see at 𝑥𝑥 is equal to 1, this 𝑥𝑥 is equal
to 1, we have 0 membership value and at 𝑥𝑥 is equal to 2 as the generic variable we have
membership value 0.3. So, you can see here, here 0.3.

So, this way we can find the intersection of any two fuzzy sets based on the Yager’s class
of T-norm for 𝑤𝑤 is equal to 1. Now, for the same set of fuzzy sets we find the intersection
based on the Yager’s class of T-norm for 𝑤𝑤 = 2.

470
(Refer Slide Time: 23:09)

So, when we increase the value of 𝑤𝑤 = 2, we see what is happening. You see that on
applying the Yager’s class of T-norm expression, we are getting here two elements of the
resultant fuzzy set which is coming, because of the intersection here of the fuzzy set 𝐴𝐴 and
fuzzy set 𝐵𝐵, you can see here. So, this way these two elements if we see we find in the
previous case where we had 𝑤𝑤 = 1, this term had the membership value 0 whereas, here
we see the membership value is not 0.

So, membership value for the first term here is 0.051 for 𝑥𝑥 is equal to 1 and the other term
has the membership value as 0.46 at 𝑥𝑥 is equal to 2. So, you see here both the terms have
been written here and this is the resultant of the intersection of fuzzy set 𝐴𝐴 and fuzzy set
𝐵𝐵. So, we can clearly see that by increasing the value of w, we see that the membership
values are also increasing. Now, let us take the value of 𝑤𝑤 = 3 and see that this value is
the membership value of the resultant fuzzy set is again increased.

471
(Refer Slide Time: 24:50)

So, this way we see that when we increase the value of 𝑤𝑤, the membership values of the
resultant of the intersection based on the Yager’s class of T-norm also increases.

(Refer Slide Time: 25:04)

So, now let us compare this. So, I would just like to repeat what I have said again here. So,
we see that while we took the Dombi’s class of T-norm, we see that in Dombi’s class of
T-norm we have lambda and you see here this is for 𝜆𝜆 = 1.

472
(Refer Slide Time: 25:29)

And when we increase the value of 𝜆𝜆 = 2. So, we see that the membership values of the
corresponding terms of the resulting fuzzy set out of the intersection of fuzzy set 𝐴𝐴 and
fuzzy set 𝐵𝐵 increase.

(Refer Slide Time: 25:51)

Similarly, here 𝜆𝜆 = 3, we have once again we see there is increase in the membership
values.

473
(Refer Slide Time: 26:01)

Similarly, when we talk of Dubois Prade’s class of T-norm with the increase of the 𝛼𝛼 as
the parameter we increase here since the change is not very significant in alpha the so the
significant change is also not seen. So, in the you see the membership values of the
resulting fuzzy set out of the intersection of fuzzy set 𝐴𝐴 and 𝐵𝐵. So, we don’t see any
significant change here for 𝛼𝛼 = 0.1, 0.2, 0.3.

(Refer Slide Time: 26:41)

474
And when we see the Yager’s class of T-norm, we see that when we increase the value of
𝑤𝑤, we see the change in the membership values of the resulting fuzzy set which is out of
the intersection of fuzzy set 𝐴𝐴 and fuzzy set 𝐵𝐵.

(Refer Slide Time: 27:01)

(Refer Slide Time: 27:10)

We can see here, this is for 𝑤𝑤 = 1 and this is for 𝑤𝑤 = 2 and here we have 𝑤𝑤 = 3. And we
can see that the membership values for the corresponding generic variable values are
increasing.

475
(Refer Slide Time: 27:21)

So, this way we see that we have covered the parameterized T-norms and in the next lecture
we will discuss the parameterized S-norms are T-conorms.

Thank you.

476
 Fuzzy Sets, Logic and Systems and Applications
Prof. Nishchal K. Verma
Department of Electrical Engineering
Indian Institute of Technology, Kanpur

Lecture – 27
Parameterized S-norm Operators

Welcome to lecture number 27 of Fuzzy Sets, Logic and Systems and Applications. Today
we will be discussing the Parameterized S-norm Operators.

(Refer Slide Time: 00:30)

And here we have Dombi’s class of S-norm, Dubois-Prade’s class of S-norm and Yager’s
class of S-norm. The parameterized S-norm we can call this as parameterized T-conorm.

477
(Refer Slide Time: 00:50)

So, let us go one by one and discuss the first Dombi’s class of S-norm. So, if we have two
fuzzy sets let us say 𝐴𝐴 and 𝐵𝐵 within universe of discourse capital 𝑋𝑋. So, the Dombi’s class
of S-norm is defined by this expression where 𝑆𝑆𝜆𝜆 is the Dombi’s class of S-norm operator.
1
So, 𝑆𝑆𝜆𝜆 �𝜇𝜇𝐴𝐴 (𝑥𝑥), 𝜇𝜇𝐵𝐵 (𝑥𝑥)� = .
−𝜆𝜆 −𝜆𝜆 −1/𝜆𝜆
1 1
1+�� −1� +�𝜇𝜇 (𝑥𝑥)−1� �
𝜇𝜇𝐴𝐴 (𝑥𝑥) 𝐵𝐵

So, this is defined by this and the lambda values here, will be in between 0 and infinite.
So, as I have already mention that the 𝑆𝑆𝜆𝜆 �𝜇𝜇𝐴𝐴 (𝑥𝑥), 𝜇𝜇𝐵𝐵 (𝑥𝑥)� here is basically termed as you
know the S-norm operated Dombi’s class of S-norm operator applied on the 2 membership
values. And these 2 membership values are from 2 different fuzzy sets whose union are to
be found.

So, here this has to be noted that normally we use S-norm, when we are interested in
finding the union of two fuzzy sets. So, here also we take two fuzzy sets and if we are
interested in union of these two fuzzy sets. So, we can either use the basic S-norm where
we take the max where we take just the union by taking max, but in parameterized norm
we have the expression here, which is given by Dombi’s class of S-norm and Dombi’s
class of S-norm as I have mentioned here and this involves a parameter lambda and this
lambda will be in between 0 and ∞.

478
(Refer Slide Time: 03:41)

So, let us take an example here where we have two discrete fuzzy sets 𝐴𝐴 and 𝐵𝐵 you can
see here discrete fuzzy set 𝐴𝐴 is shown here and discrete fuzzy set 𝐵𝐵 is also shown here we
are interested in the union of these two fuzzy sets 𝐴𝐴 and 𝐵𝐵. And the discrete fuzzy set 𝐴𝐴
has 2 elements here discrete fuzzy set 𝐵𝐵 also has 2 elements here and both the elements
you can see have the membership values for corresponding generic variable values.

(Refer Slide Time: 04:36)

So, when we take the union of these two we can use the S-norm and that to with the
Dombi’s class of S-norm. So, when we use Dombi’s class of S-norm we apply the

479
expression that I just discussed and this expression is nothing but the 𝑆𝑆𝜆𝜆 �𝜇𝜇𝐴𝐴 (𝑥𝑥), 𝜇𝜇𝐵𝐵 (𝑥𝑥)� =
1
. So, when we apply this expression for all possible pairs of
−𝜆𝜆 −𝜆𝜆 −1/𝜆𝜆
1 1
1+�� (𝑥𝑥)−1� +�𝜇𝜇 (𝑥𝑥)−1� �
𝜇𝜇𝐴𝐴 𝐵𝐵

the membership values of fuzzy set 𝐴𝐴 and fuzzy set 𝐵𝐵.

So, we see that the resultant fuzzy set which is here. So, the first term is like this it is
0.709 / 1 + 0.833 / 2. So, have two terms and this basically is a fuzzy set of the resulting
fuzzy set out of the union of fuzzy set 𝐴𝐴 and fuzzy set 𝐵𝐵. So, we can clearly see that we
had two discrete fuzzy sets 𝐴𝐴 and 𝐵𝐵 and both the fuzzy sets have two elements and hence
the union of these two will also have two elements.

So, here we take the union of fuzzy set 𝐴𝐴 and fuzzy set 𝐵𝐵 and this is what is resulting here
as the fuzzy set. And we see that when we take the parameter 𝜆𝜆 = 0. So, please note that
we can take any value of 𝜆𝜆 in between 0 and ∞ depending upon our suitability or
requirement these values are used for finding the different sets of the union of the fuzzy
sets 𝐴𝐴 and fuzzy set 𝐵𝐵.

So, we see that when we take lambda is equal to 1 the resulting fuzzy set here we get as
which has 2 terms and which has been shown here.

(Refer Slide Time: 07:34)

And now if we change the value of 𝜆𝜆 to 2 so, let us see what happens. So, we see that when
we increase the value of 𝜆𝜆 from 1 to 2 the membership values of the corresponding generic

480
variable values are reducing or I would say decreasing. So, when we take the union of
fuzzy set 𝐴𝐴 and fuzzy set 𝐵𝐵 for the value of lambda is equal to 2 we see the fuzzy set which
has the increased value of the membership for corresponding generic variable values.

(Refer Slide Time: 08:26)

So, when we take the value of 𝜆𝜆 = 3. Let us see what we are getting here we again see that
once again I would like to tell you that when we increase the value of 𝜆𝜆 the membership
values are reducing. You can see here in the previous example where we took 𝜆𝜆 = 1, the
union of the fuzzy sets 𝐴𝐴 and fuzzy set 𝐵𝐵. We have the membership values that are reducing
when we increase the value of 𝜆𝜆.

So, for value of 𝜆𝜆 = 3 here also we see the decrease in the membership value. So, we can
say when we increase the value of 𝜆𝜆 for the same set as input the resulting fuzzy set will
have reduced membership values of the corresponding generic variable values.

481
(Refer Slide Time: 09:31)

Now, let us discuss the Dubois-Prade’s class of S-norm. So, here we have 𝑆𝑆𝛼𝛼 and this
Dubois-Prade’s class of S-norm is represented by 𝑆𝑆𝛼𝛼 and on the same lines as we have
discussed in the previous class where we took the Dombi’s class of S-norm. So, here we
𝜇𝜇𝐴𝐴 (𝑥𝑥)+𝜇𝜇𝐵𝐵 (𝑥𝑥)−𝜇𝜇𝐴𝐴 (𝑥𝑥)×𝜇𝜇𝐵𝐵 (𝑥𝑥)−min(𝜇𝜇𝐴𝐴 (𝑥𝑥),𝜇𝜇𝐵𝐵 (𝑥𝑥),(1−𝛼𝛼))
use 𝑆𝑆𝛼𝛼 = . Please understand that the 𝛼𝛼
max��1−𝜇𝜇𝐴𝐴 (𝑥𝑥)�,�1−𝜇𝜇𝐵𝐵 (𝑥𝑥)�,𝛼𝛼�

takes the values anywhere from 0 up to 1.

(Refer Slide Time: 10:46)

482
So, let us now take the same fuzzy sets as we have taken and in the previous example and
let us find the union of two fuzzy sets using Dubois-Prade’s class of S-norm. So, have two
fuzzy sets here fuzzy set 𝐴𝐴 and fuzzy set 𝐵𝐵. 𝐴𝐴 is defined by this expression and 𝐵𝐵 is also
defined by here, these expression two these two fuzzy sets are discrete fuzzy sets.

(Refer Slide Time: 11:17)

So, when we apply the expression that I just discussed for 𝑆𝑆𝛼𝛼 which is for Dubois Prade’s
class of S-norm. So, when we use this for finding the membership values of the resulting
fuzzy set, which is out of the union of fuzzy set 𝐴𝐴 and fuzzy set 𝐵𝐵. So, we are getting two
terms here in the resulting fuzzy set. So, we see that we get two terms. And these two terms
will have its membership values for corresponding generic variable values 0.7/1 +
0.8 / 2.

So, here if we take union of these two fuzzy sets we are getting a fuzzy set which will have
two elements which is shown here and this is when you take the alpha is equal to 0.1.
Similarly, when we increase the value of 𝛼𝛼 let’s see what we are getting.

483
(Refer Slide Time: 12:37)

So, when we take the union again of fuzzy set 𝐴𝐴 and fuzzy set 𝐵𝐵 and here we get resultant
of this union of fuzzy set, which is again a discrete fuzzy set which has two terms and we
see that there is not any significant change here in the membership values of the resulting
fuzzy set which is out of the union of fuzzy set 𝐴𝐴 and fuzzy set 𝐵𝐵. So, here this case is
when we have taken 𝛼𝛼 = 0.2 . So, this result is same as the 𝛼𝛼 = 0.1.

(Refer Slide Time: 13:27)

And then when we again go ahead and we use 𝛼𝛼 = 0.3 we see that again we do not see
any significant increase in the membership values of the resulting fuzzy set. So, of course

484
here alpha is not playing a big role or I would say 𝛼𝛼 is here with the membership values
of the fuzzy sets that we have taken.

And it depends upon the combination of the membership values and then the 𝛼𝛼. So, here
in our case even when we increase the value of 𝛼𝛼 we see that we do not have any significant
increase in the values of the membership of the resulting fuzzy set. So, this way even when
we increase the values of 𝛼𝛼 we do not see any significant change.

(Refer Slide Time: 14:30)

Now, let us discuss the another class which is Yager’s class of S-norm. So, we have for
Yager’s class of S-norm for finding the union of the two fuzzy sets. So, by using this is
Yager’s class of S-norm we can operate this Yager’s class on any two membership value
of the fuzzy sets that we are interested in finding out the union of and if we have a fuzzy
set 𝐴𝐴 and fuzzy set 𝐵𝐵.

So, we will have the corresponding membership values as 𝜇𝜇𝐴𝐴 (𝑥𝑥), 𝜇𝜇𝐵𝐵 (𝑥𝑥) so when we apply
Yager’s class of S-norm on these two membership values 𝜇𝜇𝐴𝐴 (𝑥𝑥) 𝑎𝑎𝑎𝑎𝑎𝑎 𝜇𝜇𝐵𝐵 (𝑥𝑥) and this will
𝑤𝑤 𝑤𝑤 1/𝑤𝑤
be equal to 𝑚𝑚𝑚𝑚𝑚𝑚 �1, ��𝜇𝜇𝐴𝐴 (𝑥𝑥)� + �𝜇𝜇𝐵𝐵 (𝑥𝑥)� � �. Here w is a parameter of a Yager’s class

of S-norm and this 𝑤𝑤 is going to be in between 0 and ∞. So, the w can take any value in
between 0 and ∞.

485
(Refer Slide Time: 16:10)

Let us, now understand this also Yager’s class of S-norm also by taking an example. So,
here we take two discrete fuzzy sets A and B and here we will be taking the union of these
two fuzzy sets and we will be using the Yager’s class of S-norm to find the union and let’s
see what happens when we use Yager’s class of S-norm.

(Refer Slide Time: 16:48)

So, we have two fuzzy set as I mentioned and when we take union of the fuzzy set 𝐴𝐴 and
fuzzy set 𝐵𝐵 so obviously the membership values of fuzzy set 𝐴𝐴 membership values of
fuzzy set 𝐵𝐵 we’ll collect. And then we will use Yager’s class of S-norm on these the pairs

486
of the membership values that are coming from fuzzy set 𝐴𝐴 and fuzzy set 𝐵𝐵. So, when we
do that, we use Yager’s class of S-norm.

We find another fuzzy set as a result which will have two elements. So, one of the elements
will be 0.8 / 1 + 1.0 / 2. So, we see here that when we take the union of fuzzy set 𝐴𝐴 and
fuzzy set 𝐵𝐵. We get another fuzzy set which is the resultant of the fuzzy set of the union
of fuzzy sets A and fuzzy set B we have two elements in this fuzzy set.

And this element has 0.8 as the membership value corresponding to the generic variable
value as 1 plus we have 1 as the membership value corresponding to the generic variable
value 2 and both of these have been plotted here you can see and this is for w is equal to
1. So, 𝑤𝑤 is an important parameter here, in Yager’s class of S-norm and when we change
the value of 𝑤𝑤 let’s see what happens and obviously as I mentioned the 𝑤𝑤 can be any value
in between 0 to ∞.

(Refer Slide Time: 18:53)

So, let us now use again both the fuzzy sets and let us find the union of these two fuzzy
sets 𝐴𝐴 and 𝐵𝐵 and we take the value of w, 2. So, 𝑤𝑤 = 2 we are going to get here some
change in the membership values of the resulting fuzzy set which is coming out of the
union of fuzzy set 𝐴𝐴 and fuzzy set 𝐵𝐵. So, we can clearly see that the values of the
membership are changing as we are changing 𝑤𝑤 means the values of the membership are
reducing here.

487
So, earlier we had 0.8 and then now we have 0.707 for the generic variable 1 when we
have changed the value of 𝑤𝑤 from 1 to 2. So, this means that when we increase the value
of 𝑤𝑤 the membership values of the resulting fuzzy set are reducing.

(Refer Slide Time: 20:05)

So, similarly when we again take the value of 𝑤𝑤 as 3 here you can see the 𝑤𝑤 = 3. So, when
we take 𝑤𝑤 = 3 again there is some decrease in the values of the membership of the
corresponding generic variable values of the resulting fuzzy set which is coming out of the
union of fuzzy set 𝐴𝐴 and fuzzy set 𝐵𝐵.

(Refer Slide Time: 20:40)

488
So, this way we can say that we have the parameterized S-norms this is also called as T-
conorm so, we can either say we can either call the parameterized S-norm or parameterized
T-conorm And we have 3 parameterized S-norms as we discuss Dombi’s class of S-norm,
Dubois Prades class of S-norm and Yager’s class of S-norms. And let us compare these S-
norms for the parameters that are involved here in this three parameterized S-norms.

So, let us see what happens when we are increasing the values of 𝜆𝜆 in case of Dombi’s
class of S-norm and 𝛼𝛼 in case of Dubois-Prade’s class of S-norm and 𝑤𝑤 when we take the
case of Yager’s class of S-norm.

(Refer Slide Time: 21:45)

So, this we have already discussed, but I would like to just summarize and conclude here
that you see when we increase the value of 𝜆𝜆 here. So, we can see here this for 𝜆𝜆 is equal
to 1.

489
(Refer Slide Time: 22:03)

And then this 𝜆𝜆 is equal to 2 this for the Dombi’s class of S-norm and then when we take
the 𝜆𝜆 is equal to 3. So you see the output. So we see that the output the resulting fuzzy set
that is you know out of the fuzzy set 𝐴𝐴 union fuzzy set 𝐵𝐵. So, we see that as we increase
the value of 𝜆𝜆 the membership values of the resulting fuzzy set for the corresponding
generic variable values are reducing.

So, this is very clearly visible and when we see this in Dubois-Prade’s class of S-norm
here in Dubois-Prade’s class of S-norm we have 𝛼𝛼 as the parameter. So, when we increase
the value of 𝛼𝛼 we see that there is not much change or I would say not a significant change
in the output even when we change the when we increase the values of the 𝛼𝛼. So, this is 𝛼𝛼
is equal to 0.1 and then we have 𝛼𝛼 is equal to 0.2 and then when we have 𝛼𝛼 is equal to 0.3.

So, here once again I would like to mention that we have Dubois-Prade’s class of S-norm
and here even when we increase the value of 𝛼𝛼. We do not see any significant change in
the membership values of the corresponding generic variable values of the resulting fuzzy
set which is coming out of the union of fuzzy set 𝐴𝐴 and fuzzy set 𝐵𝐵. Now in case of Yager’s
class of S-norm on the again on taking the union when we use Yager’s class of S-norm we
see that when we take 𝑤𝑤 is equal to 1.

490
(Refer Slide Time: 24:23)

So, we see that there is the decrease in the membership value. So, as we increase the value
of 𝑤𝑤 here 𝑤𝑤 is a parameter in Yager’s class of S-norm.

(Refer Slide Time: 24:40)

We see that the membership values are decreasing. So, these membership values of the
corresponding generic variable values of the resulting fuzzy set which is coming out of the
union of fuzzy set 𝐴𝐴 and fuzzy set 𝐵𝐵. So, this way we are same as to how the parameters
of the parameterized S-norms are playing an important role in the resulting fuzzy set.

491
(Refer Slide Time: 25:16)

So, today we have discussed the parameterized S-norm in detail and in the next lecture we
will be discussing the another topic that is fuzzy relation.

Thank you.

492
 Fuzzy Sets, Logic and Systems and Applications
Prof. Nishchal K. Verma
Department of Electrical Engineering
Indian Institute of Technology, Kanpur

Lecture – 28
Fuzzy Relation

Welcome to lecture number 28 of Fuzzy Sets, Logic and Systems and Applications.

(Refer Slide Time: 00:32)

So in this lecture we will discuss a Fuzzy Relation. Before we move to fuzzy relation, let
us first understand the crisp relation because this is very important for understanding fuzzy
relation, and crisp relation also we need to know first the Cartesian product of crisp sets.
So, if we have two arbitrary crisp sets with the universe of discourse capital 𝑋𝑋 and capital
𝑌𝑌, respectively.

The Cartesian product of A and B can be written as the A cross B and this will be equal to
the set of all pairs of 𝑥𝑥 and 𝑦𝑦, such that 𝑥𝑥 ∈ 𝑋𝑋 and 𝑦𝑦 ∈ 𝑌𝑌.

493
(Refer Slide Time: 01:38)

So, this can be understood better by this example of course, this is a very basic
understanding, you must have done this in your earlier classes of mathematics. But here
this becomes little important to understand before we move to the crisp relation and then
the fuzzy relation. So, if we take an example here, where we have 𝐴𝐴 and 𝐵𝐵 two crisp sets.
You can see here, 𝐴𝐴 is a set is a crisp set where we have two elements; in 𝐴𝐴{0,1} and in
𝐵𝐵{𝑎𝑎, 𝑏𝑏, 𝑐𝑐}.

(Refer Slide Time: 02:36)

494
So, if we are interested in finding the Cartesian product, what do we do? We simply you
know, we multiply or we take the cross product of the two sets 𝐴𝐴 and 𝐵𝐵 and this is going
to give us the ordered pair of all the elements from 𝐴𝐴 and 𝐵𝐵. So, but it should be within the
universe of discourse.

So, if I have to let us say find 𝐴𝐴 × 𝐵𝐵, this is going to be like this like A cross B is going
to give us a set which will contain (𝑥𝑥, 𝑦𝑦) means all the elements in the pair is coming from
the set 𝐴𝐴 and the 𝑦𝑦 is coming from the set 𝐵𝐵. And of course, this 𝑥𝑥 and 𝑦𝑦 forms a pair here
and small 𝑥𝑥 is from the universe of discourse 𝑋𝑋, 𝑦𝑦 from the universe of discourse 𝑌𝑌. And
we all know that in Cartesian product of two sets crisp sets, we follow the order of the
elements while making the pairs here.

So, the first element will be from the set 𝐴𝐴 and then the second element of the pair will be
from the element 𝐵𝐵. And then we collect all the formed pairs in the set, and this is 𝐴𝐴 × 𝐵𝐵.
Now when we are interested in finding 𝐵𝐵 × 𝐴𝐴, the this will be a set of all the pairs of 𝑦𝑦 and
𝑥𝑥; means all the pairs here, where the first element is coming from the set 𝐵𝐵 and then the
second element will be from the set 𝐴𝐴. So, here this is to be noted that, the order is very
important in the Cartesian product, the order in which the pairs are formed. Like in 𝐴𝐴 × 𝐵𝐵,
we have 𝑥𝑥, 𝑦𝑦 as a pair; but in 𝐵𝐵 × 𝐴𝐴, we have 𝑦𝑦, 𝑥𝑥 as a pair, right.

So, similarly if we are interested in finding the 𝐴𝐴 × 𝐴𝐴, you see here that all the pairs will
be like this, like both the elements will be from the set 𝐴𝐴. Similarly, if you are interested
in finding the 𝐵𝐵 × 𝐵𝐵, so we will have the collection of pairs in this Cartesian product set,
and all the elements will be from the same set 𝐵𝐵. And of course, it is needless to say that,
they will have to follow this 𝑦𝑦, 𝑥𝑥 and 𝑦𝑦 will have to follow and the condition of universe
of discourse; means 𝑥𝑥 should be from the universe of discourse 𝑋𝑋 and small 𝑦𝑦 from the
universe of discourse 𝑌𝑌.

495
(Refer Slide Time: 06:14)

So, here in this example if we see that, 𝐴𝐴 × 𝐵𝐵 as we have defined here, if we take 𝐴𝐴 set
like this, like 𝐴𝐴 has two elements 0 𝑎𝑎𝑎𝑎𝑎𝑎 1, 𝐵𝐵 has three elements 𝑎𝑎, 𝑏𝑏, 𝑐𝑐. So, we see here
that, if we take the cross product, if we take the Cartesian product here as 𝐴𝐴 × 𝐵𝐵 so
Cartesian product of the set 𝐴𝐴. Please understand that this set is a crisp set. So, once again
I am saying here that, the Cartesian product of crisp set 𝐴𝐴 and crisp set 𝐵𝐵 is going to be all
the ordered pairs that are formed elements from 𝐴𝐴 and 𝐵𝐵.

So, we see that 𝐴𝐴 × 𝐵𝐵 is going to give us first pair (0, 𝑎𝑎); second pair (0, 𝑏𝑏); third pair
(0, 𝑐𝑐) and then the fourth pair is (1, 𝑎𝑎); fifth pair is (1, 𝑏𝑏); sixth pair is (1, 𝑐𝑐). So, we see
that we have six elements, when we are taking the Cartesian product of capital 𝐴𝐴 and
capital 𝐵𝐵, where these capital 𝐴𝐴 and capital 𝐵𝐵 are two crisp sets. So, this way we get the
Cartesian product of two crisps sets.

496
(Refer Slide Time: 07:51)

Now, let us change the order and let us find 𝐵𝐵 × 𝐴𝐴. So, we write 𝐵𝐵 here first and then we
write 𝐴𝐴. And as we know that 𝐵𝐵 has three elements 𝑎𝑎, 𝑏𝑏 and 𝑐𝑐, and 𝐴𝐴 has two elements.

Let us now find the Cartesian product of crisp set 𝐵𝐵 and crisp set 𝐴𝐴. So, here we will first
start with this element 𝑎𝑎, and this element will combine with all the elements of set 𝐴𝐴. So,
this is going to give us (𝑎𝑎, 0) as the first pair, and then the second pair will be (𝑎𝑎, 1) here.
So, like that we will be getting other pairs. So, we will have then (𝑏𝑏, 0) and then (𝑏𝑏, 1);
and then finally, with 𝑐𝑐 we are going to get (𝑐𝑐, 0) and (𝑐𝑐, 1). So, this way we are getting
six element in the set which is the Cartesian product of 𝐵𝐵 and 𝐴𝐴.

So, what actually we are doing here is that we are forming the ordered pairs of the elements
that are coming from the first set and then the second set. And please understand that, these
pairs will belong to the universe of discourse that are created by the Cartesian product of
the separate universe of discourses 𝑋𝑋 and capital 𝑌𝑌; or in this case if we take 𝑌𝑌 first then
capital 𝑋𝑋, then 𝑌𝑌 cross 𝑋𝑋.

(Refer Slide Time: 10:06)

497
So, here this 𝐵𝐵 × 𝐴𝐴 is, if we see that we are getting a set of six pairs of elements. So, we
can clearly say that, 𝐵𝐵 × 𝐴𝐴 that we have seen here is different from 𝐴𝐴 × 𝐵𝐵. The order is
changed; so that is why we can say that, 𝐴𝐴 × 𝐵𝐵 ≠ 𝐵𝐵 × 𝐴𝐴. Now let us take let us find 𝐴𝐴 ×
𝐴𝐴. So, in 𝐴𝐴 × 𝐴𝐴, on the same lines if we move we see that, 𝐴𝐴 × 𝐴𝐴 we are getting as;
because in 𝐴𝐴 we have two elements 0 and 1. So, we first take 0 and then we combine, we
make pair of the elements that are written here in the other 𝐴𝐴. So, we have two sets 𝐴𝐴 and
A and when we take the Cartesian product, we are getting four elements.

And these four elements, basically we having elements from the same set 𝐴𝐴; so we get
(0, 0), then (0, 1), then (1, 0) and then (1, 1). So, this way we get 𝐴𝐴 × 𝐴𝐴.

(Refer Slide Time: 11:21)

498
Now let us find 𝐵𝐵 × 𝐵𝐵. So, when we are interested in finding 𝐵𝐵 × 𝐵𝐵, means we are taking
two sets both the sets are 𝐵𝐵 × 𝐵𝐵 only. So, when we make pairs of the elements from the
first set as B and the second set also as 𝐵𝐵, so we see that, we are getting here is nine
elements like this, so (𝑎𝑎, 𝑎𝑎); (𝑎𝑎, 𝑏𝑏); (𝑎𝑎, 𝑐𝑐); (𝑏𝑏, 𝑎𝑎); (𝑏𝑏, 𝑏𝑏); (𝑏𝑏, 𝑐𝑐); (𝑐𝑐, 𝑎𝑎); (𝑐𝑐, 𝑏𝑏); (𝑐𝑐, 𝑐𝑐).

So, like that we have nine pairs of elements from 𝐵𝐵 set only. So, this way we have seen
that, we can get the Cartesian product of 𝐵𝐵 and 𝐵𝐵. And as we have already seen that, the
order is very important while we make pairs.

(Refer Slide Time: 12:20)

So, that is why the 𝐴𝐴 × 𝐵𝐵 ≠ 𝐵𝐵 × 𝐴𝐴. And please note that these two sets A and B are the
crisp set here. So, this we have already seen that, 𝐴𝐴 × 𝐵𝐵 was coming out to be like this
and then 𝐵𝐵 × 𝐴𝐴 is coming out to be this. And that is how we can with example we can see
that, 𝐴𝐴 × 𝐵𝐵 ≠ 𝐵𝐵 × 𝐴𝐴. And it is because of the order in which the pair is made.

So, in 𝐴𝐴 × 𝐵𝐵, the first element of the pairs are coming from crisp set 𝐴𝐴, whereas in 𝐵𝐵 × 𝐴𝐴
the first element of the pair is coming from set 𝐵𝐵. So, that is why when we compare 𝐴𝐴 × 𝐵𝐵
and 𝐵𝐵 × 𝐴𝐴, these both of these are not equal to each other.

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(Refer Slide Time: 13:28)

Now, let us move towards the Cartesian product of the 𝑛𝑛 crisp sets. So, in general we can
write the Cartesian product of the arbitrary 𝑛𝑛 crisp sets like this like 𝐴𝐴1 , if we represent the
𝑛𝑛 crisp sets like this like 𝐴𝐴1 , 𝐴𝐴2 , 𝐴𝐴3 … … . 𝐴𝐴𝑛𝑛 , so we can see here.

And then if we are interested in Cartesian product of all these n crisp sets. So, we can say
this is the Cartesian product of 𝐴𝐴1 , 𝐴𝐴2 , 𝐴𝐴3 … . 𝐴𝐴𝑛𝑛 can be represented by the 𝐴𝐴1 × 𝐴𝐴2 ×
𝐴𝐴3 … . . 𝐴𝐴𝑛𝑛 = {(𝑥𝑥1 , 𝑥𝑥2 , 𝑥𝑥3 … . . 𝑥𝑥𝑛𝑛 )}|𝑥𝑥1 ∈ 𝑋𝑋1 , 𝑥𝑥2 ∈ 𝑋𝑋2 … . 𝑥𝑥𝑛𝑛 ∈ 𝑋𝑋𝑛𝑛 .

So, in other words we can say that, the cross product of n crisp sets 𝐴𝐴1 , 𝐴𝐴2 , 𝐴𝐴3 … . 𝐴𝐴𝑛𝑛 is
nothing, but the collection of 𝑛𝑛 − 𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡; 𝑥𝑥1 , 𝑥𝑥2 , 𝑥𝑥3 … . . 𝑥𝑥𝑛𝑛 . So, all such n tuples will be
there in the Cartesian product, when we take the Cartesian product of n sets and crisp sets.
So, when we represent a Cartesian product like this, now this Cartesian product contains
all the n tuples that are formed from the elements that are coming from 𝐴𝐴1 , 𝐴𝐴2 , 𝐴𝐴3 … . 𝐴𝐴𝑛𝑛 .
And please understand here that these n tuples will belong to the universe of discourse
the 𝑋𝑋1 × 𝑋𝑋2 × 𝑋𝑋3 … . .× 𝑋𝑋𝑛𝑛 .

So, when we have the Cartesian product which is nothing, but the ordered n tuples here,
in case of the Cartesian product of 𝑛𝑛 crisp sets. Now let us understand the relation among
the crisp sets, so here when we are interested in finding the relation, so normally we say
the Cartesian product set is a relation set. But here we need to know that, relation set is
based on certain conditions, so this I will be discussing in the next slide. But before that, I
would like to define the relation set by 𝑄𝑄(𝐴𝐴1 ) here 𝑄𝑄(𝐴𝐴1 , 𝐴𝐴2 , 𝐴𝐴3 … … 𝐴𝐴𝑛𝑛 ).

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So, this is here, this Q denotes the relation set. And this relation set will always be a subset
of the Cartesian product of the 𝐴𝐴1 , 𝐴𝐴2 , 𝐴𝐴3 … . 𝐴𝐴𝑛𝑛 . So, relation set of 𝐴𝐴1 , 𝐴𝐴2 , 𝐴𝐴3 , 𝐴𝐴𝑛𝑛 is a part
of the overall Cartesian product of the sets 𝐴𝐴1 , 𝐴𝐴2 , 𝐴𝐴3 , 𝐴𝐴𝑛𝑛 .

(Refer Slide Time: 17:39)

This can be understood by an example here. So, let us take an example to understand the
crisp relation. So, here we are taking two crisp sets 𝐴𝐴 and 𝐵𝐵 and let us first find the
Cartesian product of these two sets.

(Refer Slide Time: 18:02)

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So, 𝐴𝐴 × 𝐵𝐵 can be written as the collection of all the pairs of the elements from 𝐴𝐴 and 𝐵𝐵
sets and this way 𝐴𝐴 × 𝐵𝐵 is going to be a set here, the Cartesian product of 𝐴𝐴 and 𝐵𝐵. So,
we see that, we have here, since we have four elements in 𝐴𝐴 and three elements in 𝐵𝐵 crisp
set, so, we are going to get 12 pairs here; like this, here the first pair is (1, 2); second pair
is (1, 3); third pair is (1, 4); fourth pair is (2, 2) and like that we get the twelfth pair is
(4, 4). So, this set contains all the ordered pairs which are possible in, possible out of the
Cartesian product of 𝐴𝐴 × 𝐵𝐵.

So, we see that this is the overall space, overall population that is possible here as the
Cartesian product of 𝐴𝐴 and 𝐵𝐵. Now let us see what is a relation, if we are interested by
putting some condition. So, let’s say we are interested in a particular relation and the
relation here is that, we are interested in a set let us say 𝑄𝑄(𝐴𝐴, 𝐵𝐵); so this is a relation set
and this defines some relation based on some condition. So, 𝑄𝑄(𝐴𝐴, 𝐵𝐵) is based on the
condition here is that, first element is greater than, so first element of the pair that are
formed is greater than the second element in the Cartesian product of 𝐴𝐴 and 𝐵𝐵.

So, we have already seen that we have Cartesian product here, where we have twelve pairs.
Now we are putting the condition that, the first element in all the pairs should be greater
than the second element. So, if we put this condition, will find few pairs where this is true.
So, if we are putting this condition and based on that, we pick those pairs for which this
condition is satisfied. And 𝑄𝑄(𝐴𝐴, 𝐵𝐵) relation is based on that condition, if that is true, then
we can write 𝑄𝑄(𝐴𝐴, 𝐵𝐵) is equal to a set of these three pairs for which the first element is
greater than the second element.

So, we can see here that we have three pairs that is (3, 2); (4, 2); (4, 3) where the first
element of the pair is greater than the second element in all the three pairs. So, 𝑄𝑄(𝐴𝐴, 𝐵𝐵)
represent here a relation set. And we see that, this 𝑄𝑄(𝐴𝐴, 𝐵𝐵) is coming from the total
population, which we have got from the Cartesian product of 𝐴𝐴 and 𝐵𝐵. So, Cartesian
product of 𝐴𝐴 and 𝐵𝐵 is here, and 𝑄𝑄(𝐴𝐴, 𝐵𝐵) is here and we can clearly see that (3, 2); (4, 2);
(4, 3); all these three pairs are drawn from the Cartesian product of A and B after applying
the condition, the first element is greater than the second element.

So, since 𝑄𝑄(𝐴𝐴, 𝐵𝐵) is a set and it has some elements which are drawn from the Cartesian
product of crisps set 𝐴𝐴 and 𝐵𝐵, so we can write here that 𝑄𝑄(𝐴𝐴, 𝐵𝐵) is the subset of the
Cartesian product of 𝐴𝐴 and 𝐵𝐵, where 𝐴𝐴 and 𝐵𝐵 both are the crisps set. So, this way the

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statement that I made in the previous slide that the relation you can say here, the relation
𝑄𝑄(𝐴𝐴, 𝐵𝐵) is the subset of here the relation set is a subset of the Cartesian product of the sets.

So, this way we can write that relation set, the relation set is always a subset of the
Cartesian product. Or in other words if we say the relation set in between 𝐴𝐴 and 𝐵𝐵 here in
this case a subset of the Cartesian product of set 𝐴𝐴 and 𝐵𝐵.

(Refer Slide Time: 23:45)

Now, let us take another example here, where we take crisp set 𝐴𝐴 and crisp set 𝐵𝐵 here. And
we have the universe of discourse here as for 𝐴𝐴 we have 𝑋𝑋 and for 𝐵𝐵 we have capital Y.
So, in this example we are supposed to first find the Cartesian product of crisp set 𝐴𝐴 and
crisp set 𝐵𝐵, that is 𝐴𝐴 × 𝐴𝐴. And then we are supposed to find the relation matrix 𝑄𝑄1 (𝐴𝐴, 𝐵𝐵),
such that the first element is greater than the second element as we have seen in the
previous example.

So, this very easy, we can first find the Cartesian product, so that we have the space, the
whole population of the Cartesian product and then from that we pick the elements based
on the condition that is given and this will be our relation matrix 𝑄𝑄1 (𝐴𝐴, 𝐵𝐵). And then next
in the third case we are supposed to find relation matrix 𝑄𝑄2 (𝐴𝐴, 𝐵𝐵), such that second element
is greater than the first element. So, here based on another condition, we are supposed to
find another relation set 𝑄𝑄2 (𝐴𝐴, 𝐵𝐵) and based on that we can find a relation matrix.

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In the fourth case we are supposed to find 𝑄𝑄3 (𝐴𝐴, 𝐵𝐵), such that first element is equal to the
second element for 𝐴𝐴 × 𝐵𝐵. So, let’s now quickly move ahead and first find the Cartesian
product of set 𝐴𝐴 and set 𝐵𝐵, here both the sets are crisp sets.

(Refer Slide Time: 25:42)

So, 𝐴𝐴 × 𝐵𝐵 as we have already done, we can quickly find here a set like this and this
represents the Cartesian product of this set 𝐴𝐴, where we have four elements 1, 2, 3, 4 and
then we have set 𝐵𝐵, where we have three elements 2, 3 and 4. And 𝐴𝐴 × 𝐵𝐵 the Cartesian
product of 𝐴𝐴 and 𝐵𝐵 here is a collection of all the elements which are ordered pairs; so
collection of all the ordered pairs of the elements coming from set 𝐴𝐴 and set 𝐵𝐵,
respectively. So, we are getting here 4 × 3, that is 12 ordered pairs as elements of the
Cartesian product of set 𝐴𝐴 and set 𝐵𝐵.

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(Refer Slide Time: 26:42)

Then when we have this as the whole population here of the order pairs that are possible.
Now let us go ahead and apply the condition for the first relation. So, for the first
relation𝑄𝑄1 (𝐴𝐴, 𝐵𝐵) here, 𝑄𝑄1 (𝐴𝐴, 𝐵𝐵) let us apply the first condition; first condition here is that,
the first element is greater than the second element. So, first element is greater than the
second element means the we have to look into all the ordered pairs for the first element,
which should be greater than the second element in the same pair.

So, we find here three pairs as elements of this set, 𝑄𝑄1 (𝐴𝐴, 𝐵𝐵). So, 𝑄𝑄1 (𝐴𝐴, 𝐵𝐵) is giving us 3
ordered pairs, where we see that the first element is greater than the second element, no
other pair in 𝐴𝐴 × 𝐵𝐵 in the Cartesian product of 𝐴𝐴 and 𝐵𝐵 is giving any pair which satisfies
this condition.

We can see here one by one, so we see we have the first pair (1, 2), where 1 ≤ 2, so the
first element is less than the second element, so this is not qualified. Then if we keep
moving ahead we see that, here 3 and 2 satisfies, the pair which is consisting of 3 and 2
which gives us this condition satisfied. So, we have collected this element here.

And then (4, 2) is another pair which satisfies this condition and then we have (4, 3) also
which satisfies this condition. So, this way we get three elements here, in this set 𝑄𝑄1 (𝐴𝐴, 𝐵𝐵)
which is relation set, where the first element is greater than the second element. So, this is
how the relation set is formed by the crisp sets.

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(Refer Slide Time: 29:11)

Now, let us form a matrix here. So, relation sets can also be represented by the relational
matrix. So, if we have 𝐴𝐴 like this and 𝐵𝐵 like this, like we have rows and columns and based
on that the relation we can write in form of matrix.

So, we see 𝑄𝑄1 (𝐴𝐴, 𝐵𝐵), so we have (3, 2), (4, 2), (4, 3). So, if we write 1, 2, 3, 4 in as a
column and then 2, 3 in the row. So, we see that, we have 1 here, we can write here 1,
because (3, 2) is existing as pair in 𝑄𝑄1 (𝐴𝐴, 𝐵𝐵) and then we have (4, 2) and then we have
(4, 3). So, in this 𝑄𝑄1 relation set, we have only three elements, so we have only for this we
write once, otherwise all other elements will be 0. Here this has to be noted that, why are
we writing 1, because the logic is Boolean logic. So, since this pair is completely present,
so that is why we are writing 1 for all the existing pairs and 0 for non-existing pairs.

So, that is why we have this matrix here as the relational matrix. So, this is 𝑄𝑄1 (𝐴𝐴, 𝐵𝐵).

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(Refer Slide Time: 30:50)

Next is when we apply the second condition that was given and this relation set is
represented by 𝑄𝑄1 (𝐴𝐴, 𝐵𝐵). So, we already have the set of the Cartesian product of set 𝐴𝐴 and
𝐵𝐵 here. And now we look for the pairs which satisfy this condition; what is this condition?
Condition is that the second element is greater than that of the first element. So, we see
that here, we have got (1, 2) as the first pair which satisfies this condition.

So, first pair in 𝑄𝑄2 (𝐴𝐴, 𝐵𝐵). So, 𝑄𝑄2 (𝐴𝐴, 𝐵𝐵) is the set of all such pairs which satisfy the
condition of the second element is greater than the first element. So, (1, 2) is the first
element of 𝑄𝑄2 and then we have (1, 3) also this pair which satisfies this condition, then we
have (1, 4) which also satisfies this condition. And then we have (2, 2) which is not
satisfying this condition, because 2 and 2 both are equal. So, here the second element is
not greater than the first element, so this is not satisfying.

And then (2, 3) is satisfying the condition, (2, 4) is satisfying this condition. Similarly we
see that here (3, 4) is satisfying the condition. So, this way we see that we have 6 elements
or I would say the six pairs in 𝑄𝑄2 (𝐴𝐴, 𝐵𝐵), means the relation set 𝑄𝑄2 has 6 elements which
satisfy this the given condition that is the second element is greater than the first element.

507
(Refer Slide Time: 32:55)

So, let us form a relational matrix. So, as we have done this in the previous case, so we
can write a relational matrix like this, like we have (1, 2) existing in 𝑄𝑄2 (𝐴𝐴, 𝐵𝐵).

So, we take (1, 2) here and we see that, for (1, 2) we have written 1. And please understand
that the column here is coming as the set 𝐴𝐴 and the row as the set 𝐵𝐵. So, since we have 1,
2 pair already existing in 𝑄𝑄2 , so we write here 1. And similarly for (1, 3) we write here 1,
and then (1, 4) we write 1 and then (2, 3) we write 1; (2, 4) we write 1; and (3, 4) we
write 1. So, all this six elements now have been represented in a form of a matrix and this
way you know all these elements have been included in the 𝑄𝑄2 (𝐴𝐴, 𝐵𝐵) set.

And since no other element exists apart from this six, so for all other elements, for all other
combinations we are writing 0. And this way we have written the relational matrix
𝑄𝑄2 (𝐴𝐴, 𝐵𝐵).

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(Refer Slide Time: 34:21)

And now we have to find the 𝑄𝑄3 (𝐴𝐴, 𝐵𝐵) based on the condition, the first element is equal to
the second element. So, since we already have the Cartesian product of set 𝐴𝐴 and set 𝐵𝐵 and
we apply this condition, the condition that is given which is first element is equal to the
second element, if we apply this. So, we get (2, 2); (3, 3); (4, 4) you know the pairs as the
elements of 𝑄𝑄3 (𝐴𝐴, 𝐵𝐵) here and, which satisfies the condition that first element of this pairs
is equal to the second element.

So, 𝑄𝑄3 (𝐴𝐴, 𝐵𝐵) has three elements only (2, 2); (3, 3) and (4, 4). So, first element, second
element and then the third element; and this (2, 2) is coming from here, (3, 3) coming
from here and (4, 4) is coming from here. So, this way we have another relation set, 𝑄𝑄3
based on the condition that was given, and now further we can write this in form of a
relational matrix. So, this relational matrix can be again on the same lines we can write.

So, we see that since we have (2, 2) presents, so for 2 here and 2 here, since (2, 2) is
present in 𝑄𝑄3 , so these two are forming one pair, so we are writing here 1. And then we
have (3, 3); (3, 3); means 3 of set 𝐴𝐴 and 3 of set 𝐵𝐵 these two also are forming one of the
elements of 𝑄𝑄3 , so we are writing 1 and then (4, 4) also, so we see that here writing 1.
Now since we have only 3 elements no other elements are present in 𝑄𝑄3 , so all other
elements will be put as 0 of this matrix and this is called a relational representation of
𝑄𝑄3 (𝐴𝐴, 𝐵𝐵) based on the condition that is the first element is equal to the second element.

509
So, this way we have seen that as to how we can find the Cartesian product of two crisp
sets. And from the Cartesian product of the two sets based on certain conditions, we find
the relation matrix, and based on the certain conditions we find the relation set, and this
can be represented in the matrix form, so we call this as the relational matrix. And also the
relation set that we get here is always the subset of the set which is coming out by the
Cartesian product of the sets.

So, in today’s lecture we have understood that, how do we find the Cartesian product of
two crisp sets and then how do we get the relation set based on certain conditions. And we
have seen this by taking couple of examples. And this way we have understood the
Cartesian product and relation and then relational matrix. And in the next lecture, we will
continue with the fuzzy relations.

Thank you.

510
 Fuzzy Sets, Logic and Systems and Applications
Prof. Nishchal K. Verma
Department of Electrical Engineering
Indian Institute of Technology, Kanpur

Lecture - 29
Fuzzy Relation

Welcome to lecture number 29 of Fuzzy Sets, Logic and Systems and Applications. So, in
this lecture today we will discuss Fuzzy Relations. Before discussing fuzzy relations, let
us first discuss the Cartesian product of fuzzy sets and then from this Cartesian product of
fuzzy sets we will find the fuzzy relation set and accordingly you know we will move
ahead.

(Refer Slide Time 01:06)

In the previous lecture we have seen the Cartesian product of a crisp sets. And, in the
Cartesian product of a crisp sets we had a set 𝐴𝐴 × 𝐵𝐵 that is actually the collection of all
the elements. And, these elements are the ordered pairs of the elements from 𝐴𝐴 and 𝐵𝐵.
Again, this pair must belong to the universe of discourse that is 𝑋𝑋 × 𝑌𝑌.

So, we can speak it like this like the Cartesian product of crisp sets 𝐴𝐴 and 𝐵𝐵 with the
universe of discourse 𝑋𝑋 and 𝑌𝑌 respectively is the crisp set of all ordered pairs 𝑥𝑥, 𝑦𝑦, such
that 𝑥𝑥 belongs to X and 𝑦𝑦 belongs to 𝑌𝑌 and it is denoted by 𝐴𝐴 × 𝐵𝐵. So, this all the ordered
pairs 𝑋𝑋, 𝑌𝑌 must belong to the universe of discourse 𝑋𝑋 × 𝑌𝑌 which is here.

511
So, this must be understood that all these ordered pairs 𝑥𝑥, 𝑦𝑦 will be from the universe of
discourse that has been created from 𝑋𝑋 and 𝑌𝑌 by taking the Cartesian product. So, this was
all about the Cartesian product of crisp sets.

(Refer Slide Time 02:47)

Now, let us see the Cartesian Product of Fuzzy Sets. So, if we have 𝐴𝐴 and 𝐵𝐵 two fuzzy sets
on the universe of discourse 𝑋𝑋 and 𝑌𝑌 respectively.

The Cartesian product between fuzzy sets 𝐴𝐴 and fuzzy set 𝐵𝐵 will be represented by 𝐴𝐴 × 𝐵𝐵
same as we have had in crisp set case. So, the resulting fuzzy set can be written as 𝐴𝐴 × 𝐵𝐵
and of course, this 𝐴𝐴 × 𝐵𝐵 is the resulting fuzzy set because this 𝐴𝐴 × 𝐵𝐵 is going to take the
form of a fuzzy set. So, 𝐴𝐴 × 𝐵𝐵 can be written by this expression here please understand
and we can write it like this like A cross B is equal to collection of all the ordered pairs
𝑥𝑥, 𝑦𝑦.

Along with you see here along with the membership value and this is the membership
value corresponding to 𝑥𝑥, 𝑦𝑦, this is ordered pair corresponding to the ordered pair element.
So, 𝜇𝜇𝐴𝐴×𝐵𝐵 (𝑥𝑥, 𝑦𝑦) is important here. So, we see that we have another element here which is
along with the ordered pair you see 𝑥𝑥, 𝑦𝑦 and so you see 𝑥𝑥, 𝑦𝑦 here which was there in case
of crisp Cartesian product.

So, this you know in case of Cartesian product of crisp sets, we had only the ordered pairs
means x y. Now, here we have along with 𝑥𝑥, 𝑦𝑦 we have the 𝜇𝜇𝐴𝐴×𝐵𝐵 (𝑥𝑥, 𝑦𝑦). So, this is because

512
it is a fuzzy set and mu is important because all the ordered pairs will have it is
corresponding membership values. So, that is why this 𝜇𝜇𝐴𝐴×𝐵𝐵 has been added along with
the ordered pairs.

So, this is a very important point that has to be noted this we have already discussed when
we were discussing the difference between a crisp set and the fuzzy sets. So, this is now
clear and now the membership function values that means the membership values
corresponding to the generic variable values arising from the 𝑥𝑥, 𝑦𝑦, that is I know this
membership value depends on both the generic variable values 𝑥𝑥 is for 𝑦𝑦.

So, you see here and this is going to be computed. So, that is very important here to note
that 𝜇𝜇𝐴𝐴×𝐵𝐵 (𝑥𝑥, 𝑦𝑦) = min(𝜇𝜇𝐴𝐴 (𝑥𝑥), 𝜇𝜇𝐵𝐵 (𝑦𝑦)). So, since we already have a fuzzy set 𝐴𝐴 and fuzzy
set 𝐵𝐵. So, we will be getting this 𝜇𝜇𝐴𝐴 (𝑥𝑥) from fuzzy set 𝐴𝐴 and then here we will get the
𝜇𝜇𝐵𝐵 (𝑦𝑦) from fuzzy set 𝐵𝐵. So, this is very important point that has to be noted.

And, this way by taking mean of these two values corresponding to the generic variable
value 𝑥𝑥,y we get the 𝜇𝜇𝐴𝐴×𝐵𝐵 (𝑥𝑥, 𝑦𝑦). So, this way we compute the membership value, of the
corresponding generic variable values, that is 𝑥𝑥, 𝑦𝑦 and this is nothing, but the ordered pair
element of 𝐴𝐴 × 𝐵𝐵. So, with this we now know that how 𝐴𝐴 × 𝐵𝐵 is going to look like.

So, 𝐴𝐴 × 𝐵𝐵 is here the Cartesian product and this Cartesian product is going to be the fuzzy
set again. So, Cartesian product of two fuzzy sets is again going to be a fuzzy set. And,
that is why we have the element and it is membership value associated with it as a element
of the 𝐴𝐴 × 𝐵𝐵 set.

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(Refer Slide Time 07:32)

Let us take an example here to understand this thing better. So, if we take an example like
this like here we have two fuzzy sets 𝐴𝐴 and 𝐵𝐵. And, here we have the universe of discourse
as 𝑋𝑋, which is equal to the 𝑥𝑥1 , 𝑥𝑥2 , 𝑥𝑥3 means we have in the universe of discourse 3 elements
𝑥𝑥1 , 𝑥𝑥2 , 𝑥𝑥3 and in the universe of discourse 𝑌𝑌 we have 𝑦𝑦1 and 𝑦𝑦2 .

So when we have this the fuzzy set 𝐴𝐴 𝐵𝐵 and the universe of discourse is already given. So,
then let us find the Cartesian product of fuzzy sets 𝐴𝐴 and 𝐵𝐵. So, we apply the expression
that we have just discussed and substitute the values of fuzzy set 𝐴𝐴 and fuzzy set 𝐵𝐵, which
is given here. So, since 𝐴𝐴 and 𝐵𝐵 are given as 0.2 / 𝑥𝑥1 + 0.5 /𝑥𝑥2 + 1 / 𝑥𝑥3 .

So, this is fuzzy set 𝐴𝐴 and here 𝐵𝐵 is equal to 0.3 / 𝑦𝑦1 + 0.9 / 𝑦𝑦2 . So, this our fuzzy set 𝐵𝐵
and please note that both of these fuzzy sets 𝐴𝐴 and 𝐵𝐵 are the discreet fuzzy sets.

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(Refer Slide Time 09:03)

Now, let us quickly find 𝐴𝐴 × 𝐵𝐵. A is given here as a discreet fuzzy set and 𝐵𝐵 is given as
another discreet fuzzy set. So, let us find the 𝐴𝐴 × 𝐵𝐵 which is the Cartesian product. So,
just apply the syntax of the set that we have discussed.

So, what we do here is we find all the ordered pairs of 𝐴𝐴 and 𝐵𝐵 elements. So, we find the
ordered pair generic variable values from 𝐴𝐴 and 𝐵𝐵. So, this way we get here since we have
in fuzzy set 𝐴𝐴, we have 𝑥𝑥1 , 𝑥𝑥2 , 𝑥𝑥3 , you can see here 𝑥𝑥1 , 𝑥𝑥2 , 𝑥𝑥3 and in B fuzzy set we have
𝑦𝑦1 , 𝑦𝑦2 . So, when we take the Cartesian product of A and B, that is A cross B, we are getting
here as 𝑥𝑥1 𝑦𝑦1 as one of the ordered pair elements.

And, then this ordered pair element will have it is associated membership value and I told
you as to how you are going to get this value, by taking the min of the corresponding
membership values from 𝐴𝐴 and 𝐵𝐵 corresponding to 𝑥𝑥1 and 𝑦𝑦1 respectively. So, here we are
writing 𝜇𝜇𝐴𝐴×𝐵𝐵 (𝑥𝑥1 , 𝑦𝑦1 ). So, this is going to be one of the elements of the resulting fuzzy set.

And, similarly now we’ll have 𝑥𝑥1 𝑦𝑦2 , so we have 𝑥𝑥1 𝑦𝑦2 and then the corresponding
membership value. Similarly, we’ll have 𝑥𝑥2 𝑦𝑦1 and then the corresponding membership
value. We will have 𝑥𝑥2 𝑦𝑦2 then corresponding membership value, 𝑥𝑥3 𝑦𝑦1 then the
corresponding membership value, 𝑥𝑥3 𝑦𝑦2 then the corresponding membership value. Since,
we have in fuzzy set A 𝑥𝑥1 , 𝑥𝑥2 , 𝑥𝑥3 in fuzzy set B 𝑦𝑦1 , 𝑦𝑦2 .

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So, we have three elements in fuzzy set 𝐴𝐴, 2 elements in fuzzy set 𝐵𝐵. So, we will have total
of 6 elements that is 3 into 2, that means 6 elements in the 𝐴𝐴 × 𝐵𝐵 set, that is the Cartesian
product of fuzzy set 𝐴𝐴 and fuzzy set 𝐵𝐵. So, then when we compute 𝜇𝜇𝐴𝐴×𝐵𝐵 (𝑥𝑥1 , 𝑦𝑦1 ) ,
similarly𝜇𝜇𝐴𝐴×𝐵𝐵 (𝑥𝑥1 , 𝑦𝑦2 ) all of these, that I have just mentioned as the membership values.

So, when we use the min criteria to find the respective membership values. So, when we
use this you see here, that for 𝑥𝑥1 , 𝑦𝑦1 we have to take the min(𝜇𝜇(𝑥𝑥1 ), 𝜇𝜇(𝑦𝑦1 )). Similarly, all
other corresponding membership values will be computing using the min criteria as I just
mentioned here and it is also written here you can see. So, this way our 𝐴𝐴 × 𝐵𝐵 becomes
the collection of all the elements.

And, these elements are nothing, but the ordered pair along with the it is corresponding
membership values, which are calculated by using min criteria. So, for 𝑥𝑥1 , 𝑦𝑦1 we have 0.2,
because if we take the min here you can see that when we take min of 0.2 and 0.3 we get
0.2. Similarly, for 𝑥𝑥1 , 𝑦𝑦2 we get 0.2 for 𝑥𝑥2 , 𝑦𝑦1 we get 0.3, for 𝑥𝑥2 , 𝑦𝑦2 we get 0.5, for 𝑥𝑥3 , 𝑦𝑦1
we get 0.3, for 𝑥𝑥3 , 𝑦𝑦1 we get 0.9 as the associated membership values. So, now, very easily
we have found out the Cartesian product of two fuzzy sets. So, in this fuzzy set we have
the ordered pair elements along with it is associated membership values.

(Refer Slide Time 13:41)

Now, the same can be written here by this equation and then this can also take the matrix
form. So, in the matrix form, if we are interested in writing the same equation the Cartesian
product we can use this matrix form here.

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So, where we have the column elements as the generic variable values of the fuzzy set 𝐴𝐴
and then we have the row as the elements the generic variable values of the fuzzy set 𝐵𝐵.
And, when we have these two as row and columns values, then corresponding to 𝑥𝑥1 , 𝑦𝑦1 we
write here as the 𝜇𝜇(𝑥𝑥1 , 𝑦𝑦1 ) like this and here we write the mu of 𝑥𝑥2 , 𝑦𝑦1 .

So, like that all the values are represented here. And of course this is nothing, but this is
from the Cartesian product of 𝐴𝐴 and 𝐵𝐵, that is 𝐴𝐴 × 𝐵𝐵. So, this is very easy to represent.
And, similarly here also we have 𝑥𝑥1 , 𝑦𝑦2 and this is corresponding to 𝜇𝜇(𝑥𝑥1 , 𝑦𝑦2 ). Since, this
is from the 𝐴𝐴 × 𝐵𝐵 set that is the Cartesian product we can also write 𝐴𝐴 × 𝐵𝐵 here also we
can write 𝐴𝐴 × 𝐵𝐵.

So, this way we see as to how all these membership values associated with the combination
of generic variable values from fuzzy set A and fuzzy set B that is 𝑥𝑥1 , 𝑥𝑥2 , 𝑥𝑥3 , 𝑦𝑦1 , 𝑦𝑦2 . So, we
have listed here all the membership values as elements of the matrix of A cross B that is
Cartesian product of 𝐴𝐴 and 𝐵𝐵. So, this is another way of writing this equation here. So, the
same is represented in this form.

And, this is actually easier to understand here by just looking at the elements we can
understand. And, these values once again I am telling that these values are here the these
values these elements of the matrix are nothing, but the associated membership values.
And, these membership values are computed by taking the min of the corresponding
membership values from fuzzy set A and fuzzy set B.

(Refer Slide Time 16:25)

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So, once we now know the Cartesian product of two fuzzy sets A and B. Now, let us
understand the fuzzy relations on the same lines as we have seen in the crisp relation and
we will understand as to how we find the fuzzy relation from the Cartesian product.

(Refer Slide Time 16:48)

So, here let us first take 𝐴𝐴 and 𝐵𝐵 two crisp sets with the universe of discourse 𝑋𝑋 and 𝑌𝑌,
respectively and here we are interested in finding the relation between 𝐴𝐴 and 𝐵𝐵. In terms
of the approachability among the cities, means if I am interested to move or approach from
one city to another, what is the relation? Right.

So, here we have two crisp sets, the crisp set 𝐴𝐴 here is the set of cities from US, that are
Los Angeles, Washington DC, Seattle and then we have another crisp set 𝐵𝐵 which is set
of Indian cities like Mumbai, New Delhi, Kanpur. So, here we have two crisp sets 𝐴𝐴 and
𝐵𝐵.

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(Refer Slide Time 17:49)

So, here the term Approachability does not have clear boundaries. So, here we cannot
simply define the crisps relation, between elements of crisp set A and crisp set B in binary
terms. That is approachable, 1 or non-approachable 0, means 1 for approachable and 0 for
non-approachable. So, when we talk of approachability from one city to another. So, we
cannot express this thing in 0 and 1.

So this is very clear that Boolean logic is not sufficient here to express the approachability
among the cities. Now, here when we use fuzzy logic we can see that, this approachability
can be expressed very easily and now let us see how it is done. So, in such cases the relation
can be represented by fuzzy relational matrix when we fuzzy relation.

519
(Refer Slide Time 18:52)

So, you see here as I have already mentioned that we have two sets the 𝐴𝐴 and 𝐵𝐵. So, both
the sets are crisp sets, but when we talk of approachability this cannot be very well
expressed, but then when we use a fuzzy relation matrix as you have seen in the previous
slides, that we have 𝐴𝐴 as the set of cities and 𝐵𝐵 another set of cities. So, the elements here
are nothing, but the membership values.

The elements of this matrix the relational matrix nothing but the membership value is like
0.1, 0.7, 0.6 in first column; 0.9, 0.8, 0.7 the second column; 0.1, 0.3, 0.2 in another
column. So, this is here is nothing, but A fuzzy set. So, when we use fuzzy logic we can
represent the approachability in a very convenient fashion. And, here what this represents
is like if we are interested in traveling from the Los Angeles to Mumbai we have 0.8 as
the degree of membership.

This means that you know it is approachability is very good that is 0.8 or if we want to
compare with the other cities like Washington and Mumbai which is 0.7. So, this means
that the traveling from Washington to Mumbai is not as good as traveling from the Los
Angeles to Mumbai, when it comes to approachability.

So, here we have all these membership degrees as the degree of approachability see here.
So, this is in a more convenient fashion we can represent when we take 𝐴𝐴 and 𝐵𝐵 both are
the fuzzy sets. So, if we take only crisp sets, it is very difficult because this kind of things
cannot be represented in Boolean logic.

520
(Refer Slide Time 21:13)

So, as I have already mentioned that when we talk of fuzzy relation normally, fuzzy
relation is the Cartesian product as we have seen. So, the Cartesian product of two fuzzy
sets result basically a fuzzy set again and this is nothing but the fuzzy relation unless
otherwise some other condition is satisfied and if some other condition is satisfied, then
the relation set based on certain conditions will be a subset of the Cartesian product of the
two sets 𝐴𝐴 and 𝐵𝐵.

And, of course, here this has to be noted that we have a universe of discourse which is
again the Cartesian product of 𝑋𝑋 and 𝑌𝑌. So, we have a space that we have the Cartesian
product space capital 𝑋𝑋 × 𝑌𝑌 and this represents the you know the universe of discourse of
the Cartesian product. And, since we have already used 𝑄𝑄 for representing the relation set
like in crisp sets we have seen.

So here also we are using 𝑄𝑄(𝑥𝑥, 𝑦𝑦). One more thing has to be noted here, that when we are
going for Cartesian product.

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(Refer Slide Time 22:12)

So, we need to understand here that Cartesian product fuzzy set here will have the
increased dimension, like fuzzy set 𝐴𝐴 which is defined in the universe of discourse 𝑋𝑋 and
the generic variable value here is 𝑥𝑥, whereas, fuzzy set 𝐵𝐵 which is defined in the universe
of discourse 𝑌𝑌 and here the generic variable value is 𝑦𝑦.

So, both these sets are defined in single dimension 𝑋𝑋 and 𝑌𝑌 respectively, but when you
combine the them together, when you take the Cartesian product of these two fuzzy sets.
What we get here is a resulting fuzzy set, which is a surface, which is nothing, but it is a
increased dimension it is a two dimensional 𝑋𝑋 and 𝑌𝑌, and then when we take a another
dimension as the membership values of the corresponding 𝑋𝑋 and 𝑌𝑌 then we get a 3-D
representation.

So that is how it is normally called as the surface. So, surface is a 3-D surface is represented
in 3 3 D space. So, that is why 𝑄𝑄(𝑥𝑥, 𝑦𝑦) here is a fuzzy set. And, this is again if it is based
on certain condition, we get this set if there is no condition then the complete ordered pair
elements along with the membership values are included here in this set.

And, as I already mentioned that this fuzzy set is a 3-D is represented in 3-D space. So,
two dimensions are for 𝑋𝑋 and 𝑌𝑌, generic variable values and then mu of 𝑋𝑋 and 𝑌𝑌 will be
the third dimension, which is here for representing the membership degree. So, that is why
it is a 3-D representation.

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(Refer Slide Time 24:28)

Now, like in crisp relation we have seen for n crisp sets the crisp relation for n or crisp
relation of 𝑛𝑛 crisp sets.

Here also let us see, how the fuzzy relation for n fuzzy sets look like. So, here also we have
𝐴𝐴1 , 𝐴𝐴2 , 𝐴𝐴3 … . . 𝐴𝐴𝑛𝑛 , 𝑛𝑛 number of fuzzy sets. So, let us take the Cartesian product of this 𝑛𝑛
number of fuzzy sets. So, how will these look like? So, again like you have seen in the
previous slide, that if we take the Cartesian product of two fuzzy sets we get the increased
dimension here, also we get the dimension increase small n number of times.

So, here we have the ordered pairs as you see here 𝑥𝑥1 , 𝑥𝑥2 . So, this is not called ordered pair
this is called basically 𝑛𝑛 tuples, 𝑛𝑛 tuples are here like
𝑥𝑥1 , 𝑥𝑥2 , 𝑥𝑥3 … . 𝑥𝑥𝑛𝑛 , 𝜇𝜇𝑄𝑄 (𝑥𝑥1 , 𝑥𝑥2 , 𝑥𝑥3 … . 𝑥𝑥𝑛𝑛 ). And, so, that is how it looks like and then again
this 𝑥𝑥1 , 𝑥𝑥2 , 𝑥𝑥3 … . 𝑥𝑥𝑛𝑛 this will be from the Cartesian product of capital 𝑋𝑋1 , 𝑋𝑋2 , 𝑋𝑋3 and so on
up to capital 𝑋𝑋𝑛𝑛 is space.

So, this has to be noted that here also we have a fuzzy set, which is arising out of the
Cartesian product of n fuzzy sets. And, of course, the dimension here will be the n
dimensional fuzzy set.

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(Refer Slide Time 26:12)

Now, let us move ahead and see some operations on fuzzy relation. Please understand
fuzzy relation is a fuzzy set. Fuzzy relation is the fuzzy set in an increased dimension.

So, like any other fuzzy set the fuzzy relation is also fuzzy set. So, if this can be represented
by 𝑅𝑅 and S. So, and again if we define the space the universe of discourse as 𝑋𝑋 crosswise.
So, these relations will have the operations as given below here as the union of fuzzy
relations.

So, we have the union of these fuzzy relations, and then intersection of the fuzzy relations,
complement of the fuzzy relation, and then the containment of the fuzzy relation. So, with
this we will stop here in this lecture.

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(Refer Slide Time 27:13)

And, in the next lecture we will cover the these operations first on crisps and then on fuzzy
relation sets.

Thank you.

525
 Fuzzy Sets, Logic and Systems and Applications
Prof. Nishchal K. Verma
Department of Electrical Engineering
Indian Institute of Technology, Kanpur

Lecture – 30
Operations on Crisp and Fuzzy Relations

Welcome to lecture number 30, Fuzzy Sets, Logic and Systems and Applications. So, in
this lecture we will discuss the Operations on Crisp and Fuzzy Relations. And of course
as I already mentioned in my previous lecture that fuzzy relation set is also a fuzzy set.

(Refer Slide Time 00:50)

So, in the previous class I mentioned that related to these fuzzy relation set, we have the
operations as union, intersection, complement and containment.

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(Refer Slide Time 00:58)

And, let us first understand, what is the union of relations? So, let us first take the union
of crisp set. So, if we have two crisp sets 𝑅𝑅 and 𝑆𝑆 and these sets are the relation sets. So,
we have in other words we can say, the 𝑅𝑅 and 𝑆𝑆 be the crisp relations defined on the space
𝑋𝑋 × 𝑌𝑌. And, then the union is defined by another crisp set that is 𝑇𝑇 = 𝑅𝑅 ∪ 𝑆𝑆, you can see
here and this 𝑇𝑇 is said to be the union of 𝑅𝑅 and 𝑆𝑆.

And, here this 𝑇𝑇 set will contain the ordered pairs as elements in this set. So, for every
ordered paired ∀(𝑥𝑥, 𝑦𝑦) ∈ 𝑅𝑅 𝑜𝑜𝑜𝑜 ∀(𝑥𝑥, 𝑦𝑦) ∈ 𝑆𝑆. So, then what is happening to the elements of
the set 𝑇𝑇. So, here also for every (𝑥𝑥, 𝑦𝑦), that is the ordered pairs, that is belonging into
fuzzy relations 𝑇𝑇 such that for every ordered pair (𝑥𝑥, 𝑦𝑦) is belonging into 𝑋𝑋 × 𝑌𝑌.

So, this was basically for the crisp relation sets 𝑅𝑅 and 𝑆𝑆 and then we had out of the union,
we have another fuzzy set which is 𝑇𝑇, which is coming out of the 𝑅𝑅 ∪ 𝑆𝑆.

527
(Refer Slide Time 02:52)

Now, let us discuss the union of fuzzy relations. So, as we have seen earlier also that there
is a difference in crisp and fuzzy relation. And, what is that difference? Difference here is
that here in fuzzy relation, and as of course, I have already told you, that fuzzy relation is
a fuzzy set finally. So, the fuzzy relations set will have it is corresponding membership
values.

So, that is, what is the difference here other than this there is no difference. So, in fuzzy
relations set, we will have apart from the ordered pairs we will have the corresponding
membership values. Here we have the ordered pair and then we have the 𝜇𝜇 𝑇𝑇 (𝑥𝑥, 𝑦𝑦) means,
the corresponding membership value. So, this way we have the union of fuzzy relations
and these are the fuzzy relations sets. And, as I have already mentioned that these 𝑅𝑅 and 𝑆𝑆
basically fuzzy relations, but these are fuzzy sets.

Now, how to get this 𝜇𝜇 𝑇𝑇 (𝑥𝑥, 𝑦𝑦) means 𝜇𝜇 𝑇𝑇 of the ordered pair is the corresponding
membership value? So, how to get that in case of union? So, in case of union you see here
𝜇𝜇 𝑇𝑇 (𝑥𝑥, 𝑦𝑦) = max(𝜇𝜇𝑅𝑅 (𝑥𝑥, 𝑦𝑦), 𝜇𝜇𝑆𝑆 (𝑥𝑥. 𝑦𝑦)). So, this means what? This means, you see here we
take the corresponding membership value, which is present in the fuzzy relation set 𝑅𝑅 and
then fuzzy relation set 𝑆𝑆.

And, when we are taking the union of it the 𝑅𝑅 and 𝑆𝑆, then we take the max of these two
membership values and then we term this as the 𝜇𝜇 𝑇𝑇 (𝑥𝑥, 𝑦𝑦). So, this way we have find the
corresponding membership value corresponding to the order pair. And, of course, it is

528
needless to mention here that for every (𝑥𝑥, 𝑦𝑦) ∈ 𝑆𝑆 such that for every (𝑥𝑥, 𝑦𝑦) ∈ 𝑋𝑋 × 𝑌𝑌. And,
this is because 𝑅𝑅 and 𝑆𝑆 are already defined in this space capital 𝑋𝑋 × 𝑌𝑌, because 𝑅𝑅 and 𝑆𝑆
are the fuzzy relations set.

(Refer Slide Time 05:35)

So, when we talk of the intersection of crisp relations? We will go similar on similar lines
you see that the when we talk of crisp relations? So, you see here for crisp relation you
take the intersection and when you take the intersection, you take only the common
elements right.

So, 𝑇𝑇 said to be the intersection of 𝑅𝑅 and 𝑆𝑆 if for every 𝑥𝑥, 𝑦𝑦, that is belonging into 𝑅𝑅 and
belonging into 𝑆𝑆. And, the resulting set here will be the 𝑥𝑥, 𝑦𝑦 that will belong into the 𝑇𝑇,
that is the ordered paired element, which will belong into the 𝑇𝑇, which is the outcome of
the intersection of 𝑅𝑅 and 𝑆𝑆.

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(Refer Slide Time 06:30)

Now, let us understand the intersection of fuzzy relations with respect to here again the
fuzzy relations we have to have an additional term here in the set. So, 𝑇𝑇 set here we will
have the associated membership values, which was not there in the crisp set. And, you
know why I have already explained a couple of times that here in a fuzzy set we have to
have this 𝜇𝜇 𝑇𝑇 (𝑥𝑥, 𝑦𝑦) or the associated membership values, associate and otherwise you know
we may not be aware we may not be knowing as to with what membership value a
particular element is adjusting in the fuzzy set.

So, let us now quickly define this. So, 𝑇𝑇 here is the fuzzy set and here the element will be
the 𝑥𝑥, 𝑦𝑦 which is ordered pair. And, then it is corresponding membership value 𝜇𝜇 𝑇𝑇 (𝑥𝑥, 𝑦𝑦)
and let us see as to how we can compute 𝜇𝜇 𝑇𝑇 (𝑥𝑥, 𝑦𝑦). So, here instead of max as in the
previous case where we were taking union here we since we are having we are interested
in finding the intersection. So, we take min of the two membership values.

So, when we are taking the intersection of fuzzy relations we use the min criteria. So, when
we take the min of these two membership values. The resulting value will be termed as
𝜇𝜇 𝑇𝑇 (𝑥𝑥, 𝑦𝑦). And, which is coming because of the intersection of 𝑅𝑅 and 𝑆𝑆 fuzzy relation set.

So, it is again needless to mention here that for every 𝑥𝑥, 𝑦𝑦, which is belonging into 𝑅𝑅 and
𝑆𝑆 and then this x and 𝑥𝑥, 𝑦𝑦 will also be coming from 𝑥𝑥 × 𝑦𝑦. So, 𝑅𝑅 𝑆𝑆 both are the fuzzy
relation set and this is again drawn from 𝑋𝑋 × 𝑌𝑌, which is the Cartesian product space.

530
(Refer Slide Time 09:00)

So, now coming to complement of crisp relation. So, as we have seen union and
intersection here, the complement of crisp relation we’ll first discuss and then we will go
to the complement of fuzzy relations. So, as we have already seen in the case of crisp set,
how to find the complement of any crisp set. Here also since we are discussing about the
crisp relations. So, crisp relation again is a crisp set.

So, let R be the crisp relation defined on the space 𝑋𝑋 × 𝑌𝑌, then the complement of relation
capital R is defined as, if the ordered pair 𝑥𝑥, 𝑦𝑦 ordered paired element, which is not
belonging into R, then x comma y will belong into the 𝑅𝑅� ; 𝑅𝑅� is the complement set.

And, this complement of relation 𝑅𝑅 will be basically the collection of all the ordered pairs
elements like 𝑥𝑥, 𝑦𝑦, such that for every 𝑥𝑥, 𝑦𝑦 is not belonging into the set 𝑅𝑅, that is the crisp
relation set 𝑅𝑅 that we have taken. So, it is very easy to understand that we will include all
the ordered pairs, which are not there in 𝑅𝑅. And, all the ordered pairs means the all the
ordered pairs that are existing in the Cartesian product space. So, this way we have
understood the complement of crisp relation. Now, let us go to the complement of a fuzzy
relation.

531
(Refer Slide Time 10:52)

So, as I have already mentioned that here the difference is that we include the membership
values along with the ordered pair elements. So, here we have 𝜇𝜇𝑅𝑅� (𝑥𝑥, 𝑦𝑦). So, this is
complement of relation 𝑅𝑅 is represented by 𝑅𝑅� , which is here you see. So, this is basically
the collection of these equal to the set which is collection of all the ordered pair elements.

And, these elements are those elements which are not existing in the set that we have taken,
but these are existing in the Cartesian product space, 𝑋𝑋 × 𝑌𝑌. And, this along with the
membership values. So, how to find this membership value? 𝜇𝜇𝑅𝑅� (𝑥𝑥, 𝑦𝑦) see here. So, this
very easy we here take a very basic complement, otherwise you can take other
complements also like we have done in previous lectures.

So, we are discussing only the basic complement here which is 1 − 𝜇𝜇𝑅𝑅 (𝑥𝑥, 𝑦𝑦). So, if we
apply this we will get we will compute the 𝜇𝜇𝑅𝑅� (𝑥𝑥, 𝑦𝑦). So, this way it is very easy to compute
the complement of fuzzy relation.

532
(Refer Slide Time 12:21)

Now, coming over to the containment of crisp relation. So, if we have any two fuzzy sets
𝑅𝑅 𝑎𝑎𝑎𝑎𝑎𝑎 𝑆𝑆. So, here let us first before we move to fuzzy sets let us first understand the crisp
relation, then we see the transition from crisp relation to fuzzy relation.

So, if we have 𝑅𝑅 𝑎𝑎𝑎𝑎𝑎𝑎 𝑆𝑆 as crisp relation set. So, then the containment is defined by the set,
which is let us say it 𝑇𝑇 = 𝑅𝑅 ⊂ 𝑆𝑆. So, 𝑅𝑅 is contained in 𝑆𝑆 if for every 𝑥𝑥, 𝑦𝑦, which is the
ordered pair element belonging into 𝑅𝑅(𝑥𝑥, 𝑦𝑦) again belonging into capital 𝑆𝑆.

So, this way then 𝑅𝑅(𝑥𝑥, 𝑦𝑦) ≤ 𝑆𝑆(𝑥𝑥, 𝑦𝑦)|∀(𝑥𝑥, 𝑦𝑦) ∈ 𝑋𝑋 × 𝑌𝑌. So, let us now move to the
containment of fuzzy relation.

533
(Refer Slide Time 13:40)

So, here as I just mentioned initially, so if we have 𝑅𝑅 and 𝑆𝑆 as fuzzy relation sets how to
represent the containment here. So, containment if we have the containment right like 𝑅𝑅 is
contained in 𝑆𝑆. So, that is possible only when if for every ordered pair elements 𝑥𝑥, 𝑦𝑦 is
belonging into 𝑅𝑅 and 𝑆𝑆.

And then 𝜇𝜇𝑅𝑅 (𝑥𝑥, 𝑦𝑦) ≤ 𝜇𝜇𝑆𝑆 (𝑥𝑥, 𝑦𝑦)|∀(𝑥𝑥, 𝑦𝑦) ∈ 𝑋𝑋 × 𝑌𝑌.

(Refer Slide Time 14:29)

534
So, this can be very well understood by taking one example here. And, in this example
here we have taken two crisp sets 𝐴𝐴 and 𝐵𝐵. So, let us first understand that here we have
two crisp sets 𝐴𝐴 and 𝐵𝐵 and the universe of discourse of both the sets like 𝑋𝑋 and 𝑌𝑌 are also
the same. So, let us first point the Cartesian product of 𝐴𝐴 and 𝐵𝐵 that means, 𝐴𝐴 × 𝐵𝐵.

(Refer Slide Time 15:00)

So, as we have already seen that how we can get the Cartesian product of crisp set 𝐴𝐴 and
𝐵𝐵? So, we can very easily find the Cartesian product of two crisp sets 𝐴𝐴 and 𝐵𝐵, you can
see here. So, we have all of these elements. So, very easy to quickly find and this way
when we have found this then now let us go to the relation matrix.

535
(Refer Slide Time 15:26)

So, let us put some condition as I have already mentioned in case of crisp sets in case of
crisp relations. So, here we have the complete population we have the ordered pair
elements in 𝐴𝐴 × 𝐵𝐵. So, now, let us put some condition here and the condition that we are
putting here is the first element is greater or equal to the second element.

So, if we put this condition here, that is this case and let this be represented by 𝑄𝑄4 . So,
𝑄𝑄 here is co relation so 𝑄𝑄4 (𝐴𝐴, 𝐵𝐵) here. So, this represents the relation in crisp set 𝐴𝐴, 𝐵𝐵. So,
we have collected here all those elements which follow the condition that has been stated
here, like the first element is greater or equal to the second element like (2, 2); (3, 2);
(3, 3); (4, 2); (4, 3); (4, 4) all these have been included.

And, the same can be represented by the relational matrix. So, we can see that we have
few 1s and few 0s. So, the elements that are existing the pairs that are existing here like,
from 𝐴𝐴 2 as the element and from 𝐵𝐵 2 also as the element both are forming the ordered
pair in 𝑄𝑄4 (𝐴𝐴, 𝐵𝐵).

So, that is why this is existing here. So, that is why one has been put here as one of the
elements of the relation matrix. Similarly, here (3, 2) is also existing, then (4, 2) is also
existing and then (3, 3) is existing, (4, 3) is existing and then (4, 4) is also existing in the
𝑄𝑄4 (𝐴𝐴, 𝐵𝐵) set no other elements are existing. So, that is why other elements have been put
as 0.

536
(Refer Slide Time 17:27)

Now, when we have this 𝑄𝑄4 (𝐴𝐴, 𝐵𝐵) since this is a crisp set as the relation set. So, if we are
interested in finding the complement of this relation set, we can quickly see as to how we
����
can find that. So, if it is the complement relation. So, we represent this by the 𝑄𝑄4 you can

see here.

���4 (𝐴𝐴, 𝐵𝐵) and this is equal to you know A relational representation here, we see
So, this is 𝑄𝑄
that we change 1 into 0 and 0 into 1 means, those elements which were not present in
𝑄𝑄4 (𝐴𝐴, 𝐵𝐵) are present here in this set ���
𝑄𝑄4 (𝐴𝐴, 𝐵𝐵). So, this way we find the complement of a
crisp relation.

537
(Refer Slide Time 18:24)

Now, another relation set that is 𝑄𝑄5 (𝐴𝐴, 𝐵𝐵) such that the second element is greater than or
equal to the first element. So, on the same lines we can find this set here you can just try.
And, then we find the relational matrix here as I have described in the previous case.

(Refer Slide Time 18:45)

And, if we are interested in finding the complement we can quickly get the complement
by just changing 1 to 0 and 0 to 1. So, this way the complement is found.

538
(Refer Slide Time 18:57)

Now, as the fourth case here, we are interested in finding the union of 𝑄𝑄4 and 𝑄𝑄5 and then
the intersection of 𝑄𝑄4 and 𝑄𝑄5 . So, we have 𝑄𝑄4 here 𝑄𝑄4 relation set the crisp relation set and
then we have 𝑄𝑄5 as the another crisp relation set. So, we have two crisp religion sets. Let
us now find the union first.

So, when we take the union you see here, we again see that we have those elements which
are present in the set 𝑄𝑄4 (𝐴𝐴, 𝐵𝐵). And, again those elements which are present in 𝑄𝑄5 (𝐴𝐴, 𝐵𝐵)
you see. So, all those elements have are being accounted. So, this way all the elements are
present here. So, 𝑄𝑄4 𝑄𝑄5 we look at the relational matrix and then we keep all once, which
are in both the relation sets that are 𝑄𝑄4 𝑄𝑄5 .

539
(Refer Slide Time 20:08)

Now, when we are taking the intersection, so in intersection we only take the common 1s.
So, if we see here in 𝑄𝑄4 (𝐴𝐴, 𝐵𝐵) these 1s are present in 𝑄𝑄4 (𝐴𝐴, 𝐵𝐵) and this 1s are also present
in 𝑄𝑄5 (𝐴𝐴, 𝐵𝐵). No other 1s are present in both that relations matrix 𝑄𝑄4 and 𝑄𝑄5 . So, that is
why only these are kept. So, this is how we get the crisp relation operations done.

(Refer Slide Time 20:48)

And, similarly if we are interested applying the operations on fuzzy relation set. So, we
can also do that and as we have already seen that, we have fuzzy relations set like this if

540
we have an 𝑅𝑅 fuzzy relation set and another fuzzy relation set S. So, we can represent this
𝑅𝑅 fuzzy relation set like this.

And, 𝑆𝑆 fuzzy relation set like this and if we are interested in the applying the operations
on in the fuzzy relations these 𝑅𝑅 and 𝑆𝑆 sets. So, let us now see as to how we can move
ahead.

(Refer Slide Time 21:32)

So, here basically our intention is to find the union of these two fuzzy relations. So, 𝑅𝑅 is
the fuzzy relation I am writing here set and 𝑆𝑆 also. So, both of these are the fuzzy relation
sets. Now, please recall as to how we can find the union of these two fuzzy relation sets.
So, if you recall we see that we have fuzzy relation set, which is out of the union of the 2
𝑅𝑅 and 𝑆𝑆.

541
(Refer Slide Time 22:16)

And, the membership values we represent this by 𝜇𝜇 𝑇𝑇 (𝑥𝑥, 𝑦𝑦). So, this way we represent.

(Refer Slide Time 22:24)

And, when we are taking the union here. So, in union what we do, we recall that we have
applied the max criteria and we take the max of the membership values corresponding to
the ordered paired elements. So, here we see that when we take the union. So, we see that
we have 0.3 here we have 0.3 here. And, if we take the union we use the max. So, here we
apply max criteria. And, this way we get the 𝜇𝜇 𝑇𝑇 (𝑥𝑥, 𝑦𝑦).

542
So, all the elements of mu T can be computed by applying max very quickly. And, all these
have been associated along with the generic variable values or the ordered pair elements
like 𝑥𝑥, 𝑦𝑦 in the resulting fuzzy religion set 𝑇𝑇.

(Refer Slide Time 23:29)

And, so if this was the union when we take the intersection? So, in intersection instead of
max we take the minimum and when we take minimum let us say we take this element and
this element and corresponding to the ordered pair values we take min we are getting here
0.1.

So, this is nothing but the mu is as 𝜇𝜇𝑆𝑆 (𝑥𝑥, 𝑦𝑦) and then this is 𝜇𝜇𝑅𝑅 (𝑥𝑥, 𝑦𝑦) and this is here is
𝜇𝜇 𝑇𝑇 (𝑥𝑥, 𝑦𝑦). So, all these corresponding values can be very easily computed and this way we
find the intersection of two fuzzy relations.

543
(Refer Slide Time 24:17)

And, then when it comes to complement of fuzzy relation then we apply this criteria we
simply subtract the corresponding membership values from 1, which is again the basic
complement. And, if you wish you can apply any other complements that I have already
taught in the previous lectures. So, if you want to get the complement of fuzzy relation
capital 𝑅𝑅 set. So, you can quickly write the fuzzy relation 𝑅𝑅 set here.

And, then how to get this 𝑅𝑅� is just a subtract all the corresponding elements here
corresponding elements means you see these are the membership values and these
membership values are subtracted from 1. So, when you subtract this value from 1 so that
means, that we are subtracting 0.3 from 1 and this is going to give us the value which is
0.7. So, likewise all other values of the complement of fuzzy relation set we get and this
way we managed to get the complement of any fuzzy relation set.

Similarly, we can get the complement of fuzzy relation set 𝑆𝑆 and which is represented by
𝑆𝑆̅. Here also if we see let us say we take 0.1 and the corresponding the element in 𝑆𝑆̅ will
be 0.9, because if I subtract 0.1 from 1 we are going to get 0.9. So, this is how we get the
corresponding a membership value which is the membership value of the complement of
a fuzzy relations set.

544
(Refer Slide Time 26:13)

So, this is how we get the these operations applied on fuzzy relations and now when it
comes to the containment for fuzzy relations. So, similarly if fuzzy relation set 𝑅𝑅 ⊂ 𝑆𝑆 then
you see if this is the case then 𝜇𝜇𝑅𝑅 (𝑥𝑥, 𝑦𝑦) ≤ 𝜇𝜇𝑆𝑆 (𝑥𝑥, 𝑦𝑦). And, this is for every 𝑥𝑥 ∈ 𝑋𝑋 and 𝑦𝑦 ∈
𝑌𝑌. So, if we have two sets you see here. So, with these two fuzzy relation sets that we have
if we put this condition here, so we find that R is not a subset of S, because this condition
is not satisfied; that means, all the membership values of set 𝑅𝑅 is not less than or equal to
the membership values of relation set 𝑆𝑆. So, that is why we can say that 𝑅𝑅 ⊄ 𝑆𝑆. So, this is
mentioned over here. So, therefore, we can say 𝑅𝑅 is not contained in 𝑆𝑆.

(Refer Slide Time 27:28)

545
So, this way we have seen that we have understood operations they complement
intersection union, containment with respect to crisp sets and fuzzy sets in today’s lecture
and not only fuzzy sets, but I would say we have studied these operations on the fuzzy
relation set of course, the fuzzy relation set is also a fuzzy set. We would like to stop here
and in the next lecture, we will discuss the following the projection of fuzzy relation,
cylindrical extension of projection, properties of fuzzy relations.

Thank you.

546
 Fuzzy Sets, Logic and Systems and Applications
Prof. Nishchal K. Verma
Department of Electrical Engineering
Indian Institute of Technology, Kanpur

Lecture – 31
Projection of Fuzzy Relation Set

So, welcome to lecture number 31 of Fuzzy Sets, Logic and Systems and Applications. In
this lecture we will cover projection of fuzzy relation set. So, let us consider a fuzzy
relation first represented by capital 𝑅𝑅.

(Refer Slide Time: 00:37)

And this 𝑅𝑅 ⊆ 𝐴𝐴 × 𝐵𝐵; that means R basically is either a subset of A cross B or R is equal
to the Cartesian product of A and B; that means, 𝐴𝐴 × 𝐵𝐵.

So, we have already seen that how this 𝑅𝑅 looks like. So, when we take the Cartesian
product, 𝐴𝐴 of course is defined in the universe of discourse in 𝑋𝑋 and 𝐵𝐵 is defined in the
universe of discourse 𝑌𝑌. So when we define 𝑅𝑅, when we define a fuzzy relation you can
see here that is 𝑅𝑅.

So, 𝑅𝑅 = 𝐴𝐴 × 𝐵𝐵 = ��(𝑥𝑥, 𝑦𝑦), 𝜇𝜇𝑅𝑅 (𝑥𝑥, 𝑦𝑦)�|∀(𝑥𝑥, 𝑦𝑦) ∈ 𝑋𝑋 × 𝑌𝑌�

547
So, 𝑅𝑅 is a multidimensional fuzzy set. So, here in this case we have 𝑅𝑅, 𝐴𝐴 × 𝐵𝐵 and 𝐴𝐴 × 𝐵𝐵
as I mentioned is in terms of 𝑥𝑥 and 𝑦𝑦. Means, we have the universe of discourse x and y.
So, the Cartesian space that we have here is the 𝑋𝑋 × 𝑌𝑌.

Now, let us discuss the projection of fuzzy relation; what is a projection of fuzzy relation?
As I have already mentioned that a fuzzy relation is a multidimensional fuzzy set. When
we say multidimensional, it means that we have the relation fuzzy set is defined on
multiple universe of discourse like 𝑋𝑋 and 𝑌𝑌 are even more even further like 𝑋𝑋, 𝑌𝑌, 𝑍𝑍 and so
on.

So, when we have fuzzy relation 𝑅𝑅, if we are interested in projecting this 𝑅𝑅 to its
constituents sets, like here in this case, we have 𝐴𝐴 and 𝐵𝐵. So, if we are interested in
projecting the relation set 𝑅𝑅 on 𝐴𝐴 and we know that the universe of discourse of the fuzzy
set 𝐴𝐴 is 𝑋𝑋.

So, before I move to the membership function of this, let me tell you that the projection of
any fuzzy set reduces the dimensionality. What does this mean? This means that if we have
any multidimensional set and if we project this fuzzy set to some other set, the projection
is going to reduce the dimensionality of the original set and it reduces the dimensionality
when we project.

So, let us now understand the projection. So, projection basically here when we talk of
there are fuzzy relation; fuzzy relation is defined in the universe of discourse 𝑋𝑋 × 𝑌𝑌. So, it
means its a multidimensional fuzzy set.

When we are going to project this 𝑅𝑅 either on 𝐴𝐴, which is a constituent fuzzy set or on 𝐵𝐵
fuzzy set which is again of constituent fuzzy set; so, we can project this fuzzy relation set
either on A or on B.

So, when we say project fuzzy relation set 𝑅𝑅 on fuzzy set 𝐴𝐴 it means, we are reducing the
dimensionality. And when we are reducing the dimensionality, it means that when we are
projecting 𝑅𝑅 on 𝐴𝐴, it means we are retaining its resulting fuzzy set. It means we are
retaining the universe of discourse 𝑋𝑋 because we are projecting 𝑅𝑅 on 𝐴𝐴.

Similarly, when we are projecting the relation fuzzy set 𝑅𝑅 on 𝐵𝐵, so this means that we are
going to retain the universe of discourse 𝑌𝑌, of the resulting fuzzy set. Let this be

548
represented by the projection of 𝑅𝑅 on 𝐴𝐴 is represented by 𝑅𝑅𝐴𝐴 . And here similarly, the
projection of 𝑅𝑅 on 𝐵𝐵 is represented by 𝑅𝑅𝐵𝐵 . So as I mentioned, that the resulting fuzzy set
is going to be a fuzzy set with reduced dimensionality, it means that the universe of
discourse is going to be reduced.

So, in this case since we are taking capital 𝑅𝑅, which is defined in the universe of discourse
𝑋𝑋 and 𝑌𝑌, so if we are projecting the relation fuzzy set 𝑅𝑅 on 𝐴𝐴, it means we are going to
have only one universe of discourse of the resulting set. And that is again since this is the
projection is on 𝐴𝐴, the universe of discourse will be only 𝐴𝐴.

So, this is here the expression by which we can reduce the dimensionality, we can get the
projection of 𝑅𝑅 and the resulting fuzzy sets membership values can be computed by this
expression.

Similarly, when we talk of the projection of relation fuzzy set 𝑅𝑅 on 𝐵𝐵, we use this
expression for reducing the dimensionality; that means, the projection of 𝑅𝑅 on 𝐵𝐵 that is
𝑅𝑅𝐵𝐵 . And here, the membership values can be computed by this expression and that is why
we are writing here as 𝜇𝜇𝑅𝑅𝐵𝐵 .

(Refer Slide Time: 08:02)

So, when we have the membership values of the resulting set when we get by this
expression and here in the other case when we are projecting 𝑅𝑅 on 𝐵𝐵, we get by this

549
expression. So, when the membership values are known, obviously it is easier to write the
resulting fuzzy set.

So, if let’s say I am interested in writing the resulting fuzzy set in continuous domain. So,
this is going to be like this. So of course, this 𝑅𝑅𝐴𝐴 will be only the function of 𝑅𝑅, means we
will have only the universe of discourse 𝑋𝑋 here and this is in continuous form this will be
like this, 𝜇𝜇𝑅𝑅𝐴𝐴 (𝑥𝑥) and then we have small x.

So, this is the continuous form. So, the resulting fuzzy set 𝜇𝜇𝑅𝑅𝐴𝐴 (𝑥𝑥) after projection we are
going to get like this. Similarly, here we are going to get 𝑅𝑅𝐵𝐵 like this, here also I have the
universe of discourse only 𝑌𝑌.

So, we can write 𝑦𝑦 here and 𝑌𝑌 as the universe of discourse and 𝜇𝜇𝐵𝐵 (𝑦𝑦) and then 𝑦𝑦. So, this
is how we can represent the resulting fuzzy set. This exactly the same here if you are
interested in writing this 𝜇𝜇𝑅𝑅𝐴𝐴 (𝑥𝑥).

So, let us now better understand this by taking a very interesting example. So, let us assume
that we have multidimensional relation fuzzy set. Here, in our case we have fuzzy set
represented by the surface. So, or in other words, I would say we have let us say a
continuous relation fuzzy set 𝑅𝑅 and which is defined in two universe of discourses like 𝑋𝑋
and 𝑌𝑌.

So, we have here the generic variable value 𝑥𝑥 and 𝑦𝑦, and of course, then we have the
universe of discourse as 𝑋𝑋 × 𝑌𝑌. So, this is our 𝑅𝑅 the relation fuzzy set. Now you see here
this fuzzy set is a projection of 𝑅𝑅 on fuzzy set 𝐴𝐴. So you see, as I already mentioned that
this is represented by 𝜇𝜇𝑅𝑅𝐴𝐴 (𝑥𝑥). So this is nothing but, the projection and you see this is
coming in only one dimension.

Similarly, this will be our 𝜇𝜇𝑅𝑅𝐵𝐵 (𝑦𝑦). So, this means that the membership values can be
represented by 𝜇𝜇𝑅𝑅𝐵𝐵 (𝑦𝑦) only. And here in this case, when we have the projection of 𝑅𝑅 on
𝐴𝐴, the membership values will be represented by 𝜇𝜇𝑅𝑅𝐴𝐴 (𝑥𝑥).

So, it is very interesting to note that projection here in this case, is also reducing the
dimension. So, earlier 𝑅𝑅 was defined in the universe of discourse 𝑋𝑋 × 𝑌𝑌. Now, when it is
projected on A, the universe of discourse Y is eliminated. So, you see here when we project

550
R and this R if it is projected on A, y is eliminated only x is retained. Similarly, if R is
projected on B only y is retained, x is eliminated. So, this has to be noted.

(Refer Slide Time: 12:26)

And this also can be understood by discrete fuzzy relation set example; so, we are taking
here one example where we have two discrete fuzzy sets A and B. And here we will first
form a relation fuzzy set and then, from the relation fuzzy set, we will see how the
projection of fuzzy relation R on A and projection of fuzzy relation R on B.

(Refer Slide Time: 13:00)

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So, let’s now go ahead and try this example. So, as I mentioned that we have two discrete
fuzzy sets; we already have done this thing that as to how we can quickly get from these
two fuzzy sets A and B, how we can get a fuzzy relation set.

So, our relation set is here you can see. So, we will just have to get the Cartesian product
of these two A and B fuzzy sets. And, here we write our relation fuzzy set R which is again
this relation fuzzy set is basically is defined in the universe of discourse X and Y.

So, this can also be written as 𝑅𝑅(𝑥𝑥, 𝑦𝑦). Remember this R is of multidimensional fuzzy set.
So, if we look at A fuzzy set we see that it is a one dimensional fuzzy set. Although, it is
represented in two dimensional space because the we have the generic variable values
𝑥𝑥1 , 𝑥𝑥2 , 𝑥𝑥3 and so on and then we have the corresponding membership values like
𝜇𝜇(𝑥𝑥1 ), 𝜇𝜇(𝑥𝑥2 ), 𝜇𝜇(𝑥𝑥3 ) and so on.

So, we represent this in two-dimensional space, but so single dimensional fuzzy set.
Similarly, the B also and here our relation fuzzy set that is 𝑅𝑅(𝑥𝑥, 𝑦𝑦) becomes this way the
three-dimensional fuzzy set. Actually, it is a two-dimensional fuzzy set because only x and
y are the generic variable values. But, the representation of this fuzzy set is in three-
dimensional because one more dimension which is needed here is apart from 𝑥𝑥 and 𝑦𝑦, we
need 𝜇𝜇(𝑥𝑥, 𝑦𝑦) for the associated membership values.

So, as we see here that we have the 𝑅𝑅(𝑥𝑥, 𝑦𝑦) as the fuzzy relation set 𝑅𝑅 here. Now, this can
be represented in the form of fuzzy relation matrix, the same we know as to how we can
manage to convert this into the matrix form.

You see here, the R we have the column and then we have rows. So, column is basically
𝑥𝑥1 , 𝑥𝑥2 , 𝑥𝑥3 and rows are 𝑦𝑦1 , 𝑦𝑦2 , 𝑦𝑦3 , and then here these are the min values.

So, we know as to how we can get the relation matrix. We have already done this so I am
not going to discuss this again. We now represent this whole thing this relation fuzzy set
into the matrix form. So, we call this as the fuzzy relation matrix.

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(Refer Slide Time: 16:00)

So, now when we have the fuzzy relation matrix R, let us now find out the projection of
fuzzy relation R on A as we have already discussed. So, when we say projection of fuzzy
relation R on A, it means that we have to retain only x generic variable and we have to
eliminate y generic variable value. And when we apply the expression for doing this, this
means that the R which is going to be the resulting fuzzy set will look like this.

So, we call when we say if projection of fuzzy relation R on A, we represent this by 𝑅𝑅𝐴𝐴
and also this 𝑅𝑅𝐴𝐴 now we will retain only the x as the generic variable and this 𝑅𝑅𝐴𝐴 will be
basically if we represent this in continue in discrete form this will be like this.

So, 𝜇𝜇𝑅𝑅𝐴𝐴 (𝑥𝑥) and then 𝑥𝑥. So this will, the whole fuzzy set will look like this and now how
to get this 𝜇𝜇𝑅𝑅𝐴𝐴 (𝑥𝑥). So, this 𝜇𝜇𝑅𝑅𝐴𝐴 (𝑥𝑥) can be, we can get from here. So, when we use this
expression, the expression is 𝜇𝜇𝑅𝑅𝐴𝐴 (𝑥𝑥) = max 𝜇𝜇𝑅𝑅 (𝑥𝑥, 𝑦𝑦).
𝑦𝑦

So, we see that we have a fuzzy relation matrix R. So, when we apply this we see that, for
the first row corresponding to 𝑥𝑥1 we see that we have to take the maximum of all the
elements in the first row. So similarly, when we take maximum all the values, all the
elements in 𝑥𝑥2 row and 𝑥𝑥3 row, we find this matrix which is the column matrix.

So, this is called 𝑅𝑅𝐴𝐴 and this is equal to the column matrix 0.2, 0.4, 0.5, and this can be
written as in the normal form normal form is these summation form. So, when we write

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the summation form we can write it like this like 𝑅𝑅𝐴𝐴 is nothing but this R A is equal to
0.2 / 𝑥𝑥1 + 0.4 / 𝑥𝑥2 + 0.5 /𝑥𝑥3 .

So, this is how we get the projection of fuzzy relation set on A and this is represented by
the 𝑅𝑅𝐴𝐴 (𝑥𝑥).

(Refer Slide Time: 19:02)

Now similarly, when it comes to projecting the same fuzzy relation capital R on the same
lines we move ahead. So, we have here the relation fuzzy set which is again the defined
on 2 generic variables. So as I have discussed, for the projection of fuzzy relation R on B
on the same lines when we apply this expression, this criteria on the relation fuzzy set we
get a row matrix.

Here, since we are projecting the relation fuzzy set on B, we are eliminating x we are
retaining only y. So, that is why we see that we are taking the max for all the column
values, max of all the column values means over all x values.

So, that is why all the 3 rows are reduced to only single row. So, we see that we get these
3 elements only and this is nothing but 𝑅𝑅𝐵𝐵 and which is defined on only y. And this can be
represented by the normal fuzzy representation form and which is actually 0.5 /𝑦𝑦1 +
0.1 / 𝑦𝑦2 + 0.5 / 𝑦𝑦3 .

So, this way we have got the projection of fuzzy relation R on A and on B. And in both
the cases we have seen that the original fuzzy relation set R was the multidimensional, that

554
means both the R was on x and y, but here when we have projected this either A or B we
saw that we have reduced the dimension by 1.

(Refer Slide Time: 21:00)

So, this way we can generalize the projection of fuzzy relation R, so, R in n-dimension.
So, let us understand this here better. So, the projection of a fuzzy relation let us say R can
be extended to n-dimension. So, let relation R is defined among fuzzy sets 𝐴𝐴1 , 𝐴𝐴2 , 𝐴𝐴3 ,
𝐴𝐴4 … . . 𝐴𝐴𝑛𝑛 , where small n is the total number of generic variables.

So accordingly, we will have a space here. So, the universe of discourse space will be 𝑋𝑋1 ×
𝑋𝑋2 × 𝑋𝑋3 … … 𝑋𝑋𝑛𝑛 . So, we can project this relation to 𝑋𝑋1 × 𝑋𝑋2 × 𝑋𝑋3 … … 𝑋𝑋𝑘𝑘 . So, we are taking
some multiple dimensions where let us say up to 𝑘𝑘 is the dimension where we are
projecting.

So, when we project the n-dimensional fuzzy relation set into k-dimension k-dimensional
a space. So, what we are getting is here. So, we represent this the resulting relation fuzzy
set by capital 𝑅𝑅𝐴𝐴1 ×𝐴𝐴2×𝐴𝐴3……𝐴𝐴𝑘𝑘 . It means here we are retaining only this universe of discourse
means 𝑘𝑘 universe of discourse.

So, the membership values of this projected relation can also be computed and this is
computed by this relation here. So, this is self-explanatory, so, we define the membership
value of the resulting fuzzy set and this resulting fuzzy set of course will have k universe
of discourse.

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I should not say k universe of discourse, but I will say the universe of discourse will be the
Cartesian product of k generic variable. So, here 𝜇𝜇𝑅𝑅𝐴𝐴1×𝐴𝐴2×𝐴𝐴3……𝐴𝐴 (𝑥𝑥1 , 𝑥𝑥2 , … . . , 𝑥𝑥𝑘𝑘 ) =
𝑘𝑘

max 𝜇𝜇𝑅𝑅 (𝑥𝑥1 , 𝑥𝑥2 , … . , 𝑥𝑥𝑛𝑛 ) ∀(𝑥𝑥1 , 𝑥𝑥2 , … . . , 𝑥𝑥𝑛𝑛 ∈ 𝑋𝑋1 × 𝑋𝑋2 × … . .× 𝑋𝑋𝑛𝑛 .
𝑋𝑋1 ,𝑋𝑋2 ,…𝑋𝑋𝑚𝑚

So this way, it is defined and here this k plus m. So, here it is to be noted that the total
number of dimensions are the generic variables that are there, the total number is here n.

So, we have a small n, total number of generic variables, the relation fuzzy set and this
relation fuzzy set is projected in the k universe of discourse or k generic variable based
fuzzy set. So, what is the elimination here is the m, the m generic variable values are
eliminated.

So, what does this mean? It means that the k plus m is going to be equal to m, means the
total generic variables that we had initially is n, and then the resulting fuzzy set after
projection we have small k and then the small m here is the generic variable values that
are eliminated by this projection.

So this way, we have understood as to how this projection of fuzzy relation in n dimension
is done and this is also to be noted here that the relation that I have just mentioned among
the universe of discourse or the generic variable values here.

So, this can be understood by the union. So, here the total universe of discourse R, I would
say the generic variable values is equal to the union of the k generic variables and then the
m generic variables.

So this way, we have understood as to how we can project fuzzy relation in the reduced
dimension. So, as I have already mentioned that the projection of fuzzy relation always
gives a resulting fuzzy set, which will have the dimension less than the dimension of the
fuzzy relation set.

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(Refer Slide Time: 26:46)

So, this way we have understood the projection of fuzzy relation very clearly and the
remaining part, that is the cylindrical extension of fuzzy set will be a covered in the next
lecture.

Thank you.

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 Fuzzy Sets, Logic and Systems and Applications
Prof. Nishchal K. Verma
Department of Electrical Engineering
Indian Institute of Technology, Kanpur

Lecture – 32
Cylindrical Extension of Fuzzy Set

So welcome to lecture number 32 of Fuzzy Sets, Logic and Systems and Applications in
this lecture, we will discuss Cylindrical Extension of Fuzzy Sets. So, let us first take the
normal fuzzy set which is defined in the universe of discourse 𝑋𝑋.

(Refer Slide Time: 00:33)

So, if we have a fuzzy set 𝐴𝐴 let us see as to how we can find the cylindrical extension of
this fuzzy set. If we have a fuzzy set 𝐴𝐴 is defined here as we have already seen in either
continuous domain or discrete domain. So, what is a cylindrical extension of fuzzy set 𝐴𝐴?
So, if it is continuous fuzzy set 𝐴𝐴 we see that the cylindrical extension of fuzzy set is
defined by 𝐶𝐶(𝐴𝐴) this is the cylindrical extension here which is defined by 𝐶𝐶(𝐴𝐴), and 𝐶𝐶(𝐴𝐴)
is nothing but it is equal to the fuzzy set which is in continuous domain and this is
expressed in the increased dimension or increased universe of discourse.

So, here in this case we see that the 𝐶𝐶(𝐴𝐴) which is cylindrical extension of 𝐴𝐴 is defined in
two universe of discourse so the original fuzzy set that was taken here is 𝐴𝐴 is defined in
the universe of discourse 𝑋𝑋. When we take cylindrical extension of 𝐴𝐴 you see we have

558
added one more universe of discourse that is 𝑌𝑌. So, or in other words we say that we have
added one more generic variable 𝑦𝑦. So, the universe of discourse basically becomes capital
𝑋𝑋 × 𝑌𝑌.

And since this is continuous fuzzy set so, we represent this fuzzy set with the integral sign
and the membership values here you see as 𝜇𝜇𝐶𝐶(𝐴𝐴) (𝑥𝑥, 𝑦𝑦) and these membership values are
corresponding to (𝑥𝑥, 𝑦𝑦) points, which is represented of course, in the universe of discourse
𝑋𝑋 × 𝑌𝑌. So, this way we have increased the dimension we have added one more generic
variable here and that is how the extension happens and this is why we call the cylindrical
extension of fuzzy set 𝐴𝐴. So, cylindrical extension of fuzzy set 𝐴𝐴 is 𝐶𝐶(𝐴𝐴) which is defined
by

� 𝜇𝜇𝐶𝐶(𝐴𝐴) (𝑥𝑥, 𝑦𝑦)/(𝑥𝑥, 𝑦𝑦)

𝑋𝑋×𝑌𝑌

And when 𝐴𝐴 is a discrete fuzzy set for which we are taking the cylindrical extension. So,
for this we are getting

𝐶𝐶(𝐴𝐴) = � 𝜇𝜇𝐶𝐶(𝐴𝐴) (𝑥𝑥, 𝑦𝑦)/(𝑥𝑥, 𝑦𝑦)

𝑋𝑋×𝑌𝑌

So, this is very simple and we can clearly understand as to how the dimension, extension
happens. And by the name itself it is very clear that the extension is happening the one
more dimension is happening every time whenever we use the cylindrical extension over
any fuzzy set.

So, if we will look at the fuzzy set 𝐴𝐴 that we have taken it has only one generic variable in
both the cases either discrete or continuous, but when we have taken the cylindrical
extension we have added one more generic variable that is y and it is needless to say that
the initially 𝐴𝐴 had the universe of discourse as capital 𝑋𝑋, but after the cylindrical extension
the universe of discourse becomes 𝑋𝑋 × 𝑌𝑌 as you know the space of the universe of
discourse.

So, it is very interesting to note here that the increased membership values corresponding
to the increased dimension as to how we can get the values of membership in this new
generic variable or the combination of generic variables.

559
So, we see that its very simple here what is happening is that we take the original
membership value which is 𝜇𝜇𝐴𝐴 (𝑥𝑥) which was earlier in this generic variable space capital
𝑋𝑋 and this is going to be equal to the new membership value which is corresponding to the
point (𝑥𝑥, 𝑦𝑦). So, 𝜇𝜇𝐶𝐶(𝐴𝐴) (𝑥𝑥, 𝑦𝑦) is nothing but the original membership value corresponding
to 𝑥𝑥. So, here this does not depend upon y. So, no matter what dimension is added we just
write this and original membership value is retained this is called a cylindrical extension
because (𝑥𝑥, 𝑦𝑦) is 𝜇𝜇𝐴𝐴 (𝑥𝑥).

(Refer Slide Time: 06:33)

So, this is very important to note we’ll understand this better by one example here. If we
have a fuzzy set 𝐴𝐴 which is defined by a Gaussian membership function. So, this fuzzy set
𝐴𝐴 we see that it is here fuzzy set 𝐴𝐴. And the cylindrical extension of this fuzzy set 𝐴𝐴 will
look like this here this what is this cylindrical extension of the original fuzzy set 𝐴𝐴. So,
𝐶𝐶(𝐴𝐴) which is nothing but the 𝐶𝐶(𝐴𝐴) and this looks like this. So, we see that we have the
generic variable only 𝑥𝑥 and when we extend it cylindrically 𝑦𝑦 gets extended. So, here we
have taken the fuzzy set 𝐴𝐴 which is with the generic variable 𝑥𝑥.

So, the extension is happening in the other dimension or we are adding one more
dimension that is 𝑦𝑦.

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(Refer Slide Time: 07:41)

But if we take fuzzy set 𝐵𝐵 which is here and this fuzzy set is defined in the universe of
discourse 𝑌𝑌. So, we see that when we add one more dimension in the 𝑥𝑥 direction we see
that the cylindrical extension happens when we take the cylindrical extension of the fuzzy
set 𝐵𝐵. So, it is very simple to understand and in nutshell we increase the generic variables
in the cylindrical extension of fuzzy sets.

(Refer Slide Time: 08:15)

So, now if we have the fuzzy relation sets then what will happen? So, on the same lines as
we have discussed the cylindrical extension of fuzzy set 𝐴𝐴, 𝐵𝐵 so, on the same lines we can

561
apply the cylindrical extension on the relation fuzzy set as well. So, we can define a
relation fuzzy set by 𝑅𝑅 which is let’s say in the universe of discourse 𝑋𝑋 × 𝑌𝑌 which is here.
So, R is defined in the universe of discourse 𝑋𝑋 × 𝑌𝑌. 𝑅𝑅 is a relation set this has generic
variable variables 𝑥𝑥, 𝑦𝑦 and if we are interested in extending it the dimension extending the
generic variable let us say 𝑧𝑧.

So, we can go for it and the cylindrical extension of 𝑅𝑅 can be expressed by the 𝐶𝐶 (𝑅𝑅) which
is here and then the same relation fuzzy set is now I expressed in the three generic variable
and the universe of discourse here will be capital 𝑋𝑋 × 𝑌𝑌 × 𝑍𝑍, here this is defined by this
expression. So, this 𝐶𝐶 (𝑅𝑅) is nothing but the cylindrical extension of 𝑅𝑅 and 𝑅𝑅 as we already
know that this is defined in the 𝑋𝑋 × 𝑌𝑌 universe of discourse.

So, here also as we have seen in the previous case where we have taken 𝐴𝐴 fuzzy set and 𝐵𝐵
fuzzy sets, so here since we are taking 𝑅𝑅. So, 𝑅𝑅 is already defined with two generic
variables like 𝑥𝑥 and 𝑦𝑦 all the membership values will be based on the points in the space
(𝑥𝑥, 𝑦𝑦). So now, when we are extending the dimension let us say 𝑧𝑧 after taking the 𝐶𝐶(𝑅𝑅)
mu cylindrical extension of 𝑅𝑅. So, here the membership values will be now dependent on
the three points three generic values.

So, this 𝜇𝜇𝐶𝐶(𝑅𝑅) (𝑥𝑥, 𝑦𝑦, 𝑧𝑧) means the membership value at any point in the space (𝑥𝑥, 𝑦𝑦, 𝑧𝑧) all
these are generic values. So, here this will be equal to 𝜇𝜇𝑅𝑅 (𝑥𝑥, 𝑦𝑦) and of course, this is
needless to say that all these 𝑥𝑥, 𝑦𝑦, 𝑧𝑧 generic variable values will lie in the universe of
discourse 𝑋𝑋 × 𝑌𝑌 × 𝑍𝑍. So, here by taking the cylindrical extension of a fuzzy relation we
are adding one more dimension. And please note that by doing this we can increase any
number of dimensions. So, this is a cylindrical extension of 𝑅𝑅 which is increasing one
dimension. Now, further we can go ahead we can increase one more dimension if it is
needed.

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(Refer Slide Time: 11:36)

So, it is very simple to understand. Here if we take an example where we have a relation
continuous fuzzy set which is defined by this expression here and we have universe of
discourse 𝑋𝑋 × 𝑌𝑌 and all these corresponding membership values are defined by the

�𝑥𝑥 2 +𝑦𝑦 2 �
�− �
𝑅𝑅(𝑥𝑥, 𝑦𝑦) = � 𝑒𝑒𝑒𝑒𝑒𝑒 2𝜎𝜎 2 �
(𝑥𝑥, 𝑦𝑦)
𝑋𝑋×𝑌𝑌

So, here this two dimensional membership function and here since we have one more
dimension for membership values. So, we have three dimension, so that we see very clearly
that we have a surface the Gaussian surface being shown.

And this is defined in 𝑋𝑋 × 𝑌𝑌 universe of discourse. So, here we are taking the fuzzy relation
in the beginning and now we first try to find the projection of this fuzzy relation on 𝐴𝐴. So,
we have already studied the projection of fuzzy relation on 𝐴𝐴. So, if we do that we see that
we apply this formula and by doing this we can very easily find where the projection of
this fuzzy relation set on A and let me just remind you that this projection of any fuzzy set
reduces the dimension. So, it is eliminating the dimension. So, when we say dimension it
means the generic variable value with which the fuzzy set is defined.

So, in this case R is originally defined with generic variables (𝑥𝑥, 𝑦𝑦) and when we take the
projection of R on A it means we are eliminating the y generic variable and retaining the
𝑥𝑥 generic variable. So, you can see here as after taking the projection of R on A the

563
resulting fuzzy set will look like this and then similarly, we can go for the projection of
the original fuzzy relation here on B. So, when we do that, obviously B fuzzy set is defined
in the space of the generic variable y. So, here when we project the relation fuzzy set on
B. So, 𝑥𝑥 is eliminated means the 𝑦𝑦 generic variable is retained.

So, this way we see that we are going to get this fuzzy set is a reduced dimensional fuzzy
set as a result. So, originally we had started with the fuzzy set with two generic variables
(𝑥𝑥, 𝑦𝑦) and when we take the projection on A we are only retaining 𝑥𝑥 as the generic variable
we are eliminating 𝑦𝑦. And similarly, when we are taking the projection on B so, we are
only retaining 𝑦𝑦 generic variable and 𝑥𝑥 we are eliminating, but in both the cases here we
are eliminating one generic variables. So, this means that we are taking projection of fuzzy
set whether it is a relation fuzzy set or any fuzzy set the dimension is getting reduced. The
generic variable is getting reduced.

Now, since we have started with fuzzy relation set R which is a multidimensional multi
generic variable fuzzy set and after taking projection we have reduced the dimension. Now,
if we are further interested in increasing the dimension then we have already seen that we
can take a cylindrical extension of the fuzzy sets to increase the dimension, but in this
exercise if we do both we are not going to get the original fuzzy set, which with which we
started and we took projection and then we took cylindrical extension and during this
process we are not going to retain the original fuzzy set.

So here, but in many cases we need to do that because we are many cases for processing
of the fuzzy sets for you know inferencing we are interested in projection and cylindrical
extension, but not simultaneously. So, here the fuzzy set that we have got after taking the
projection 𝑅𝑅𝐴𝐴 and if we are let’s say increasing the dimension by taking the cylindrical
extension. So, you see how the resulting fuzzy set will look like. So, we had here only the
original A fuzzy set here. And now you see that we are extending towards y. So, this y
dimension is getting extended. So, that is how it is called cylindrical extension.

And we used this formula for extending the dimension. So, mu 𝐶𝐶𝑅𝑅𝐴𝐴 (𝑥𝑥, 𝑦𝑦) all these mu
values of the cylindrically extended fuzzy set is going to be equal to the 𝜇𝜇𝑅𝑅𝐴𝐴 (𝑥𝑥) and this
is for all (𝑥𝑥, 𝑦𝑦) belonging into the universe of discourse 𝑋𝑋 × 𝑌𝑌. So, similarly you see here
I can just over write this here just to make sure that what was 𝑅𝑅𝐵𝐵 earlier. So, this was the
𝑅𝑅𝐵𝐵 which was defined in the universe of discourse 𝑌𝑌 only. And here is the extension

564
cylindrical extension in the direction 𝑋𝑋. So, we can clearly see that whenever we increase
the dimension we see that it is the extension is here the cylindrical means whatever
membership value that we have for the generic variable let us say the same value is going
to be obtained for membership value (𝑥𝑥, 𝑦𝑦) as well.

So, we can clearly understand here that how the cylindrical extension will look like and
also before that we see the projection of a multidimensional fuzzy set is going to look like.
So, we have done a very interesting thing here that we took projection on A and projection
on B and then we further tried to extend the dimension. So, we clearly see that we are not
going to get the original relationship set with which we started. So, this is what is very
interesting to note, but what we gain here is that we are able to get the dimension extended.

(Refer Slide Time: 19:22)

Now, this was the understanding of cylindrical extension with the continuous relation
fuzzy set. Now, let us take a discrete relation fuzzy set. So, with discrete fuzzy set we have
the let us say we have a fuzzy set 𝑅𝑅 which is a relation fuzzy set and again this is defined
in the universe of discourse 𝑋𝑋 × 𝑌𝑌. So, of course, we can represent this discrete fuzzy set
in the matrix form. So, we have a relation matrix here 𝑅𝑅 and this 𝑅𝑅, I can just write that
this is dependent on 𝑥𝑥 and 𝑦𝑦. So, we see that the we have all the column values are with
respect to 𝑥𝑥1 , 𝑥𝑥2 , 𝑥𝑥3 and all the row values are with respect to 𝑦𝑦1 , 𝑦𝑦2 , 𝑦𝑦3 .

And now let us take this fuzzy set this relation fuzzy set and find the projection of this
fuzzy set on A, so that means, when we take projection of 𝑅𝑅𝐴𝐴 on 𝐴𝐴 we call this as 𝑅𝑅𝐴𝐴 . And

565
similarly projection of fuzzy relation R on B we call this as 𝑅𝑅𝐵𝐵 , and then finally, the
cylindrical extensions of both 𝑅𝑅𝐴𝐴 and 𝑅𝑅𝐵𝐵 . So, let us now go and see with this discrete
relation fuzzy set as to how we get all these projections and the cylindrical extensions.

(Refer Slide Time: 21:13)

So, here we have the relation fuzzy set that has been given to us and if we apply the formula
for finding the membership values for the resulting fuzzy set after taking the projection on
A. So, the formula is here. So, as I already mentioned that when we take the projection we
are reducing the dimension. So, you see that we take the maximum of all the membership
values over the generic variable value which is going to be eliminated. So, here since we
are taking the projection of 𝑅𝑅𝐴𝐴 it means the 𝐴𝐴 which is defined with the generic variable
𝑥𝑥. So, it means we are eliminating 𝑦𝑦 generic variable here.

So, when we are taking 𝑦𝑦 generic variable here it means we take all the for corresponding
to the generic variable value we take max and then that is how we eliminate the 𝑦𝑦
dimension you see here. So, we have original relation matrix. Now, we take the across all
the rows we take the maximum of all the elements. So, corresponding to 𝑥𝑥1 we get the
membership value here after taking max(0.2, 1.0,0.3). So, the membership value
corresponding 𝑥𝑥1 point we are getting 1.

And then similarly 𝑥𝑥2 we are getting 0.8 and corresponding 𝑥𝑥3 we are getting 1. So, this
way we see that we are reducing the dimension y and corresponding to 𝑥𝑥 we are getting
these as the membership values and which is written here in the expression of the fuzzy

566
set. So, 𝑅𝑅𝐴𝐴 will be nothing but 1 /𝑥𝑥1 which is the corresponding generic variable value.
And similarly plus 0.8 /𝑥𝑥2 which is again the corresponding generic variable value
corresponding to 𝑥𝑥2 and similarly 1 /𝑥𝑥3 means 1 is the generic variable value
corresponding to 𝑥𝑥3 . So, this way we have got here 𝑅𝑅𝐴𝐴 .

(Refer Slide Time: 23:59)

Now, on the same lines we get here 𝑅𝑅𝐵𝐵 and we can start from here the 𝑅𝑅 is in terms of 𝑥𝑥
and 𝑦𝑦 generic variables and when we use this criteria, when we use this expression for
finding the membership value in the projection we take 𝑚𝑚𝑚𝑚𝑚𝑚 over 𝑥𝑥. So, when we take the
max over x we see that the 𝑅𝑅𝐵𝐵 is going to be max(0.2, 0.4, 0.5) here with respect to the 𝑦𝑦1
value and similarly max we are getting as you see max corresponding to 𝑦𝑦2 we are getting
max(1, 0.8, 0.7).

And then 𝑦𝑦3 with reference to 𝑦𝑦3 we are getting max(0.3, 0.4,1). So, this way we are
getting 0.5 1.0 1.0 as the membership values corresponding to the generic variable values
𝑦𝑦1 , 𝑦𝑦2 , 𝑦𝑦3 and this way we can quickly get the expression of the fuzzy set here for 𝑅𝑅𝐵𝐵 which
is the projected fuzzy set. Now, when we have reduced the dimensions now as I already
mentioned that we can extend the dimension again.

567
(Refer Slide Time: 25:32)

So, but that is not always necessary here since we are understanding the concept we are
now adding the dimension on the projected fuzzy set. So, cylindrical extension of 𝑅𝑅𝐴𝐴 what
it is going to result we see here. So, we have here the 𝑅𝑅𝐴𝐴 which is which we have got after
the projection of 𝑅𝑅𝐴𝐴 . So, 𝑅𝑅𝐴𝐴 is this and then when we apply the formula for the cylindrical
extension here that means, mu 𝐶𝐶𝑅𝑅𝐴𝐴 (𝑥𝑥, 𝑦𝑦) is going to be 𝜇𝜇𝑅𝑅𝐴𝐴 (𝑥𝑥)∀(𝑥𝑥, 𝑦𝑦) ∈ 𝑋𝑋 × 𝑌𝑌.

So, this way we see that when we apply this we keep adding the 𝑦𝑦1 values to 𝑦𝑦1 values to
𝑦𝑦2 , 𝑦𝑦3 and so on. If there are many other points so, we see that this way we are adding the
dimension and whatever was the original dimension, that means, we had in 𝑅𝑅𝐴𝐴 we had here
if I represent it like this the 𝑅𝑅𝐴𝐴 fuzzy set in matrix form it was simply a column matrix. It
was like this.

And now, when we are taking the cylindrical extension, what is happening here is that this
one is getting added like for 𝑥𝑥1 we had 𝑥𝑥1 , 𝑦𝑦1 we have 1 and similarly 𝑥𝑥1 , 𝑦𝑦2 will also be
1, 𝑥𝑥1 , 𝑦𝑦3 will also be 1 which we can see here and similarly for all other elements we
extend so, 0.8, 0.8, 0.8, 1, 1, 1. So, the cylindrically extended matrix; cylindrically
extended relation matrix will have either the rows or columns you know duplicated. So,
that is the one thing that we just by looking at that we can say that this fuzzy set is out of
the cylindrical extension.

568
(Refer Slide Time: 28:03)

Similarly, now as we have got the expression of the fuzzy set or the relation matrix or the
matrix representation for fuzzy set 𝐶𝐶𝑅𝑅𝐴𝐴 we have 𝐶𝐶𝑅𝑅𝐵𝐵 also this we are getting when 𝑅𝑅𝐵𝐵 fuzzy
set which we have got out of the projecting relation fuzzy set on 𝐵𝐵. So, 𝑅𝑅𝐵𝐵 we have got
and 𝑅𝑅𝐵𝐵 is this 0.5 /𝑦𝑦1 + 1 / 𝑦𝑦2 + 1 /𝑦𝑦3 and then now when we are extending the
dimension or I can say the when we are extending the generic variable here instead of 𝑦𝑦
we are adding one more 𝑥𝑥 generic variable.

So, we are adding here 𝑥𝑥 here and then you see here we use this expression we have already
discussed this. So, once again I am telling that this cylindrically extended fuzzy set will
have its membership values corresponding to the original fuzzy set. So, whatever
membership value that we will be having here will be dependent on (𝑥𝑥, 𝑦𝑦) both, but this is
going to be equal to the membership value which was in the set that we have taken
originally and with 𝑥𝑥 only. So, we can see here that how we are getting the membership
values of the cylindrically extended fuzzy set?

(Refer Slide Time: 29:51)

569
And the same we can very easily represent in the form of a matrix. So, this way we have
seen that we can increase the dimension of a fuzzy set by taking the cylindrical extension
of the fuzzy set and similarly we can decrease the dimension of the fuzzy set by simply
projecting the fuzzy set.

So, with this I will stop here and in the next lecture we will discuss the properties of fuzzy
relations.

Thank you.

570
 Fuzzy Sets, Logic and Systems and Applications
Prof. Nishchal K. Verma
Department of Electrical Engineering
Indian Institute of Technology, Kanpur

Lecture - 33
Properties of Fuzzy Relation

So, welcome to lecture number 33 of Fuzzy Sets, Logic and Systems and Applications in
this lecture we will discuss the Properties of Fuzzy Relations.

(Refer Slide Time: 00:33)

So, here let us have a table of all the properties that normally the crisp and fuzzy sets you
know follow or un-follow. So, we have out of these properties that are listed here, we have
a law of contradiction, law of excluded middle and absorption of complement these three
properties are not followed by the fuzzy sets.

571
(Refer Slide Time: 01:10)

Whereas, these properties are followed by the crisp sets. So, as I have already discussed
this thing in my previous lectures all of these like a law of contradiction, law of excluded
middle and absorption of complement do not hold good for fuzzy sets. Now, when it comes
to the crisp relations and fuzzy relations. So, crisp since crisp relation is again a crisp set
and fuzzy relation also a fuzzy set these properties will be followed by crisp relations and
these properties will not be followed by the fuzzy relations.

So that means, when we talk of fuzzy relations. So, fuzzy relations do not hold good for
law of contradiction, law of excluded middle, absorption of complement.

572
(Refer Slide Time: 01:15)

So, we have a table here as we have already seen this table in case of you know when we
have discussed crisp set and the fuzzy sets.

(Refer Slide Time: 02:28)

So, let us discuss all these properties one by one by taking the fuzzy relations and the crisp
relations. So, for crisp relation let us say which is represented by 𝑅𝑅. So, 𝑅𝑅 ∩ 𝑅𝑅�, so, under
any universe of discourse let us say 𝑋𝑋. So, here the intersection of 𝑅𝑅 ∩ 𝑅𝑅� = 0. This means
when we take the intersection of any crisp relation and its complement this is going to be
the null set, but when it comes to the fuzzy relation 𝑅𝑅. Let us say if we take some fuzzy

573
relation 𝑅𝑅 and then we take the intersection of 𝑅𝑅�. So, this is what is the complement of
relation 𝑅𝑅 of relation 𝑅𝑅.

So, if we take the intersection of these two it is not coming as the null set. So, this is called
the law of contradiction. So, in nutshell basically this when we take a crisp relation we are
getting the intersection of crisp relation, and its complement it is coming out to be a null
set whereas, when we take fuzzy relation 𝑅𝑅 set and take the intersection of it with it’s
complement it is not going to be a null set and this is called as the law of contradiction.

(Refer Slide Time: 04:28)

So, let us understand this again by taking one example here. So, if we have two fuzzy sets
𝐴𝐴 and 𝐵𝐵 both these fuzzy sets are discreet fuzzy sets. So, let us form a relation quickly. So,
let 𝑅𝑅 be a fuzzy relation which is represented by the relation matrix here. So, this 𝑅𝑅 we
have gotten just by taking the 𝐴𝐴 × 𝐵𝐵.

So, here we have the fuzzy relation and if we see that if we take the intersection of 𝑅𝑅 and
its complement, so, then it is not going to be the null matrix. So, let us first find the 𝑅𝑅� that
means, the complement of the relation matrix means the fuzzy relation 𝑅𝑅. So, complement
of 𝑅𝑅 is 𝑅𝑅� which is here.

574
(Refer Slide Time: 05:24)

So, which we can get by just computing all its membership values by subtracting from 1.
So, when we do that if we have 𝑅𝑅 like this means the 𝑅𝑅 the fuzzy relation matrix
represented by here, 𝑅𝑅, then 𝑅𝑅� is going to be this matrix which we are getting by
subtracting it is all its elements from 1. So, you see 𝑅𝑅� is this. So, now, here we have the
complement of fuzzy relation 𝑅𝑅.

(Refer Slide Time: 06:19)

Now, let us take the 𝑅𝑅 ∩ 𝑅𝑅� . So, when we take the 𝑅𝑅 ∩ 𝑅𝑅� which is here. So, as we already
know that the basic intersection is computed by simply taking the min between all the

575
corresponding membership values. So, if we compute the 𝑅𝑅 ∩ 𝑅𝑅� and the basic min criteria
is followed. So, 𝑅𝑅 ∩ 𝑅𝑅� we are going to get here like this.

So, we get here fuzzy relation matrix or I would say we get as a result of 𝑅𝑅 ∩ 𝑅𝑅� some fuzzy
relation matrix which is not equal to 0. So, this way we can say that the law of contradiction
is verified.

(Refer Slide Time: 07:25)

So, let us now come to the 2nd property which is law of excluded middle. So, in law of
excluded middle as we all know that when we take a crisp relation we are getting if 𝑅𝑅 is
our crisp relation here, we are getting the universe of discourse when we take 𝑅𝑅 ∪ 𝑅𝑅� .

This means that if we take the union of crisp relation and its complement we are going to
get 𝐸𝐸. 𝐸𝐸 is nothing but the universe of discourse. So, this is true for the crisp relation. Now,
when we take 𝑅𝑅 as a fuzzy relation, so here if 𝑅𝑅 is a fuzzy relation and if we take 𝑅𝑅 ∪ 𝑅𝑅� ≠
𝐸𝐸. So, this is called the law of excluded middle.

(Refer Slide Time: 08:57)

576
So, let us quickly understand this by taking an example here. So, if we take an example
here where, we have two discrete fuzzy sets and we form a relation 𝑅𝑅 out of this 𝐴𝐴 𝑎𝑎𝑎𝑎𝑎𝑎 𝐵𝐵
discrete fuzzy sets. So, we have a fuzzy relation matrix which is represented by 𝑅𝑅. So, this
I can write here this is a fuzzy relation matrix alright. So, next is we have to here take the
𝑅𝑅 ∪ 𝑅𝑅� .

(Refer Slide Time: 09:41)

So, let us first find the 𝑅𝑅�. So, we know as to how we can find 𝑅𝑅� that means, the complement
of the fuzzy relation 𝑅𝑅. So, we have 𝑅𝑅 here and then we find 𝑅𝑅� by simply you know
subtracting each element from 1 and of the matrix and then we get here 𝑅𝑅�. So, this is

577
nothing but the complement of 𝑅𝑅 is complement of 𝑅𝑅 and 𝑅𝑅 is nothing but the fuzzy
relation.

(Refer Slide Time: 10:17)

Now, when we take the max of the corresponding membership values so, we use here the
basic union criteria we are getting here a fuzzy relation matrix. And we see that this is not
equal to 𝐸𝐸. So, what does this mean when we say this is equal to 𝐸𝐸. So, this would have
been equal to 𝐸𝐸, when all these element should have its values equal to 1. So, but here we
see that not all the values of this fuzzy relation matrix is equal to 1. So, in case it would
have been the 𝐸𝐸 would have been like this.

So, this is our 𝑥𝑥1 , this our 𝑥𝑥2 , this is our 𝑥𝑥3 and then 𝑦𝑦1 , 𝑦𝑦2 , 𝑦𝑦3 if you write it like this. So,
this would have been like this all the elements of the resulting fuzzy relation would have
been 1. So, we can say 𝑅𝑅 ∪ 𝑅𝑅� ≠ 𝐸𝐸.

578
(Refer Slide Time: 11:46)

(Refer Slide Time: 11:48)

So, then we have here the 3𝑟𝑟𝑟𝑟 property which is idempotency. So, let us check this and see
whether this is satisfied or not for the fuzzy relation. So, for crisp relation it is satisfied
and when we have a fuzzy relation 𝑅𝑅. So, if we take the intersection or union with 𝑅𝑅. So,
we see that this satisfied means this is coming out to be the same fuzzy relation.

579
(Refer Slide Time: 12:18)

So, let us quickly check that by taking this example here. So, here also we take two discrete
fuzzy sets and we find a relation a set from 𝐴𝐴 and 𝐵𝐵, that means, 𝐴𝐴 × 𝐵𝐵 when we take this
we have a fuzzy relation set here, which is represented in the form of a matrix, where the
matrix elements are nothing but the belongingness the membership values.

(Refer Slide Time: 12:49)

So, let us first have the intersection. So, we have the 𝑅𝑅 ∩ 𝑅𝑅. So, when we take intersection
we all know that we take the min of the respective for all the resulting membership values
which are nothing but the elements of the resulting relation matrix. So, 𝑅𝑅 ∩ 𝑅𝑅 is coming

580
out to be this. So, we will be see that the resultant of 𝑅𝑅 ∩ 𝑅𝑅 is the same matrix with which
we started. So, you see here that this matrix and this matrix both the matrix are same. Since
this fuzzy relation matrix are same. So, we can say that 𝑅𝑅 ∩ 𝑅𝑅 = 𝑅𝑅. So, this way this is
satisfied.

(Refer Slide Time: 14:52)

Now, let us take the union. So, when we take the union here 𝑅𝑅 ∪ 𝑅𝑅 for union we use max
criteria when we apply the max criteria here you see, for computing the membership values
with respect to the corresponding generic variable values. So, we find again the relation
matrix which is nothing but 𝑅𝑅. So, what does this mean this means that whether we take
the union or we take the intersection of the set 𝑅𝑅 and the union of the same set means if
we take the union of 𝑅𝑅 and 𝑅𝑅.

We are going to get 𝑅𝑅 or when we take the 𝑅𝑅 ∩ 𝑅𝑅 we are going to get the same set. So, this
way we can say that the idempotency property is satisfied for fuzzy relation set.

Now, next comes the involution property. So, here if we have any crisp relation R and for
crisp relation R if we take this complement twice, that means 𝑅𝑅�.

So, when we take crisp relation 𝑅𝑅 and if we take twice the, its complement we are going
to get the same set with which we started means we are going to get 𝑅𝑅� = 𝑅𝑅. And when it
comes to the fuzzy relation set R here. So, if we take the complement twice here also we
get the same set again means we get the double complement of fuzzy relations set 𝑅𝑅 is

581
equal to the fuzzy relation set again. So, this holds here for fuzzy relation also and this is
called the involution property. So, let us verify this property for fuzzy relation 𝑅𝑅.

(Refer Slide Time: 16:05)

So, when we do that here with the same example as we have taken in the previous property.
So, we again here we have with these two discreet fuzzy sets, we have the fuzzy relation
set and with this when we go for R bar, so, if we have this as the fuzzy relation set and this
is represented in the matrix form.

(Refer Slide Time: 16:34)

582
So, let us first get the 𝑅𝑅�; that means the 𝑅𝑅�. So, 𝑅𝑅� is here and this as we already have done
that each of the elements of this fuzzy relation matrix is found by subtracting its
membership values from 1. So, this way we get 𝑅𝑅� here. When we have R bar, now, let us
find the complement of 𝑅𝑅�; that means 𝑅𝑅�. So, 𝑅𝑅� with the same kind of subtraction, that
means, following this criteria we get double complement of 𝑅𝑅 and when we do that we are
getting the fuzzy relations set here. And which is nothing but if we equate it if we see that
if we compare this we see that we are getting the same set with which we started. Means
we are getting the set 𝑅𝑅 again means 𝑅𝑅� is going to give us 𝑅𝑅 where 𝑅𝑅 is the fuzzy relation
matrix.

(Refer Slide Time: 17:53)

So, this way we can say that the involution property holds good for the fuzzy relation set.

583
(Refer Slide Time: 18:02)

Now, let us check with the commutativity for union property. So, we all know that when
we take a two crisp relations 𝑅𝑅 and 𝑆𝑆 and we interchange their positions means 𝑅𝑅 ∪ 𝑆𝑆 =
𝑆𝑆 ∪ 𝑅𝑅.

So, this is where we see that this is commutative. So, crisp relations are commutative.
Now, when it comes to the fuzzy relations 𝑅𝑅 and 𝑆𝑆 this is also commutative means if we
have fuzzy relation set 𝑅𝑅 and if we have another fuzzy set relation set 𝑆𝑆. So, we can write
the 𝑅𝑅 ∪ 𝑆𝑆 = 𝑆𝑆 ∪ 𝑅𝑅 it means that the commutativity for union holds good for fuzzy relation
sets 𝑅𝑅 and 𝑆𝑆.

584
(Refer Slide Time: 19:07)

So, let us take an example to just verify this. So, here we have two relations sets this 1st
fuzzy relation set 𝑅𝑅 and the 2𝑛𝑛𝑛𝑛 fuzzy relation set is here 2𝑛𝑛𝑛𝑛 fuzzy relation set. So, we
have two fuzzy relations sets.

(Refer Slide Time: 19:29)

And let us now quickly take the union of these two sets means 𝑅𝑅 ∪ 𝑆𝑆. So, 𝑅𝑅 ∪ 𝑆𝑆 we get a
matrix here like this and this matrix is giving us the elements like this
1,0.7, 0.8 0.5, 0.8, 0.9, 0.7, 0.9, 0.2. So, this is 𝑅𝑅 ∪ 𝑆𝑆. Now, let us find the 𝑆𝑆 ∪ 𝑅𝑅. So, here
we get 𝑆𝑆 ∪ 𝑅𝑅 which is giving us a fuzzy relation set and if we compare these two we see

585
that both of these sets remain the same. This means that we are going to get the same fuzzy
set whether we take the 𝑅𝑅 ∪ 𝑆𝑆 or we take a 𝑆𝑆 ∪ 𝑅𝑅 both are same.

So, this way we can say that the commutativity for union for the fuzzy relation set is
verified 𝑅𝑅 holds good. So, commutativity property for union is verified is holding good
for fuzzy relations 𝑅𝑅 and 𝑆𝑆.

(Refer Slide Time: 20:55)

Now, let us check the commutativity property for intersection. So, when we take 𝑅𝑅 and 𝑆𝑆
again. So, instead of union let us check this for the intersection. So, let us quickly go ahead
with 𝑅𝑅 fuzzy relation set here this is 𝑅𝑅 fuzzy relation set, this is 𝑆𝑆 fuzzy relation set. Now,
let us take the intersection of the 2.

586
(Refer Slide Time: 21:22)

So, when we take intersection of the 2 we get here, this as the fuzzy relation set which is
𝑅𝑅 ∩ 𝑆𝑆. Now, let us take 𝑆𝑆 ∩ 𝑅𝑅. So, when we take 𝑆𝑆 ∩ 𝑅𝑅 what does this mean, this means
that we take 𝑆𝑆 first and then 𝑅𝑅 thereafter. So, when we do that, we get here this fuzzy
relation set as a result. So, when we compare these two outcomes we see that both the
outcomes remain same. So, this way we can say we are getting the same fuzzy relation set.
So, we can say that the commutativity property for intersection hold good for fuzzy
relations sets 𝑅𝑅 and 𝑆𝑆.

(Refer Slide Time: 22:15)

587
So, this way we have seen that in today’s lecture, we have discussed, we have studied the
following properties of fuzzy relations the law of contradiction, the law of excluded
middle, idempotency property, involution property and the commutative property.

So, we will stop here and in the next lecture we will study the remaining properties which
I have shown in this lecture in the previous slides.

Thank you.

588
 Fuzzy Sets, Logic and Systems and Applications
Prof. Nishchal K. Verma
Department of Electrical Engineering
Indian Institute of Technology, Kanpur

Lecture – 34
Properties of Fuzzy Relation

So, welcome to lecture number 34 of Fuzzy Sets, Logic and Systems and Applications.
This lecture is in continuation to our previous lecture, where we discussed the properties
that was relation with the Fuzzy Relation and the Crisp Relation.

(Refer Slide Time: 00:39)

So, in the previous lecture, we discussed some of the properties listed here. So, we
discussed law of contradiction, law of excluded middle, idempotency, involution and
commutativity with respect to fuzzy relations. And we saw that we had law of
contradictions with respect to the fuzzy relations, law of excluded middle with reference
to fuzzy relations.

So, these were not behaving same as the crisp relations these two properties and now the
idempotency, involution, commutativity all these three properties were satisfied, were
holding good when we took fuzzy relations.

589
(Refer Slide Time: 01:39)

So, let us now move in continuation to these properties that has come to associativity and
when we take crisp relations 𝑅𝑅, 𝑆𝑆, 𝑇𝑇. So, these crisp relation shows the associativity for
Union. So, as it is mentioned here that if we take the (𝑅𝑅 ∪ 𝑆𝑆) ∪ 𝑇𝑇 = 𝑅𝑅 ∪ (𝑆𝑆 ∪ 𝑇𝑇). So, this
is true for, this holding good for the crisp relations. Now, when it comes to fuzzy relations
𝑅𝑅, 𝑆𝑆 𝑎𝑎𝑎𝑎𝑎𝑎 𝑇𝑇, here also the associativity for union holds good. So, let us verify this by taking
one example.

(Refer Slide Time: 02:30)

590
And in this example here, we have taken 3 fuzzy relation sets. So, we have taken R fuzzy
relations set, 𝑆𝑆 fuzzy relation set and 𝑇𝑇 fuzzy relation set.

(Refer Slide Time: 02:47)

Now, when we take the 𝑅𝑅 ∪ 𝑆𝑆, we are getting here this fuzzy relation set and as we already
know that we take the maximum of the respective membership values and with this we get
the 𝑅𝑅 ∪ 𝑆𝑆 and the process is shown here you can follow. And then here, we take the
(𝑅𝑅 ∪ 𝑆𝑆) ∪ 𝑇𝑇. So, here when we take the whatever we have got here as 𝑅𝑅 ∪ 𝑆𝑆, now with
further take the union with 𝑇𝑇 here, we get here this fuzzy set as result.

(Refer Slide Time: 03:38)

591
So, this is nothing but the 𝑅𝑅 ∪ 𝑆𝑆 and then, the union of it with 𝑇𝑇. So, this is what we are
getting.

(Refer Slide Time: 03:53)

Now, let us take the 𝑆𝑆 ∪ 𝑇𝑇. So, 𝑆𝑆 ∪ 𝑇𝑇 is here;𝑅𝑅 ∪ (𝑆𝑆 ∪ 𝑇𝑇).

(Refer Slide Time: 04:17)

So, if we take this, we are getting here this fuzzy relation set and in the matrix
representation, we see that we get R union, 𝑅𝑅 ∪ (𝑆𝑆 ∪ 𝑇𝑇). So, if we compare this with the

592
previous outcome which was nothing but the (𝑅𝑅 ∪ 𝑆𝑆) ∪ 𝑇𝑇. So, we see that these two, these
both the sets are same.

So, we can say that (𝑅𝑅 ∪ 𝑆𝑆) ∪ 𝑇𝑇 = 𝑅𝑅 ∪ (𝑆𝑆 ∪ 𝑇𝑇) and these 𝑅𝑅, 𝑆𝑆 and 𝑇𝑇 all are the fuzzy
relation sets. So, this way we see that the associativity property for union holds good for
fuzzy relation sets 𝑅𝑅, 𝑆𝑆 and 𝑇𝑇. Now, let us check the associativity for intersection.

(Refer Slide Time: 05:34)

And we already know that if we take crisp relations 𝑅𝑅, 𝑆𝑆 and 𝑇𝑇, this relation this property
that means, the associativity for intersection is satisfied. So, we are not going into it. Now,
let us understand this relation that when we take fuzzy relations 𝑅𝑅, 𝑆𝑆 and 𝑇𝑇 the associativity
for intersection is also holding good. So, let us verify this by taking an example for this
property.

593
(Refer Slide Time: 06:09)

So, here also we take 𝑅𝑅, 𝑆𝑆 and 𝑇𝑇, as you can see here these 3 fuzzy relation sets, we have
taken.

(Refer Slide Time: 06:22)

And let us first find the 𝑅𝑅 ∩ 𝑆𝑆. So, when we use the min criteria here, we find here this
outcome as the (𝑅𝑅 ∩ 𝑆𝑆) ∩ 𝑇𝑇. So, when we try to find this, again the intersection of 𝑅𝑅, 𝑆𝑆 ∩
𝑇𝑇. So, we find here this outcome. So, this is nothing but the intersection of R intersection
S and T.

594
(Refer Slide Time: 07:10)

Now, let us find the 𝑆𝑆 ∩ 𝑇𝑇. So, 𝑆𝑆 ∩ 𝑇𝑇 is here and when we take the intersection of this
fuzzy set which is which we have found here as 𝑆𝑆 ∩ 𝑇𝑇, 𝑆𝑆 ∩ 𝑇𝑇 and the 𝑅𝑅 ∩ (𝑆𝑆 ∩ 𝑇𝑇) here.
So, if we find that we find this outcome.

(Refer Slide Time: 07:49)

So, now when we compare this, we see that these both the outcomes are same. So, this
means that the (𝑅𝑅 ∩ 𝑆𝑆) ∩ 𝑇𝑇 = 𝑅𝑅 ∩ (𝑆𝑆 ∩ 𝑇𝑇). So, this way we can say that associativity for
intersection is satisfied for fuzzy relation sets.

595
(Refer Slide Time: 08:22)

Now, let us move to the distributivity of union over intersection. So, when we take crisp
relations 𝑅𝑅, 𝑆𝑆 and 𝑇𝑇, we know that this is satisfied. So, we are not going into it, we are not
going to discuss anything about it further. However, if we take fuzzy relations 𝑅𝑅, 𝑆𝑆 and 𝑇𝑇
we see that the 𝑅𝑅 ∪ (𝑆𝑆 ∩ 𝑇𝑇) = (𝑅𝑅 ∪ 𝑆𝑆) ∩ (𝑅𝑅 ∪ 𝑇𝑇). So, let us now verify this. So, this holds
good for fuzzy relations 𝑅𝑅, 𝑆𝑆 and 𝑇𝑇. So, this is called distributivity of union over
intersection property and this holds good for fuzzy relation sets.

(Refer Slide Time: 09:25)

596
Let us take an example and verify this quickly. So, here we have taken 3 sets; 3 fuzzy
relation sets.

(Refer Slide Time: 09:36)

And let us take the 𝑆𝑆 ∩ 𝑇𝑇. So, 𝑆𝑆 ∩ 𝑇𝑇 which is here which is which we get like this, this the
outcome of the 𝑆𝑆 ∩ 𝑇𝑇 and let us take the 𝑅𝑅 ∪ (𝑆𝑆 ∩ 𝑇𝑇). So, when we take this, we are getting
this as the 𝑅𝑅 ∪ (𝑆𝑆 ∩ 𝑇𝑇).

(Refer Slide Time: 10:15)

597
So, lets us move further and see whether the distributivity of union over intersection is
verified here. So, in order to do that let us first find the 𝑅𝑅 ∪ 𝑆𝑆. So, 𝑅𝑅 ∪ 𝑆𝑆 outcome is here,
then we go to find the 𝑅𝑅 ∪ 𝑇𝑇. So, 𝑅𝑅 ∪ 𝑇𝑇 is here; 𝑅𝑅 ∪ 𝑇𝑇, and now we take the intersection
of these two fuzzy relations. So, let us take the intersection of these two.

(Refer Slide Time: 11:01)

So, when we take the intersection of these two here, we get as a result here, a new fuzzy
relation set which is represented by fuzzy relation matrix. So, we see that this outcome is
same as this outcome, means the 𝑅𝑅 ∪ (𝑆𝑆 ∩ 𝑇𝑇) = (𝑅𝑅 ∪ 𝑆𝑆) ∩ (𝑅𝑅 ∪ 𝑇𝑇). So, this way we can
say that the distributivity of union over intersection is verified for fuzzy relations 𝑅𝑅, 𝑆𝑆 and
𝑇𝑇.

598
(Refer Slide Time: 11:56)

So, let us now go to the next property which is Distributivity of Intersection over Union
and we all know that this holding good for crisp relations 𝑅𝑅, 𝑆𝑆 and 𝑇𝑇 and also we know
that the, this property the distributivity of intersection over union is satisfied is holding
good for the fuzzy relations 𝑅𝑅, 𝑆𝑆 and 𝑇𝑇 also. Now, let us verify this relation by taking an
example like previously we have done for other properties.

(Refer Slide Time: 12:42)

So, let us once again take the fuzzy relation sets 𝑅𝑅, 𝑆𝑆 and 𝑇𝑇.

599
(Refer Slide Time: 12:51)

And then, since we have to verify this property that means, the distributivity of intersection
over union that means, the 𝑅𝑅 ∩ (𝑆𝑆 ∪ 𝑇𝑇) = (𝑅𝑅 ∩ 𝑆𝑆) ∪ (𝑅𝑅 ∩ 𝑇𝑇). So, let us first find the 𝑆𝑆 ∪
𝑇𝑇. So, 𝑆𝑆 ∪ 𝑇𝑇 is here. By applying the max criteria, we get the 𝑆𝑆 ∪ 𝑇𝑇.

Similarly, when we take the fuzzy relation set 𝑅𝑅 and we take the 𝑅𝑅 ∩ (𝑆𝑆 ∪ 𝑇𝑇), we get this
outcome, by applying the min criteria. So, we can write this as 𝑅𝑅 ∩ (𝑆𝑆 ∪ 𝑇𝑇). So, the
outcome is here.

(Refer Slide Time: 14:02)

600
And now let us go to 𝑅𝑅 ∩ 𝑆𝑆. So, 𝑅𝑅 ∩ 𝑆𝑆 is here. When we take R and S take the intersection.
So, this is 𝑅𝑅 ∩ 𝑆𝑆 and here we have the 𝑅𝑅 ∩ 𝑇𝑇. Now, if we take the union of these two.

(Refer Slide Time: 14:27)

So, when we take union of these two, means the (𝑅𝑅 ∩ 𝑆𝑆) ∪ (𝑅𝑅 ∩ 𝑇𝑇). So, when we take this
since we are taking union, we take the max, we take we use the max criteria and when we
use this max criteria we are getting here this has the outcome.

So, we get a new fuzzy relation set and which is nothing but equal to you see here equal
to this fuzzy relation set. So, this way we can say that the outcome of the 𝑅𝑅 ∩ (𝑆𝑆 ∪ 𝑇𝑇) =
(𝑅𝑅 ∩ 𝑆𝑆) ∪ (𝑅𝑅 ∩ 𝑇𝑇). So, this way we can say that the distributivity of intersection over
union is verified for fuzzy relations 𝑅𝑅, 𝑆𝑆 and 𝑇𝑇.

601
(Refer Slide Time: 15:39)

Let us move to the next property which is absorption of union over intersection. So, here
for crisp relations 𝑅𝑅 and 𝑆𝑆, it is like this like a we take the 𝑅𝑅 ∪ (𝑅𝑅 ∩ 𝑆𝑆), we get here 𝑅𝑅. So,
this is holding good for crisp relations and when it comes to fuzzy relations 𝑅𝑅 and 𝑆𝑆, here
also the absorption of union over intersection holds good means the 𝑅𝑅 ∪ (𝑅𝑅 ∩ 𝑆𝑆) = 𝑅𝑅. So,
let us this also verify by taking an example.

(Refer Slide Time: 16:33)

602
So, here we have taken one fuzzy relation set 𝑅𝑅 and another fuzzy relation set 𝑆𝑆 also we
have taken. So, 𝑅𝑅 and 𝑆𝑆 both are the fuzzy relation sets and now let us verify the absorption
of union over intersection.

(Refer Slide Time: 16:53)

So, for this let us find the 𝑅𝑅 ∩ 𝑆𝑆. So, since here we are finding the intersection, we use min
criteria you see here and when we use min criteria the outcome is this. So, this is nothing
but the 𝑅𝑅 ∩ 𝑆𝑆. Now, this outcome is again used for union with R. So, when we take the
𝑅𝑅 ∪ (𝑅𝑅 ∩ 𝑆𝑆), now since we are taking union, we use the max criteria here and then, the
outcome here is like this.

603
(Refer Slide Time: 17:46)

So, the outcome here is like this and which is nothing but the fuzzy set R which we have
taken. So, we can say that the 𝑅𝑅 ∪ (𝑅𝑅 ∩ 𝑆𝑆) = 𝑅𝑅. So, this means that the absorption of union
over intersection is verified for fuzzy relations 𝑅𝑅 and 𝑆𝑆.

(Refer Slide Time: 18:19)

Now, let us move to another property which is absorption of intersection over union. So,
here, we have the 𝑅𝑅 ∩ ( 𝑅𝑅 ∪ 𝑆𝑆) = 𝑅𝑅. So, this is this holds good for the crisp relation sets.
And the same also holds for the fuzzy relations R and S. That means, if we have two fuzzy
relation sets R and S, so the intersection between the 𝑅𝑅 fuzzy relation set and the union of

604
fuzzy relation set 𝑅𝑅 and fuzzy relation set 𝑆𝑆, we are going to get 𝑅𝑅. So, this means that
“Absorption” of intersection over union holds good for a crisp relations as well as fuzzy
relations. So, let us verify this by taking an example.

(Refer Slide Time: 19:33)

So, here we have taken two fuzzy relation sets 𝑅𝑅 and 𝑆𝑆, which you can see here.

(Refer Slide Time: 19:48)

And then, now let us try to verify for absorption of intersection over union. So, when we
do that in order to do that, we first find the 𝑅𝑅 ∪ 𝑆𝑆. So, 𝑅𝑅 ∪ 𝑆𝑆 is here. So, we use max criteria

605
for union. So, we get the 𝑅𝑅 ∪ 𝑆𝑆 here and then we take intersection of this with R. So, we
take R first and then we take the 𝑅𝑅 ∩ (𝑅𝑅 ∪ 𝑆𝑆).

So, since we are taking the intersection here, we use min criteria and then when we use
min criteria, we are getting this as the outcome and this outcome if we compare this is
nothing but the 𝑅𝑅. So, we can clearly say here that the 𝑅𝑅 ∩ (𝑅𝑅 ∪ 𝑆𝑆) = 𝑅𝑅. So, here this
absorption of intersection over union is satisfied or we can say it is verified.

(Refer Slide Time: 21:01)

Now, let us go to the absorption of complement for union. So, here we have the relation
sets, crisp relation sets 𝑅𝑅 and 𝑆𝑆 and we see that when we take the 𝑅𝑅 ∪ (𝑅𝑅� ∩ 𝑆𝑆). So, this
comes out to be the union of 𝑅𝑅 and 𝑆𝑆.

So, this holds good for the crisp relations, but when it comes to fuzzy relations, this does
not hold good. That means, when we have fuzzy relations 𝑅𝑅 and 𝑆𝑆, if we take the 𝑅𝑅 ∪
(𝑅𝑅� ∩ 𝑆𝑆) ≠ 𝑅𝑅 ∪ 𝑆𝑆. So, this way we can say that the absorption of complement property for
union does not hold good for fuzzy relations 𝑅𝑅 and 𝑆𝑆.

606
(Refer Slide Time: 22:10)

So, let us quickly verify this by taking an example. So, we have taken here in this example
fuzzy relation set 𝑅𝑅 and fuzzy relation set 𝑆𝑆.

(Refer Slide Time: 22:23)

So, let us first find the complement of 𝑅𝑅. So, the complement of 𝑅𝑅 is here. I am writing
just complement of 𝑅𝑅 and then, let us take the 𝑅𝑅� ∩ 𝑆𝑆. So, this is the outcome. Since, we
are taking the intersection, we will use the min criteria which is clearly visible here and
then when we have taken this, then let us take the union of 𝑅𝑅 and this outcome. This

607
outcome is the 𝑅𝑅� ∩ 𝑆𝑆. So, when we do that, we are getting here 𝑆𝑆, the 𝑅𝑅 ∪ (𝑅𝑅� ∩ 𝑆𝑆). Now,
let us find the 𝑅𝑅 ∪ 𝑆𝑆.

So, we have 𝑅𝑅 we have 𝑆𝑆, now since we are taking union we use max criteria and when
we find this we see that we are getting here as the 𝑅𝑅 ∪ 𝑆𝑆. When we compare this outcome
with the 𝑅𝑅 ∪ (𝑅𝑅� ∩ 𝑆𝑆), we see that both of these are not same. So, when both of these are
not same, we can say that the absorption of complement for union is not holding good.

(Refer Slide Time: 23:58)

Now, let us check the absorption of complement for intersection. So, on the same lines we
see that the crisp relations 𝑅𝑅 and 𝑆𝑆, these relations are holding good, the absorption of
complement for intersection means when we take the 𝑅𝑅 ∪ (𝑅𝑅� ∩ 𝑆𝑆) = 𝑅𝑅 ∩ 𝑆𝑆 and if we take
the fuzzy relations R and S, so, this relation does not hold good means this property does
not hold good when we take fuzzy relation sets 𝑅𝑅 and 𝑆𝑆. So that means, if we have fuzzy
relations sets R and S and if we take the 𝑅𝑅 ∪ (𝑅𝑅� ∩ 𝑆𝑆) ≠ 𝑅𝑅 ∩ 𝑆𝑆.

608
(Refer Slide Time: 25:02)

So, let us verify this also and for this, if we take our fuzzy relation set here as 𝑅𝑅, one of
the fuzzy relation sets as 𝑅𝑅 and then, another fuzzy relation set 𝑆𝑆 here. So, when we take
these two fuzzy relation sets; first fuzzy relation set is 𝑅𝑅 and the second fuzzy relation set
is 𝑆𝑆.

(Refer Slide Time: 25:26)

Now, let us first find the complement of 𝑅𝑅 here which is coming out to be like this. So,
this is 𝑅𝑅� ∪ 𝑆𝑆. So, since we are taking union ,so we use max criteria as we can clearly see

609
and then, the outcome of 𝑅𝑅� ∪ 𝑆𝑆. So, this we right as the 𝑅𝑅� ∪ 𝑆𝑆 and then, we take the
intersection of it with 𝑅𝑅. So that means, the 𝑅𝑅 ∩ (𝑅𝑅� ∪ 𝑆𝑆). So, this is the outcome of it.

And now let us further find the 𝑅𝑅 ∩ 𝑆𝑆. So, R intersection is coming out to be this. Now,
we see that the 𝑅𝑅 ∩ (𝑅𝑅� ∪ 𝑆𝑆) ≠ 𝑅𝑅 ∩ 𝑆𝑆. This clearly shows that the absorption of
complement for intersection does not hold good when it comes to the fuzzy relations 𝑅𝑅
and 𝑆𝑆.

(Refer Slide Time: 26:55)

So, now let us go to another property which is Demorgan’s law of union. So, we already
know that when we take the �������
𝑅𝑅 ∪ 𝑆𝑆, we get the 𝑅𝑅� ∩ 𝑆𝑆̅. So, when we take fuzzy relation sets
𝑅𝑅 and 𝑆𝑆, this property also holds good for the fuzzy relation sets. That means, either we
take fuzzy relations or we take crisp relations, for both the relations Demorgan’s law of
union is holding good.

610
(Refer Slide Time: 27:41)

So, let us here also we take one example and we see as to how Demorgan’s law of union
is verified for fuzzy relations 𝑅𝑅 and 𝑆𝑆.

(Refer Slide Time: 27:54)

So, when we take the 𝑅𝑅 ∪ 𝑆𝑆 for this fuzzy relation set 𝑅𝑅 and 𝑆𝑆. So, 𝑅𝑅 and 𝑆𝑆 we are getting
like this. We use the union so that is why we take the we use max criteria and when we
use max criteria, we get 𝑅𝑅 ∪ 𝑆𝑆 and then, when we take the complement of it, we are getting
�������
this outcome. So, this is nothing but the 𝑅𝑅 ∪ 𝑆𝑆. Now, we take the complement of 𝑅𝑅 which

611
is this and then, we take the complement of 𝑆𝑆 which is this. Now, when we take the 𝑅𝑅� ∩ 𝑆𝑆̅.
So, we see that this is equal to the �������
𝑅𝑅 ∪ 𝑆𝑆.

So, this way we can say that when we take two fuzzy relation sets R and S, we take the
�������
𝑅𝑅 ∪ 𝑆𝑆. And this is nothing, but this is equal to the 𝑅𝑅� ∩ 𝑆𝑆̅ which is here. So, Demorgan’s
law of union for fuzzy relation sets is holding good.

(Refer Slide Time: 29:28)

Now, let us verify the Demorgan’s law of intersection. So, here also we see that if we take
the �������
𝑅𝑅 ∩ 𝑆𝑆 = 𝑅𝑅� ∪ 𝑆𝑆̅ and this is holding good for the crisp relation sets as well as the fuzzy
relation sets 𝑅𝑅 and 𝑆𝑆. So, let us verify this also.

612
(Refer Slide Time: 30:02)

So, when we take two fuzzy relation sets here as 𝑅𝑅 and 𝑆𝑆.

(Refer Slide Time: 30:09)

And let us find the 𝑅𝑅 ∩ 𝑆𝑆 first. So, this is the 𝑅𝑅 ∩ 𝑆𝑆 and when we take the complement of
it here, so this is the outcome when we take the �������
𝑅𝑅 ∩ 𝑆𝑆 and the 𝑅𝑅� is here, 𝑆𝑆̅ is here. So, we
see that when we take the 𝑅𝑅� ∪ 𝑆𝑆̅, we get this as the outcome. So, we get the new fuzzy
relation set as the outcome in form of the matrix here. So, this is nothing but this is equal
to the �������
𝑅𝑅 ∩ 𝑆𝑆. So, this way we can say that the Demorgan’s law of union for fuzzy relation
sets is holding good.

613
(Refer Slide Time: 31:12)

So, in today’s lecture, we have studied the following properties of fuzzy relations;
associativity property, distributivity property, absorption property, absorption of
complement property and Demorgan’s Law and this way we have seen as to how the fuzzy
relation set holds good with respect to the properties that I have already discussed in this
lecture. And in the next lecture, we will study the fuzzy extension principle.

Thank you.

614
 Fuzzy Sets, Logic and Systems and Applications
Prof. Nishchal K. Verma
Department of Electrical Engineering
Indian Institute of Technology, Kanpur

Lecture - 35
Extension Principle

So, welcome to lecture number 35 of Fuzzy Sets, Logic and Systems and Applications. In
this lecture, we will discuss Extension Principle.

(Refer Slide Time: 00:26)

So, the extension principle is very interesting and with the help of the extension principle.
We basically convert a transform of fuzzy set, which is defined in one universe of
discourse into another which is defined in the other universe of discourse. So, the extension
principle is a basic concept in the fuzzy set theory that provides a general procedure for
transforming a fuzzy set from one universe of discourse to another universe of discourse
provided the mapping function, that means the 𝑓𝑓 which connects one universe of discourse
to another universe of discourse is known.

And, this procedure generalizes a common point to point mapping of a function that is
𝑓𝑓(. ) to a mapping between fuzzy sets. More specifically 𝑓𝑓 is a function from 𝑋𝑋 universe
of discourse, that means, the universe of discourse 𝑋𝑋 to universe of discourse 𝑌𝑌. And, if

615
we take let us say to understand any fuzzy set which is defined in the universe of discourse
𝑋𝑋.

So, if we take 𝐴𝐴(𝑥𝑥) here. So, normally we write a fuzzy set by simply either 𝐴𝐴, 𝐵𝐵 or 𝐶𝐶 like
that we never write normally 𝐴𝐴(𝑥𝑥), but here since we are putting an emphasis on the
generic variable with which it is defined that means, we are indicating here that 𝐴𝐴 is
defined in the universe of discourse 𝑋𝑋. So, 𝐴𝐴 here is 𝐴𝐴(𝑥𝑥). And, this is defined by this
equation here, means this fuzzy set here where we have the membership value and then its
corresponding generic variable value.

So, 𝜇𝜇𝐴𝐴 (𝑥𝑥1 )⁄𝑥𝑥1 + 𝜇𝜇𝐴𝐴 (𝑥𝑥2 )⁄𝑥𝑥2 + ⋯ + 𝜇𝜇𝐴𝐴 (𝑥𝑥𝑛𝑛 )/𝑥𝑥𝑛𝑛 . So, here if we see 𝐴𝐴, 𝐴𝐴 is defined in
the universe of discourse 𝑋𝑋 with the generic variable 𝑥𝑥. So, if we are interested in
transforming this fuzzy set into the universe of discourse capital 𝑌𝑌; that means, if we
transform this fuzzy set 𝐴𝐴(𝑥𝑥) into say 𝐵𝐵(𝑦𝑦) where 𝑦𝑦 is the generic variable in the universe
of discourse 𝑌𝑌.

So, then how we can manage to do that is shown here. So, we see here that 𝐴𝐴(𝑥𝑥) is
transform into 𝐵𝐵, where 𝐵𝐵 is nothing but it is defined by the generic variable 𝑦𝑦 in the
universe of discourse 𝑌𝑌. And, this is defined by this fuzzy set here 𝜇𝜇𝐴𝐴 (𝑥𝑥1 )⁄𝑦𝑦1 +
𝜇𝜇𝐴𝐴 (𝑥𝑥2 )⁄𝑦𝑦2 + ⋯ + 𝜇𝜇𝐴𝐴 (𝑥𝑥𝑛𝑛 )/𝑦𝑦𝑛𝑛 . And then these 𝑦𝑦𝑖𝑖 means 𝑦𝑦1 , 𝑦𝑦2 , 𝑦𝑦3 … … , 𝑦𝑦𝑛𝑛 can be found
by substituting 𝑥𝑥𝑖𝑖 values that means, 𝑥𝑥1 , 𝑥𝑥2 , 𝑥𝑥3 , 𝑥𝑥4 and so on in 𝑓𝑓(𝑥𝑥𝑖𝑖 ), that means we use
the mapping function to manage to get the conversion of the generic variable let us say 𝑥𝑥
into 𝑦𝑦.

So, this can also be written like this. So, 𝑥𝑥𝑖𝑖 = 𝑓𝑓 −1 (𝑦𝑦𝑖𝑖 ). For every 𝑖𝑖 is equal to 1, 2, 3, 4, 5
and so on up to 𝑛𝑛. So, this way the extension principle helps us in transforming a fuzzy set
which is in a particular universe of discourse into another fuzzy set which is in a different
universe of discourse say 𝑌𝑌. So, extension principle helps us in managing this conversion

616
(Refer Slide Time: 05:59)

So, let us now understand this further in continuation to this if this 𝑓𝑓(𝑥𝑥) if this mapping
function is which is here is many to one mapping then there exist 𝑥𝑥1 , 𝑥𝑥2 that are belonging
into 𝑋𝑋 as the universe of discourse, where 𝑥𝑥1 ≠ 𝑥𝑥2 . So, in that case what will happen,
𝑓𝑓(𝑥𝑥1 ) = 𝑓𝑓(𝑥𝑥2 ). So, this is a situation where let us say we have two values of generic
variable 𝑥𝑥1 and 𝑥𝑥2 . And then their corresponding 𝑓𝑓(𝑥𝑥1 ) and 𝑓𝑓(𝑥𝑥2 ) both are if equal. So, if
both of these are equal means if 𝑓𝑓(𝑥𝑥1 ) = 𝑓𝑓(𝑥𝑥2 ). And also when we say 𝑓𝑓(𝑥𝑥1 ) = 𝑓𝑓(𝑥𝑥2 ).

So, it means it is having some values let us say that is 𝑦𝑦 ∗ , and this y star is belonging into
𝑌𝑌 universe of discourse. So, what does this mean. This means that if we have any two
generic variables if we have any two generic variable values 𝑥𝑥1 , 𝑥𝑥2 . And they are not equal
to each other, but 𝑓𝑓(𝑥𝑥1 ) = 𝑓𝑓(𝑥𝑥2 ) then in that case what we do is here. So, what we do here
in this case is, the membership value of fuzzy set B at 𝑦𝑦 = 𝑦𝑦 ∗ , 𝑦𝑦 ∗ is the value which is
𝑓𝑓(𝑥𝑥1 ) or 𝑓𝑓(𝑥𝑥2 ).

So, this y star will play an interesting role here. We see that how do we find corresponding
to this 𝑦𝑦 ∗ , 𝜇𝜇𝐵𝐵 (𝑦𝑦 ∗ ) you see here that 𝜇𝜇𝐵𝐵 (𝑦𝑦 ∗ ) = max(𝜇𝜇𝐴𝐴 (𝑥𝑥1 ), 𝜇𝜇𝐴𝐴 (𝑥𝑥2 )). So, this means that
even when the 𝑓𝑓(𝑥𝑥1 ) = 𝑓𝑓(𝑥𝑥2 ), but 𝑥𝑥1 ≠ 𝑥𝑥2 . So, in that case what happens 𝜇𝜇𝐵𝐵 (𝑦𝑦 ∗ ) =
max(𝜇𝜇𝐴𝐴 (𝑥𝑥1 ), 𝜇𝜇𝐴𝐴 (𝑥𝑥2 )). So, this can also be written as 𝜇𝜇𝐵𝐵 (𝑦𝑦) which is here.

So, the general formula can be like this. So, 𝜇𝜇𝐵𝐵 (𝑦𝑦) = max
−1
𝜇𝜇𝐴𝐴 (𝑥𝑥), which is written here.
𝑥𝑥∈𝑓𝑓 (𝑦𝑦)

So, this is followed when we come across this situation this situation means when I am

617
just repeating the situation where 𝑥𝑥1 ≠ 𝑥𝑥2 please understand here 𝑥𝑥1 ≠ 𝑥𝑥2 , but 𝑓𝑓(𝑥𝑥1 ) =
𝑓𝑓(𝑥𝑥2 ).

So, we will take an example ahead to make it more clear. So, that is how we are able to
convert or transform the membership functions from one domain to another, from one
generic variable to another, from one universe of discourse to another. So, here very clearly
we have transformed 𝐴𝐴 into 𝐵𝐵. 𝐴𝐴 is defined in the universe of discourse 𝑋𝑋 where 𝐵𝐵 here
which is transform fuzzy set is defined in the universe of discourse 𝑌𝑌. So, this is called the
extension principle.

(Refer Slide Time: 10:34)

So, let us take again here other examples we are taking this example also to make you
understand the extension principle better. So, we have taken an example here and in this
example we have taken a fuzzy set 𝐴𝐴, 𝐴𝐴 fuzzy set which is a discrete fuzzy set of course,
you can see and this discrete fuzzy set 𝐴𝐴 is defined in the universe of discourse 𝑋𝑋. So, that
is why 𝑥𝑥 is mentioned here 𝑥𝑥 is the generic variable. And which is nothing, but belonging
into the universe of discourse 𝑋𝑋.

Now, this is very simple this is very clear very easy to understand that we have a simple
discrete fuzzy set 𝐴𝐴, which has five elements. Now, our job is to convert or transform this
fuzzy set into 𝐵𝐵(𝑦𝑦). And here the new fuzzy set the transformed fuzzy set you see it is
defined in the universe of discourse 𝑌𝑌 and this is possible when the 𝑦𝑦 = 𝑓𝑓(𝑥𝑥) that is the
mapping function when this is known this is possible. So, let us do that.

618
And here this problem will include the universe of discourse which is mentioned here that
is from −10 to 10. So, let us now use the extension principle for mapping function and
let us move ahead. Mapping function is given in this problem mapping function is 𝑓𝑓(𝑥𝑥)
and 𝑓𝑓(𝑥𝑥) = 𝑥𝑥 2 + 𝑥𝑥 − 3 which is here. So, the transform fuzzy set that is 𝐵𝐵(𝑦𝑦), B of y will
look like this of course, now we have to find its membership values corresponding to its
generic variable values small 𝑦𝑦’s.

(Refer Slide Time: 12:57)

So, let us now quickly move ahead and let us see what we do to get this. So, we simply
write here the fuzzy set 𝐴𝐴 which is in the universe of discourse 𝑋𝑋. And then we write the
mapping function this is the mapping function. So, it is very easy to convert 𝐴𝐴(𝑥𝑥) into
𝐵𝐵(𝑦𝑦) and we simply write 𝐵𝐵(𝑦𝑦) in terms of 𝜇𝜇𝐵𝐵 (𝑦𝑦)/𝑦𝑦’s. And then we find the values of
𝜇𝜇𝐵𝐵 (𝑦𝑦)’s and 𝑦𝑦’s means the membership values corresponding its generic variable values.

619
(Refer Slide Time: 13:51)

So, the mapping function is here this is given and here we have the fuzzy set this is also
given. So, I am just writing that given fuzzy set in the universe of discourse 𝑋𝑋. So, since
in this discrete fuzzy set we have these generic variable values −2 ,−1, 0, 1, 2. So, let us
first compute the generic variable values, that mean, the 𝑦𝑦1 , 𝑦𝑦2 , 𝑦𝑦3 , 𝑦𝑦4 𝑎𝑎𝑎𝑎𝑎𝑎 𝑦𝑦5 in the
universe of discourse 𝑌𝑌. And this is very easy to compute because we have the mapping
function with us.

So, what we need is here let say the mapped fuzzy set that we are interested in is 𝐵𝐵(𝑦𝑦),
then what we need here is corresponding to 𝐴𝐴(𝑥𝑥) we need here is
𝜇𝜇(𝑦𝑦1 )⁄𝑦𝑦1 , 𝜇𝜇(𝑦𝑦2 )⁄𝑦𝑦2 , 𝜇𝜇(𝑦𝑦3 )/𝑦𝑦3 , 𝜇𝜇(𝑦𝑦4 )/ 𝑦𝑦4 , 𝜇𝜇 (𝑦𝑦5 )/ 𝑦𝑦5 . So, the unknowns are
𝑦𝑦1 , 𝑦𝑦2 , 𝑦𝑦3 , 𝑦𝑦4 , 𝑦𝑦5 mapping function is given. So, let us first compute 𝑦𝑦1 . So, our 𝑦𝑦1 is here.
So, since the mapping function is known we just substitute the value of 𝑥𝑥, that means, for
𝑥𝑥1 = −2 we are getting our 𝑦𝑦1 = −1.

Similarly, for 𝑥𝑥2 = −1 we are getting 𝑦𝑦2 = −3. For 𝑥𝑥 we are getting 𝑦𝑦 for 𝑥𝑥3 we are
getting 𝑦𝑦3 = −3, 𝑥𝑥3 = 0 here. So, for 𝑥𝑥3 = 0we are getting 𝑦𝑦3 = −3, 𝑥𝑥4 = 4 we are
getting 𝑦𝑦4 = −1, 𝑥𝑥5 = 2 we are getting 𝑦𝑦5 , 3. So, this means that the mapping function
helping us to compute 𝑦𝑦1 , 𝑦𝑦2 , 𝑦𝑦3 , 𝑦𝑦4 , 𝑦𝑦5 corresponding to the generic variable values
defined in the universe of discourse 𝑋𝑋.

So, when this is known then we directly substitute this in the equation 𝐵𝐵(𝑦𝑦) here this
equation this fuzzy set. So, all 𝑦𝑦, 𝑦𝑦1 , 𝑦𝑦2 , 𝑦𝑦3 , 𝑦𝑦4 , 𝑦𝑦5 are basically substituted.

620
(Refer Slide Time: 17:57)

And please understand that here this is 𝜇𝜇(𝑥𝑥1 ), 𝜇𝜇(𝑥𝑥2 ), 𝜇𝜇(𝑥𝑥3 ), 𝜇𝜇(𝑥𝑥4 ), 𝜇𝜇(𝑥𝑥5 ). So, that means,
when we substitute the values of 𝜇𝜇(𝑥𝑥1 ), 𝜇𝜇(𝑥𝑥2 ), 𝜇𝜇(𝑥𝑥3 ), 𝜇𝜇(𝑥𝑥4 ), 𝜇𝜇(𝑥𝑥5 ) and 𝑦𝑦1 , 𝑦𝑦2 , 𝑦𝑦3 , 𝑦𝑦4 , 𝑦𝑦5 .
Then we get this equation. So, this means we get a transform set like this. So, transform
fuzzy set is 𝐵𝐵(𝑦𝑦) fuzzy set.

So, I am again writing here how are we getting just to make you understand better. So, this
we have got by just substituting these values 𝜇𝜇(𝑥𝑥2 ), 𝜇𝜇(𝑥𝑥3 )/ 𝑦𝑦 3, 𝜇𝜇(𝑥𝑥4 )/ 𝑦𝑦4 then
𝜇𝜇(𝑥𝑥5 )/ 𝑦𝑦5 . So, when we substitute this these values 𝜇𝜇(𝑥𝑥1 ), 𝜇𝜇(𝑥𝑥2 ), 𝜇𝜇(𝑥𝑥3 ), 𝜇𝜇(𝑥𝑥4 ) and 𝜇𝜇(𝑥𝑥5 )
these values we have already got with the equation 𝐴𝐴(𝑥𝑥), that means in fuzzy set 𝐴𝐴(𝑥𝑥) we
already have these values. So, we do not have to compute this.

We only have to compute 𝑦𝑦1 , 𝑦𝑦2 , 𝑦𝑦3 , 𝑦𝑦4 , 𝑦𝑦5 , that means the generic variable values in the
universe of discourse 𝑌𝑌. And this has to be computed by using the generic variable values
in the universe of discourse given, that means the 𝑋𝑋 and this is possible because we have
the mapping function available, that means 𝑦𝑦 = 𝑓𝑓(𝑥𝑥) is available by using this here we get
𝑦𝑦1 , 𝑦𝑦2 , 𝑦𝑦3 , 𝑦𝑦4 and 𝑦𝑦5 .

(Refer Slide Time: 20:34)

621
And when we substitute finally we get the transform fuzzy set 𝐵𝐵(𝑦𝑦). And it is very clear
that this B of y is defined in the universe of discourse 𝑌𝑌. So, when we substitute
𝜇𝜇(𝑥𝑥1 ), 𝜇𝜇(𝑥𝑥2 ), 𝜇𝜇(𝑥𝑥3 ), 𝜇𝜇(𝑥𝑥4 ) and 𝜇𝜇(𝑥𝑥5 ) and also 𝑦𝑦1 , 𝑦𝑦2 , 𝑦𝑦3 , 𝑦𝑦4 , 𝑦𝑦5 . Then we get here this
fuzzy set, but when we rearrange this what we see is this situation. So, this situation is
situation of conflict, that means that for the generic variable here for the same generic
variable value; that means, 𝑦𝑦 = −1 we have two membership values. So, 𝑦𝑦1 = −1 we
have 0.1 as the membership value. And again here also 𝑦𝑦 this is 𝑦𝑦4 . So, 𝑦𝑦4 = −1 we have
0.9.

So, this means for the same value of the generic variable we are getting two membership
values. And this is the conflicting situation, so which one to keep because we can only
have one membership value. So, we take the max of 0.1 and 0.9. So, max of 0.1 and 0.9 is
0.9. So, that why how we avoid this conflicting situation by keeping 0.9 for the generic
variable value −1.

And then finally, we have our 𝐵𝐵(𝑦𝑦) the transformed fuzzy set in the universe of discourse
that is 𝐵𝐵(𝑦𝑦) = 0.9/(−1) + 0.8/(−3) + 0.3/3. So, here we have two conflicting situation
one was for minus 1 as the generic variable value another one for 𝑦𝑦 = −3. So, these two
have been avoided and then finally, we have the transform fuzzy set which is 𝐵𝐵(𝑦𝑦). And
this is very clear that we have got it from 𝐴𝐴(𝑥𝑥) which was given and this 𝐴𝐴(𝑥𝑥) was defined
in the universe of discourse 𝑋𝑋 and this was possible or this is possible with the help of
extension principle.

622
(Refer Slide Time: 23:37)

Now, let us take an other example here, we are taking another fuzzy set which is again
defined in the universe of discourse 𝑋𝑋. And this 𝑋𝑋 range here is given that is −50 to 50.
So, all the generic variable values will be within the limit that is given that is −50 to 50.
So, we have the fuzzy set 𝐴𝐴 and let us now use the mapping function which is also given
that is 𝑦𝑦 = −3𝑥𝑥 2 + 𝑥𝑥. So, this is the mapping function if the mapping function is not given
then the conversion is not possible.

So, let us use this mapping function 𝑦𝑦 = 𝑓𝑓(𝑥𝑥) = −3𝑥𝑥 2 + 𝑥𝑥 to compute

𝑦𝑦1 , 𝑦𝑦2 , 𝑦𝑦3 , 𝑦𝑦4 and 𝑦𝑦5 in this case.

623
(Refer Slide Time: 24:44)

So, let us now quickly compute these values.

(Refer Slide Time: 24:49)

So, for 𝑥𝑥1 = 0 we have 𝑦𝑦1 = 0, 𝑥𝑥2 = 1 we have 𝑦𝑦2 = −2, 𝑥𝑥3 = 2 we have 𝑦𝑦3 = −10,
𝑥𝑥4 = 3 we have 𝑦𝑦4 , 𝑥𝑥5 = 4 we have 𝑦𝑦5 = −44.

So, what we have here is 𝑦𝑦1 = 0, 𝑦𝑦2 = −2, 𝑦𝑦3 = −10, 𝑦𝑦4 = −24, 𝑦𝑦5 = −44. What else
do you need to write the transform fuzzy set that is 𝐵𝐵(𝑦𝑦). So, we already have
𝑦𝑦1 , 𝑦𝑦2 , 𝑦𝑦3 , 𝑦𝑦4 and 𝑦𝑦5 computed by using the mapping function.

624
So, yes so, we have to now find 𝜇𝜇𝐵𝐵 (𝑥𝑥1 ), 𝜇𝜇𝐵𝐵 (𝑥𝑥2 ), 𝜇𝜇𝐵𝐵 (𝑥𝑥3 ), 𝜇𝜇𝐵𝐵 (𝑥𝑥4 )and 𝜇𝜇𝐵𝐵 (𝑥𝑥5 ). So, let we tell
you that we do not have to do anything we do not have to do any calculation we simply
write here the 𝜇𝜇𝐵𝐵 (𝑥𝑥1 ) = 𝜇𝜇𝐴𝐴 (𝑥𝑥1 ). And as I said that this 𝜇𝜇(𝑦𝑦1 ) is going to be equal to this,
this means that we do not have to compute this we have to just keep these values just take
these values. And, this gets automatically transfered into the other universe of discourse
means 𝜇𝜇(𝑦𝑦𝑖𝑖 ) = 𝜇𝜇(𝑥𝑥𝑖𝑖 ).

So, when we apply this we see that 𝜇𝜇𝐵𝐵 (0) = 0.2. And then 𝜇𝜇𝐵𝐵 (−2) = 0.7 and 𝜇𝜇𝐵𝐵 (−10) =
0.5, 𝜇𝜇𝐵𝐵 (−24) = 0.6, 𝜇𝜇𝐵𝐵 (−44) = 0.1. So, I can write here one more equation and this is
𝜇𝜇(𝑥𝑥1 )/𝑦𝑦 1 𝑡𝑡ℎ𝑒𝑒𝑒𝑒 𝜇𝜇(𝑥𝑥2 )/𝑦𝑦2 𝑡𝑡ℎ𝑒𝑒𝑒𝑒 𝜇𝜇(𝑥𝑥3 )/𝑦𝑦3 𝑡𝑡ℎ𝑒𝑒𝑒𝑒 𝜇𝜇(𝑥𝑥4 )/𝑦𝑦4 𝑡𝑡ℎ𝑒𝑒𝑒𝑒 𝜇𝜇(𝑥𝑥5 )/𝑦𝑦5 . So, this way
we are able to get the transformed fuzzy set 𝐵𝐵(𝑦𝑦) in the other universe of discourse that is
𝑌𝑌, but here we have seen that for this transformation we should have the mapping function
which is connecting the universe of discourse 𝑋𝑋 and 𝑌𝑌 is known.

Now, let us take another example here which is the case of a continuous fuzzy set. So, here
we are taking a continuous fuzzy set instead of a discrete fuzzy set.

(Refer Slide Time: 28:49)

As we have seen in the previous example. So, here we are taking a fuzzy set mu fuzzy set
𝐴𝐴, where 𝜇𝜇𝐴𝐴 (𝑥𝑥) is gaussian like this. So, we can write it like this and let me tell you that
this fuzzy set is defined in the universe of discourse 𝑋𝑋. So, we can write it like this.

625
Now, the mapping function is also available. So, mapping function is this, this is our
mapping function. And this mapping function says 𝑦𝑦 = 𝑓𝑓(𝑥𝑥) = 𝑥𝑥 2 = 3, when the generic
variable values 𝑥𝑥 is more than 0. And it is the 𝑓𝑓(𝑥𝑥) is 𝑥𝑥 when generic variable values are
either less than or equal to 0.

So, when we use this mapping function and the fuzzy set, that is given which is defined in
the universe of discourse 𝑋𝑋. So, let us see how we write or how we find the fuzzy set B
which is defined in the universe of discourse 𝑌𝑌.

(Refer Slide Time: 30:41)

So here you see figure a is the plot of given fuzzy set A, you see here this is given that is
the Gaussian fuzzy set. So, we can write here the fuzzy set A and figure b is the plot of
given mapping function.

So, here is the mapping function which is given to us see here. So, we see that we have the
𝑓𝑓(𝑥𝑥) here we have the 𝑓𝑓(𝑥𝑥). So, we see x axis and y axis. And then we have the mapping
function defined here for all values of 𝑥𝑥 when it is greater than 0 the 𝑓𝑓(𝑥𝑥) = 𝑥𝑥 2 − 3, that
means, this part or I can say that when the 𝑥𝑥 values are more than 0 the 𝑓𝑓(𝑥𝑥) become 𝑥𝑥 2 −
3. And when the 𝑥𝑥 is either less than or equal to 0 the 𝑓𝑓(𝑥𝑥) becomes only 𝑥𝑥, means this
part.

So, f x is known to me here, now let us find B. So, we clearly see that let me first tell you
what is figure c here. So, figure c is the fuzzy set which we have obtained after employing

626
the extension principle. So, I am going to explain you has to how we are going to get this
fuzzy set 𝐵𝐵. And this 𝐵𝐵 fuzzy set is in the universe of discourse 𝑌𝑌. So, we see that if we
take up 𝑓𝑓(𝑥𝑥) we take up the mapping function here. So, for some value of 𝑥𝑥 here let us say
we have 𝑥𝑥1 and then let’s say we have 𝑥𝑥2 let us say this is 𝜇𝜇(𝑥𝑥2 ).

So, this means that when 𝑥𝑥1 and 𝑥𝑥2 these two are the different values, for these two values
𝑥𝑥1 and 𝑥𝑥2 we see that we have here the same membership value. So, when we apply the
extension principle here.

(Refer Slide Time: 33:36)

So, this says that if we have we can clearly see here if we have 𝑥𝑥1 and 𝑥𝑥2 like this which
is belonging into the universe of discourse 𝑋𝑋. And if they are not equal as we have seen in
this case, I am making this again let us say this is my 𝑥𝑥1 and let us say this is my 𝑥𝑥2 .

And we see that we are for these two different values of 𝑥𝑥, that means, 𝑥𝑥1 and 𝑥𝑥2 we have
getting same 𝑦𝑦, same 𝑦𝑦 and we can call this as y star like this we have our 𝑥𝑥1 and 𝑥𝑥2 which
are not equal, but we are getting 𝑓𝑓(𝑥𝑥1 ) is equal to 𝑓𝑓(𝑥𝑥2 ). And here y remains the same for
these 𝑥𝑥1 and 𝑥𝑥2 . So, we call this as the y star and of course, this 𝑦𝑦 ∗ should also belong into
the universe of discourse 𝑌𝑌.

So, in this case what happens in this case what we do here is that we take the max of the
𝜇𝜇𝐴𝐴 (𝑥𝑥1 ) and 𝜇𝜇𝐴𝐴 (𝑥𝑥2 ), that means, we see here if we take the mu x what is the mu x this is x
axis. So, generic variable mu here and then my 𝜇𝜇(𝑥𝑥1 ) will be this. And my here this is

627
𝜇𝜇(𝑥𝑥2 ). So, what we are doing here is we are getting two mu values two membership values
corresponding to two generic variable values, that means, 𝑥𝑥1 and 𝑥𝑥2 we are getting
𝜇𝜇(𝑥𝑥1 ), 𝜇𝜇(𝑥𝑥2 ) and 𝜇𝜇(𝑥𝑥1 ) and 𝜇𝜇(𝑥𝑥2 ) are different they are not the same. So, what we do here
is we take max of these two. So, when we take max of these two it is very clear that 𝜇𝜇(𝑥𝑥1 )
is the winner.

So, we retain 𝜇𝜇(𝑥𝑥1 ) because 𝜇𝜇(𝑥𝑥1 ) here in this case I am talking about this case only. So,
in this case 𝜇𝜇(𝑥𝑥1 ) is greater than here 𝜇𝜇(𝑥𝑥1 ) > 𝜇𝜇(𝑥𝑥2 ). So, 𝜇𝜇𝐵𝐵 (𝑦𝑦 ∗ ). So, what will be the
membership value corresponding to this generic variable value in the universe of discourse
𝑌𝑌, that means this star 𝑦𝑦 ∗ . So, 𝜇𝜇𝐵𝐵 (𝑦𝑦 ∗ ) is nothing, but it is going to be the
max(𝜇𝜇(𝑥𝑥1 ), 𝜇𝜇(𝑥𝑥2 )) and by using this what we get is here.

So, we see that the corresponding 𝑦𝑦 is here corresponding 𝑦𝑦 is 𝑦𝑦 ∗ . So, 𝑦𝑦 ∗ is here and
corresponding to this 𝑦𝑦 ∗ , the mu is here this is 𝜇𝜇𝐵𝐵 (𝑦𝑦) this is 𝜇𝜇𝐵𝐵 (𝑦𝑦 ∗ ). And this is nothing
but max(𝜇𝜇(𝑥𝑥1 ), 𝜇𝜇(𝑥𝑥2 ). So, that is how we keep getting the corresponding 𝜇𝜇𝐵𝐵 values means
the membership values.

So, here in this problem or in this example what is happening is that for the range here
which is shown by this dotted line this and this, we see that here we see that in between
this we have x values where we are getting same 𝑦𝑦 values for two 𝑥𝑥 values, means in other
words I can say that for multiple values of 𝑥𝑥 we are getting same 𝑦𝑦, I should not say
multiple but I will say the pair of 𝑥𝑥.

So, this is the range in which more than one value of 𝑥𝑥 we are getting the same 𝑦𝑦. So, that
is why here this range in this range we apply the max condition. And then after this range
we do not have to worry because we are going to get the only one to one mapping. So, that
is how when we apply these we are going get this shape. So, this is what is our 𝜇𝜇𝐵𝐵 (𝑦𝑦)
which is nothing, but the membership function in the universe of discourse 𝑌𝑌 and this is
for the fuzzy set B. So, I can write here that the resulting function B is like this. So, this
way we apply extension principle in order to map from one universe of discourse to another
for the fuzzy set A transforming into B.

628
(Refer Slide Time: 39:53)

So, now if we have let say the mapping function in n dimension means if we have a
mapping function like 𝑓𝑓(𝑥𝑥1 , 𝑥𝑥2 , 𝑥𝑥3 , … . . , 𝑥𝑥𝑛𝑛 ). So, then how can be map this to 𝑦𝑦. So,
suppose a mapping function 𝑓𝑓 is in the 𝑛𝑛 dimensional universe of discourse; that means,
(𝑋𝑋1 , 𝑋𝑋2 , 𝑋𝑋3 , … . . , 𝑋𝑋𝑛𝑛 ). And, we are mapping this to the universe of discourse 𝑌𝑌. So, this
means that we have 𝑌𝑌 is equal to 𝑓𝑓(𝑥𝑥1 , 𝑥𝑥2 , 𝑥𝑥3 , … . . , 𝑥𝑥𝑛𝑛 ).

And here let’s say we have a fuzzy set which is defined in the universe of discourse (𝑋𝑋1 ,
𝑋𝑋2 , 𝑋𝑋3 , … . . , 𝑋𝑋𝑛𝑛 ). So, how can we transform the fuzzy set B which is in the universe of
discourse 𝑌𝑌. And this is what is the mapping function that is given to us, this is mapping
function.

So, in this case the extension principle helps us in finding the membership values, that
means, the 𝜇𝜇𝐵𝐵 (𝑦𝑦) where this

max −1
[𝑚𝑚𝑚𝑚𝑚𝑚𝑖𝑖 𝜇𝜇𝐴𝐴𝑖𝑖 (𝑥𝑥𝑖𝑖 )] , 𝑖𝑖𝑖𝑖 𝑓𝑓 −1 (𝑦𝑦) ≠ 𝜙𝜙
𝜇𝜇𝐵𝐵 (𝑦𝑦) = �(𝑥𝑥𝑖𝑖1 ,𝑥𝑥𝑖𝑖2,……,𝑥𝑥𝑖𝑖𝑖𝑖)=𝑓𝑓 (𝑦𝑦)
0, 𝑖𝑖𝑖𝑖 𝑓𝑓 −1 (𝑦𝑦) = 𝜙𝜙

So, this is how we can manage to map the n dimensional space into the single dimensional
space, that means, 𝑥𝑥1 , 𝑥𝑥2 , 𝑥𝑥3 … . . , 𝑥𝑥𝑛𝑛 into 𝑦𝑦 and with this I would like to stop here.

629
(Refer Slide Time: 42:32)

And, in the next lecture we will discuss the composition of fuzzy relations.

Thank you.

630
 Fuzzy Sets, Logic and Systems and Applications
Prof. Nishchal K. Verma
Department of Electrical Engineering
Indian Institute of Technology, Kanpur

Lecture – 36
Composition of Fuzzy Relations

So, welcome to lecture number 36 of Fuzzy Sets, Logic and Systems and Applications. In
this lecture we will discuss the Composition of Fuzzy Relations.

(Refer Slide Time: 00:32)

So, fuzzy relations basically in different product spaces can be combined through a
composition operation. And there have been suggested mainly two composition strategies.
So, what are these two composition operations or strategies for fuzzy relations? So, first
one is the max-min composition and the second one is the max-product composition.

631
(Refer Slide Time: 01:05)

So, let us discuss this one by one. So, max-min composition here, let us have two fuzzy
relation sets 𝑅𝑅1 and 𝑅𝑅2 where, 𝑅𝑅1 (𝑥𝑥, 𝑦𝑦) and 𝑅𝑅2 (𝑦𝑦, 𝑧𝑧).

So, you can see here that 𝑅𝑅1 is in the universe of discourse 𝑋𝑋 and 𝑌𝑌 and 𝑅𝑅2 is the universe
of discourse 𝑌𝑌 and 𝑍𝑍. So, the max-min composition of 𝑅𝑅1 and 𝑅𝑅2 will result a new fuzzy
set which is defined by the 𝑅𝑅1 ∘ 𝑅𝑅2 . What is this ∘? ∘ is the composition operation.

So, 𝑅𝑅1 ∘ 𝑅𝑅2 , so that means the composition operation on 𝑅𝑅1 and 𝑅𝑅2 . So, this is basically
this gives us a new fuzzy set which is defined by the elements 𝑥𝑥, 𝑧𝑧 along with its
membership value like 𝜇𝜇𝑅𝑅1 ∘𝑅𝑅2 (𝑥𝑥, 𝑧𝑧).

Such that for every 𝑥𝑥, 𝑧𝑧 that is belonging into the universe of discourse 𝑋𝑋 and 𝑍𝑍, that means
the universe of discourse becomes 𝑋𝑋 × 𝑍𝑍. So, that membership values of 𝑅𝑅1 ∘ 𝑅𝑅2 can be
found by this expression. So,

𝜇𝜇𝑅𝑅1 ∘𝑅𝑅2 (𝑥𝑥, 𝑧𝑧) = max min�𝜇𝜇𝑅𝑅1 (𝑥𝑥, 𝑦𝑦), 𝜇𝜇𝑅𝑅2 (𝑦𝑦, 𝑧𝑧)� |∀(𝑥𝑥, 𝑦𝑦) ∈ 𝑋𝑋 × 𝑌𝑌 𝑎𝑎𝑎𝑎𝑎𝑎 ∀(𝑦𝑦, 𝑧𝑧) ∈ 𝑌𝑌 × 𝑍𝑍
𝑦𝑦

So, this can also be written as,

𝜇𝜇𝑅𝑅1 ∘𝑅𝑅2 (𝑥𝑥, 𝑧𝑧) =∨𝑦𝑦 [𝜇𝜇𝑅𝑅1 (𝑥𝑥, 𝑦𝑦) ∧ 𝜇𝜇𝑅𝑅2 (𝑦𝑦, 𝑧𝑧)]

632
So, let us now understand this here that what actually is happening. So, what is happening
over here is 𝑅𝑅1 is the relation which is defined in the universe of discourse 𝑋𝑋 × 𝑌𝑌, 𝑅𝑅2 is a
relation which is defined in the universe of discourse 𝑌𝑌 × 𝑍𝑍.

So, here we need to understand that 𝑅𝑅1 and 𝑅𝑅2 are two fuzzy relation sets which are defined
in the different universe of discourses. So, from this if we are interested in finding the
relation set in between 𝑅𝑅1 and 𝑅𝑅2 like this like a new fuzzy set which is defined in 𝑥𝑥 and
𝑧𝑧 only.

So, this max-min composition helps us. So, 𝑅𝑅1 ∘ 𝑅𝑅2 , that means, max-min composition if
it is applied on 𝑅𝑅1 and 𝑅𝑅2 we are going to get the third fuzzy relation which is the outcome
of 𝑅𝑅1 and 𝑅𝑅2 and if you see, the resulting fuzzy relation set will not have the dependency
on 𝑦𝑦. So, that is what is interesting here.

So, if 𝑅𝑅1 fuzzy relation set and 𝑅𝑅2 fuzzy relation set are expressed as fuzzy relation
matrices 𝑅𝑅1 ∘ 𝑅𝑅2 which is almost similar as the matrix multiplication except multiplication
and plus are replaced by the min and max respectively. So, for this reason max min
composition is also called as the max min product. So, this is the best known composition
proposed by Prof. Lotfi A Zadeh, who is father of fuzzy logic.

(Refer Slide Time: 06:33)

So, when we come to the max-product composition what happens is that the min is
changed to product, that means instead of min we multiply.

633
So, if we take the same 𝑅𝑅1 𝑅𝑅2 fuzzy relation sets and they are defined in different universe
of discourses like 𝑋𝑋 × 𝑌𝑌 and 𝑌𝑌 × 𝑍𝑍 respectively then 𝑅𝑅1 ∘ 𝑅𝑅2 is equal to the (𝑥𝑥, 𝑧𝑧), that
means the element of the fuzzy relation set along with 𝜇𝜇𝑅𝑅1 ∘𝑅𝑅2 (𝑥𝑥, 𝑧𝑧) and this (𝑥𝑥, 𝑧𝑧) is again
belonging into the universe of discourse 𝑋𝑋 × 𝑍𝑍.

Now, how to get this 𝜇𝜇𝑅𝑅1 ∘𝑅𝑅2 (𝑥𝑥, 𝑧𝑧) this? So, this can be computed by this expression. So,

𝜇𝜇𝑅𝑅1 ∘𝑅𝑅2 (𝑥𝑥, 𝑧𝑧) = max�𝜇𝜇𝑅𝑅1 (𝑥𝑥, 𝑦𝑦) ∗ 𝜇𝜇𝑅𝑅2 (𝑦𝑦, 𝑧𝑧)� |∀(𝑥𝑥, 𝑦𝑦) ∈ 𝑋𝑋 × 𝑌𝑌 𝑎𝑎𝑎𝑎𝑎𝑎 ∀(𝑦𝑦, 𝑧𝑧) ∈ 𝑌𝑌 × 𝑍𝑍.
𝑦𝑦

This can also be written by this expression here we can use the open inverted triangle for
max. So, this can be written simply by this expression also.

(Refer Slide Time: 08:25)

Let us take an example to understand this max-min composition. So, here we have three
fuzzy sets and what we will be doing here, from these three fuzzy sets we will be
developing the fuzzy relation sets these fuzzy sets A, B and C are defined in different
universe of discourses. Like fuzzy set A is defined in the universe of discourse 𝑋𝑋, B is
defined in the universe of discourse 𝑌𝑌 and C is defined in the universe of discourse 𝑍𝑍.

So, if we develop a relation set here 𝑅𝑅1 which is in between 𝑋𝑋 and 𝑌𝑌. So, we can say that
𝑅𝑅1 is a fuzzy relation set which is basically is based on 𝑥𝑥 which is related to 𝑦𝑦 and similarly
we have another fuzzy relation set which is 𝑅𝑅2 and this 𝑅𝑅2 fuzzy relation set in which the
𝑦𝑦 is related to 𝑧𝑧.

634
So, let us assume some membership values for 𝑅𝑅1 and similarly for 𝑅𝑅2 and we write the
fuzzy relation matrix 𝑅𝑅1 as we have already seen in the previous lectures. So, we have 𝑅𝑅1
is a fuzzy relation matrix, 𝑅𝑅2 is another fuzzy relation matrix. So, 𝑅𝑅1 is in terms of the
generic variable 𝑥𝑥 and 𝑦𝑦, 𝑅𝑅2 is in terms of the generic variable 𝑦𝑦, 𝑧𝑧.

(Refer Slide Time: 10:24)

So, let us go for the max-min composition of 𝑅𝑅1 and 𝑅𝑅2 first and one more thing that is
very important here is to note that the dimension of 𝑅𝑅1 here is 3 × 4, the dimension of 𝑅𝑅2
is 4 × 2. So, as we already know that there is a condition that we have to follow that the
column of the first matrix should be equal to the row of the second matrix. So, otherwise
we cannot multiply.

So, we see that we have 𝑅𝑅1 , 3 × 4, 𝑅𝑅2 , 4 × 2; so that means, we can multiply these two
matrices. So, let us do that and the outcome of this multiplication will generate another
fuzzy relation matrix and the order of this matrix will be 3 × 2.

635
(Refer Slide Time: 11:25)

So, let us represent this 𝑅𝑅1 composition 𝑅𝑅2 , and this 𝑅𝑅1 composition 𝑅𝑅2 will have the 𝑥𝑥
and 𝑧𝑧 generic variables and this can be written by this matrix here and we already know
that this is going to be a 3 × 2 matrix because the 𝑅𝑅1 is 3 × 4, 𝑅𝑅2 is 4 × 2.

(Refer Slide Time: 12:03)

So, let us know multiply this. So, 𝑅𝑅1 composition 𝑅𝑅2 we have 𝑅𝑅1 here, we have 𝑅𝑅2 here
and let us now multiply we take the first row of 𝑅𝑅1 and the first column of 𝑅𝑅2 . So, when
we take first row of 𝑅𝑅1 here and then first column of 𝑅𝑅2 like this.

636
So, like a normal matrix multiplication we will multiply this, but instead of using the
multiplication sign we use min and instead of plus sign we use max. So, here 0.7 × 0.6.
So, when we use multiplication, we simply take the product of 0.7 and 0.6. So, here we
will not take the product rather than we will take min of it. So, we used the min sign here
the open triangle and then again what we do here is that we again go to the second element
of the first row of the 𝑅𝑅1 and then second element of the first column of 𝑅𝑅2 .

We have 0.6 and 0.9, so instead of taking the product we will take min again. Similarly,
here instead of taking product of 0.3 and 0.4 we take the min, similarly here the 0.4 and
0.2.

So, this way we see that we have multiplied the first row and first column of 𝑅𝑅1 and 𝑅𝑅2
respectively. Now, when we have taken the respective minimum values, then whatever
values that we have now we take the max. So, when we are multiplying we were just
adding, but instead of adding here we take the max. So, when we take max we are getting
0.6.

So, likewise here as we have taken the first row of 𝑅𝑅1 and first column of 𝑅𝑅2 , now we will
take first row of 𝑅𝑅1 and second column of 𝑅𝑅2 . So, when we do that we find 0.7 after taking
max-min and then similarly we pick the second row of 𝑅𝑅1 .

So, when we take second row of 𝑅𝑅1 and first column of 𝑅𝑅2 . So, we see that here we have
0.9 min of 0.9, 0.6 and then min of 0.4, 0.9 and then we have min of 0.2, 0.4, then min of
0.7, 0.2. So, like that and then we have 4 values coming after taking mins. So, then we take
max of these 4 values and then when we take max of these 4, we get 0.6. So, that is how
we take the composition of the second row and first column.

Now, similarly the we use the second row of 𝑅𝑅1 and second column of 𝑅𝑅2 and when we
do that we again get the max single value and then similarly we take the third row of 𝑅𝑅1
and take first column of 𝑅𝑅2 and then third row of 𝑅𝑅1 and second column of 𝑅𝑅2 . So, this
way we are getting all these values.

637
(Refer Slide Time: 16:14)

Now, when we have found all these max min values, then we let us arrange and when we
arrange this we are getting a matrix as a result which is of 3 cross 2. So, this gives us the
max-min composition of 𝑅𝑅1 and 𝑅𝑅2 and let us recall that this 𝑅𝑅1 is defined in terms of x
and y means the fuzzy relation set 𝑅𝑅1 is in the space in the universe of discourse 𝑋𝑋 × 𝑌𝑌
and here 𝑅𝑅2 is in the universe of discourse 𝑌𝑌 × 𝑍𝑍.

But the max min composition is giving us a new fuzzy relation set which is let us say R
dash. So, this 𝑅𝑅′ is going to be in 𝑋𝑋, 𝑍𝑍 and this is nothing, but the max min composition
of 𝑅𝑅1 and 𝑅𝑅2 .

638
(Refer Slide Time: 17:28)

So, now, on the same lines we can get max-product composition here the difference is
nothing, but only instead of taking min we take the product at rest other things remain the
same.

So, if we take the same example here means 𝑅𝑅1 fuzzy reason set we have here as it is
mentioned as it is written here and then similarly the 𝑅𝑅2 also and here we see that again
we have 3 × 4 matrix and then we have 4 × 2 matrix. So, we see that we can multiply
these two matrices and the resulting matrix will again be 3 × 2.

(Refer Slide Time: 18:19)

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So, what we do here is we first take the first row of 𝑅𝑅1 and then first column of 𝑅𝑅2 and
here we multiply we take the product of the corresponding elements like 0.7 and 0.6 we
take the product of it. In earlier case where we discussed max-min we were taking only
the minimum of 0.7 and 0.6, here we will be multiplying 0.7 and 0.6.

So, that is what is the difference in between the max-product and max-min. So, since we
are dealing with max-product composition. So, here we will multiply 0.7 and 0.6, then we
will multiply 0.6 and 0.9, we will multiply 0.3 and 0.4, we will multiply 0.4, 0.2.

Now, these multiplications will result 4 values and when we take max of it we are going
to get 0.54. So, similarly this first row of 𝑅𝑅1 and then second column of 𝑅𝑅2 will result 0.49
and then the second row of 𝑅𝑅1 and first column of 𝑅𝑅2 will result 0.54, second row of 𝑅𝑅1
and second column of 𝑅𝑅2 will result 0.63.

The third row of 𝑅𝑅1 and first column of 𝑅𝑅2 will result 0.81 and finally, the third row of R
1 and second column of 𝑅𝑅2 will give us 0.4. So, now, we have got all these values of the
membership and these values that we have found these are the membership values which
are with the resulting fuzzy relations set and this is resulting out of the max-product
composition of 𝑅𝑅1 and 𝑅𝑅2 .

(Refer Slide Time: 20:50)

So, let us know substitute these values and we get the max-product composition of 𝑅𝑅1 and
𝑅𝑅2 here like this where we have the 0.54, 0.49, 0.54, 0.63, 0.81, 0.4 as the values of the

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resulting fuzzy set. And of course, as I already mentioned that we have the resulting fuzzy
relation set the order of this matrix is 3 × 2.

And it is clearly visible here that the matrix that we have got here is out of the relation
between 𝑥𝑥 and 𝑧𝑧. So, this means that the max product composition of 𝑅𝑅1 and 𝑅𝑅2 is giving
us a new fuzzy relation set and this new fuzzy relation set is defined in terms of 𝑥𝑥 and 𝑧𝑧.

So; that means, here the new fuzzy relation set is defined in the universe of discourse 𝑋𝑋 ×
𝑍𝑍. So, this is very important point that has to be noted that if we have any fuzzy relation
set which is in the universe of discourse of 𝑋𝑋 × 𝑌𝑌 and another fuzzy relation set which is
defined in another universe of discourse 𝑌𝑌 × 𝑍𝑍. Then by doing this exercise either max-
product composition or max-min composition we can find the fuzzy relation set in the
universe of discourse 𝑋𝑋 × 𝑍𝑍. So, this is a very important point that has to be noted.

(Refer Slide Time: 23:10)

So, this way we have understood the max-min composition of 𝑅𝑅1 and 𝑅𝑅2 and max-product
composition of 𝑅𝑅1 and 𝑅𝑅2 and we have seen this with a couple of examples and we have
understood this very clearly. So, I will stop here for this lecture and then in the next lecture
we will study the properties of composition of fuzzy relations.

Thank you.

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 Fuzzy Sets, Logic and Systems and Applications
Prof. Nishchal K. Verma
Department of Electrical Engineering
Indian Institute of Technology, Kanpur

Lecture – 37
Properties of Composition of Fuzzy Relations

So, welcome to lecture number 37 of Fuzzy Sets, Logic and Systems and Applications. In
this lecture, we will discuss Properties of Composition of Fuzzy Relations.

(Refer Slide Time: 00:26)

As we already know that the composition of fuzzy relations are based on the either max-
min criteria or max-product criteria. And here I would like to mention that the fuzzy
relation 𝑅𝑅 fuzzy relation 𝑆𝑆1 , fuzzy relation 𝑆𝑆2 and fuzzy relation 𝑇𝑇 are the fuzzy relations
that are defined on the spaces. That means, their universe of discourses 𝑋𝑋 × 𝑌𝑌, 𝑌𝑌 × 𝑍𝑍, 𝑌𝑌 ×
𝑍𝑍 and 𝑍𝑍 × 𝑊𝑊 respectively.

So, this means that relation 𝑅𝑅 which is here; this relation 𝑅𝑅 is defined in the universe of
𝑋𝑋 × 𝑌𝑌. Similarly, 𝑆𝑆1 and 𝑆𝑆2 are defined in the universe of discourses 𝑌𝑌 × 𝑍𝑍, T is defined
in the universe of discourse 𝑍𝑍 × 𝑊𝑊. So, as I have already mentioned that 𝑅𝑅, 𝑆𝑆1 , 𝑆𝑆2 and 𝑇𝑇
are fuzzy relations.

And we have already discussed the composition of fuzzy relations in the previous lecture.
So, here we will be discussing as to how these composition of fuzzy relations hold these

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properties. So, first property is the associativity, then the second property is distributivity
over union and then we have weak distributivity over intersection. And finally, we have
monotonicity. And all these properties are defined here in this column where we have the
composition of 𝑅𝑅 and 𝑆𝑆1 .

So, when we say composition; this composition can be either max-min or max-product.
So, the composition of 𝑅𝑅 and 𝑆𝑆1 and then whatever we get here out of this (𝑅𝑅 ∘ 𝑆𝑆1 ) ∘ 𝑇𝑇.
We are going to get 𝑅𝑅 ∘ (𝑆𝑆1 ∘ 𝑇𝑇). So, these things must be understood very clearly that the
associativity with respect to the composition is satisfied or in other words we can say that
the composition of fuzzy relations 𝑅𝑅, 𝑆𝑆1 and 𝑇𝑇 are holding good with respect to the
composition.

Similarly, distributivity over union. So, we see here that we have a composition here and
please understand this small o is for the composition, this is a symbol for composition. So,
we have a ∘ so, R is small o and then we have the composition of 𝑅𝑅 ∘ (𝑆𝑆1 ∪ 𝑆𝑆2 ). So, this
can also be written as or in other words we can say 𝑅𝑅 ∘ (𝑆𝑆1 ∪ 𝑆𝑆2 ) = (𝑅𝑅 ∘ 𝑆𝑆1 ) ∪ (𝑅𝑅 ∘ 𝑆𝑆2 ).

Similarly the other property, the third one, the third property is weak distributivity over
intersection. So, here we have the 𝑅𝑅 ∘ (𝑆𝑆1 ∩ 𝑆𝑆2 ). 𝑅𝑅 ∘ (𝑆𝑆1 ∩ 𝑆𝑆2 ) ⊆ (𝑅𝑅 ∘ 𝑆𝑆1 ) ∩ (𝑅𝑅 ∘ 𝑆𝑆2 ).
Similarly, here we have the fourth property that is a monotonicity and here if we take 𝑆𝑆1
and 𝑆𝑆2 fuzzy relations and 𝑆𝑆2 is the subset of 𝑆𝑆2 .

So, then in that case the 𝑅𝑅 ∘ 𝑆𝑆1 ⊆ 𝑅𝑅 ∘ 𝑆𝑆2 . So, this is called the monotonicity here. And as I
have already mentioned that this small o represents the composition symbol and this
composition can either be the max-min or max-product. So, let us go through these
properties one by one.

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(Refer Slide Time: 05:46)

And as I have already mentioned here that for fuzzy relations 𝑅𝑅, 𝑆𝑆1 and 𝑇𝑇 this relation this
associativity relation this holds good. And I have already mentioned about 𝑅𝑅 also, 𝑅𝑅 is
defined in the universe of discourse 𝑋𝑋 × 𝑌𝑌. Similarly, 𝑆𝑆1 is defined in the universe of
discourse 𝑌𝑌 × 𝑍𝑍 and 𝑇𝑇 here is defined in the universe of discourse 𝑍𝑍 × 𝑊𝑊.

So, this needs to be understood very clearly and this is called the associativity property for
composition of fuzzy relations. So, this means that we have (𝑅𝑅 ∘ 𝑆𝑆1 ) ∘ 𝑇𝑇 = 𝑅𝑅 ∘ (𝑆𝑆1 ∘ 𝑇𝑇).
So, this is what this means and this is called the associativity property.

(Refer Slide Time: 07:00)

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So, let us take an example to understand this better with respect to the fuzzy relations. So,
let us understand this associativity for composition of fuzzy relations. So, here we have
taken the fuzzy relation 𝑅𝑅 first. We see here this is 𝑅𝑅 fuzzy relation and then we have
another fuzzy relation here as 𝑆𝑆1 and then we have the third one is the fuzzy relation 𝑇𝑇.
And we clearly see that 𝑅𝑅 is defined in the universe of discourse 𝑋𝑋 × 𝑌𝑌. Similarly, 𝑆𝑆1 is
defined in the universe of discourse 𝑌𝑌 × 𝑍𝑍 and here 𝑇𝑇 is defined in the universe of
discourse 𝑍𝑍 × 𝑊𝑊.

(Refer Slide Time: 07:59)

So, let us now compose 𝑅𝑅 and 𝑆𝑆1 first, we already have done this exercise in the previous
lecture. So, based on that we’ll try to generate the composition matrix. So, we have the
𝑅𝑅 ∘ 𝑆𝑆1 which is this. And this is going to be equal to all these elements, means we have
3 × 3 elements here in this composition of fuzzy relation matrix.

And here, please understand that when we compose a fuzzy relations. So, all these
exercises must be in conversant with the order of the matrix. This means that we need to
have the suitable order of the fuzzy relation matrices that is needless to say. So, here now
as I have already mentioned that we have followed this criteria for max-min composition,
max-min composition.

Here we could also follow max product if it is set. So, but in this case we will only be
interested in max-min. So, we have followed max-min criteria and based on this when we
apply this criteria, we get 𝜇𝜇𝑅𝑅∘𝑆𝑆1 (𝑥𝑥1 , 𝑧𝑧1 ) like this. We have already done this exercise in the

645
previous lecture. So, I am not going to explain this again. If you have any doubt, you can
go through the previous lecture to understand this better.

So, the max of all the mins here and then when we take a max of these the outcome of the
mins then we are going to get 1 here in this case. So, similarly all combinations of elements
of 𝑅𝑅 ∘ 𝑆𝑆1 .

(Refer Slide Time: 10:16)

We are going to get here this matrix, 𝑅𝑅 ∘ 𝑆𝑆1 . So, you can just try this and you are going to
get the composition of fuzzy relation matrix 𝑅𝑅 and 𝑆𝑆1 here, which is 1, 0.7, 0.8, 0.5, 0.6,
0.8, 0.7, 0.7, 0.9. And this outcome is again you can clearly see that is defined in the
universe of discourse 𝑋𝑋 × 𝑍𝑍.

So, we have now at this moment, the composition of fuzzy relation 𝑅𝑅 and fuzzy relation
𝑆𝑆1 . Now, let us go further. So, this part is done here, so now let us go further and whatever
is the outcome here is we compose with 𝑇𝑇.

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(Refer Slide Time: 11:21)

So, let us take 𝑇𝑇 here and when we compose this 𝑇𝑇 with 𝑅𝑅 ∘ 𝑆𝑆1 . So, means we take first
(𝑅𝑅 ∘ 𝑆𝑆1 ) ∘ 𝑇𝑇. So, both 𝑅𝑅 ∘ 𝑆𝑆1 and 𝑇𝑇, both are fuzzy relations. Because, 𝑅𝑅 ∘ 𝑆𝑆1 is going to
give us another fuzzy relation. So, here we compose these two here.

And when we compose this what we are going to get upon applying the max-min criteria
is this. So, 𝑅𝑅 ∘ 𝑆𝑆1 and whatever is the outcome here is composed with 𝑇𝑇 is going to give
us a new fuzzy relation matrix which is defined in the universe of discourse 𝑋𝑋 × 𝑊𝑊. So,
this is what is the outcome out of this exercise. So, this is the LHS, I can say this is the
LHS because left hand side is coming like this. Now, in the property let us now try to find
the RHS.

So, we see that this is the RHS part and this is the LHS part. So, we have done LHS, now
let us find the RHS part using the same set of fuzzy relation matrices.

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(Refer Slide Time: 13:09)

So, let us now find the 𝑆𝑆1 ∘ 𝑇𝑇. So, when we apply the max-min criteria as we see here with
the same max-min criteria, if we compose 𝑆𝑆1 and 𝑇𝑇 which is here we get this matrix.

So, we get a new fuzzy relation matrix which is defined in the universe of discourse of
𝑌𝑌 × 𝑊𝑊. Now, in the RHS, if we see we have another fuzzy relation 𝑅𝑅 with which this
outcome needs to be composed. So, when we compose, when we take the composition of
𝑅𝑅 ∘ (𝑆𝑆1 ∘ 𝑇𝑇). So, 𝑆𝑆1 ∘ 𝑇𝑇 we already have now, let us take 𝑅𝑅 and make a composition of this
and let us see what we are getting.

(Refer Slide Time: 14:06)

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So, here when we do that we are getting this outcome. And we clearly see that we are
getting a new fuzzy relation matrix out of these compositions which is defined in the
universe of discourse 𝑋𝑋 × 𝑊𝑊. So, this is nothing, but the RHS part.

(Refer Slide Time: 14:36)

So, when we write LHS here and RHS here means both LHS and RHS here. We see that
both the fuzzy relation matrices are same, are equal. So, we can say that the associativity
property holds good for the composition of fuzzy relations and with this example we are
able to understand that if we take any 3 fuzzy relations and then when we apply the
associativity property.

So, with either the max-min or max-product, this property is well satisfied or in other
words we can say that the associativity property for composition of fuzzy relations hold
good.

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(Refer Slide Time: 15:29)

Now, the second property is the distributivity over union. And this is also for the
composition of fuzzy relation. So, when we say composition of fuzzy relations. So, this
distributivity property over union is also holding good. And what is this? This is nothing
but if we have fuzzy relations 𝑅𝑅, 𝑆𝑆1 and 𝑆𝑆2 . And this 𝑅𝑅, 𝑆𝑆1 , 𝑆𝑆2 are defined in the universe
of discourses as we have already discussed.

So, in that case this distributivity over union we have the 𝑅𝑅 ∘ (𝑆𝑆1 ∪ 𝑆𝑆2 ) = (𝑅𝑅 ∘ 𝑆𝑆1 ) ∪ (𝑅𝑅 ∘
𝑆𝑆2 ). And if since these 2 are equal. So, we can say the distributivity property over union
for composition of fuzzy relations hold good.

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(Refer Slide Time: 16:47)

So this also let us understand properly by taking an example. So, here also we are taking
fuzzy relations 𝑅𝑅, 𝑆𝑆1 and 𝑆𝑆2 . Again, these are defined on spaces 𝑋𝑋 × 𝑌𝑌, 𝑌𝑌 × 𝑍𝑍, 𝑌𝑌 × 𝑍𝑍
respectively. So, means 𝑅𝑅 is fuzzy relation set is defined in the universe of discourse 𝑋𝑋 ×
𝑌𝑌. And 𝑆𝑆1 is defined in the universe of discourse 𝑌𝑌 × 𝑍𝑍.

Similarly, 𝑆𝑆2 is defined in the universe of discourse 𝑌𝑌 × 𝑍𝑍. So, this way we see that we
have 𝑅𝑅, 𝑆𝑆1 and 𝑆𝑆2 as 3 fuzzy relation matrices.

(Refer Slide Time: 17:52)

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Now, let us try to see whether the distributivity over union property holds good for these
fuzzy relations 𝑅𝑅, 𝑆𝑆1 and 𝑆𝑆2 . So, in this process we have first the LHS and then we have
RHS.

And let us move ahead, first take the LHS part. So, LHS is here this is our LHS. So, for
computing LHS we first need to find the union of 𝑆𝑆1 ∪ 𝑆𝑆2 . So, we already know how to
find the union of 𝑆𝑆1 ∪ 𝑆𝑆2 . So, when we take the union of 𝑆𝑆1 and 𝑆𝑆2 , we simply take the
max of the corresponding membership values. So, when we do this exercise we get here
as union of 𝑆𝑆1 and 𝑆𝑆2 .

Now, I am writing just this is the matrix which is 𝑆𝑆1 ∪ 𝑆𝑆2 . So, when we take the
composition of R and the union of 𝑆𝑆1 and 𝑆𝑆2 here, we get this as the outcome. So, this is
nothing but the LHS part. So, this is my LHS part.

(Refer Slide Time: 19:30)

Now, let us move ahead and we see that in the right hand side, the RHS we have the union
of two compositions. So, let us first find both the compositions, the first composition here
is the composition of 𝑅𝑅 and 𝑆𝑆1 which is this. This by applying you know the max-min
composition as I said before that it depends if you have been asked to find the 𝑅𝑅 ∘ 𝑆𝑆1 by
max-product, then you can use the max-product criteria. But here we are taking we are
using the max-min criteria, so the outcome of 𝑅𝑅 ∘ 𝑆𝑆1 is this and then 𝑅𝑅 ∘ 𝑆𝑆2 is this.

652
Now let us take the union of these two. So, when we comes to the union, then we apply
the max criteria again and we take max of the corresponding membership values of the
both fuzzy relations matrices.

(Refer Slide Time: 20:47)

So, we see that here when we take the max criteria which is this, then the outcome is this.
So, this is nothing but the RHS part. So this is the outcome of the (𝑅𝑅 ∘ 𝑆𝑆1 ) ∪ (𝑅𝑅 ∘ 𝑆𝑆2 ).

(Refer Slide Time: 21:11)

So then when we write the outcome of LHS and the outcome of RHS we see that both of
these the fuzzy relation matrices that are the outcome of the 𝑅𝑅 ∘ (𝑆𝑆1 ∪ 𝑆𝑆2 ) and the

653
(𝑅𝑅 ∘ 𝑆𝑆1 ) ∪ (𝑅𝑅 ∘ 𝑆𝑆2 ). So, both are same. So, in this way we can say the LHS is equal to
RHS.

So, this means that the matrices that we took. The fuzzy relations sets that we took holds
good for the distributivity over union property. So, in other words we can say that the
distributivity property over union holds good for the composition of fuzzy relations.

(Refer Slide Time: 22:10)

And now the next is the weak distributivity over intersection. So, when we have fuzzy
relations 𝑅𝑅, 𝑆𝑆1 and 𝑆𝑆2 and if we have the 𝑅𝑅 ∘ (𝑆𝑆1 ∩ 𝑆𝑆2 ). So if we have this thing, this is
going to be the (𝑅𝑅 ∘ 𝑆𝑆1 ) ∩ (𝑅𝑅 ∘ 𝑆𝑆2 ). This is called the weak distribution over intersection.

So what essentially we are doing here is that we are taking the intersection of the 2
composition in the right hand side. So, we first have the composition of 2 fuzzy relations,
means the 𝑅𝑅 ∘ 𝑆𝑆1 and 𝑅𝑅 ∘ 𝑆𝑆2 we are taking the intersection of it here and we have then the
𝑅𝑅 ∘ (𝑆𝑆1 ∩ 𝑆𝑆2 ). So, if we have this 𝑅𝑅 ∘ (𝑆𝑆1 ∩ 𝑆𝑆2 ), this is going to be the subset of the
outcome of the RHS.

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(Refer Slide Time: 23:31)

So, let us take an example again to understand the week distributivity over intersection
again. So, here we have 3 fuzzy relation matrices 𝑅𝑅, 𝑆𝑆1 and 𝑆𝑆2 and again I would like to
mention here that 𝑅𝑅 is defined in the universe of discourse, 𝑋𝑋 × 𝑌𝑌 and 𝑆𝑆1 is defined in the
universe of discourse 𝑌𝑌 × 𝑍𝑍 and 𝑆𝑆2 is also defined in the universe of discourse 𝑌𝑌 × 𝑍𝑍.

(Refer Slide Time: 24:12)

So, let us now first take LHS here and for LHS we have to have the 𝑅𝑅 ∘ (𝑆𝑆1 ∩ 𝑆𝑆2 ). So, let
us first find this. So, let us first find the 𝑆𝑆1 ∩ 𝑆𝑆2 and when we take intersection we know
that we use min criteria we take min of the corresponding membership values from 𝑆𝑆1 and

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𝑆𝑆2 . So, when we do that we are getting here this as the outcome. This is nothing but, the
𝑆𝑆1 ∩ 𝑆𝑆2 .

So, when we have this fuzzy relation matrix as 𝑆𝑆1 ∩ 𝑆𝑆2 . Now, let us take the max-min
composition of 𝑅𝑅 and this matrix which is here. So, when we do that. So, since we are
using max-min criteria, we are getting the final outcome as this where the elements are
0.5, 0.6, 0.8, 0.5, 0.6, 0.2, 0.5, 0.6, 0.7. So, this way we get the LHS computed for our
example using 𝑅𝑅, 𝑆𝑆1 and 𝑆𝑆2 . Now, when we have LHS computed.

(Refer Slide Time: 25:51)

Let us now find the intersection of the (𝑅𝑅 ∘ 𝑆𝑆1 ) and (𝑅𝑅 ∘ 𝑆𝑆2 ). So, here we have the (𝑅𝑅 ∘ 𝑆𝑆1 )
and (𝑅𝑅 ∘ 𝑆𝑆2 ). So, this is in the process of computing the LHS for finding RHS.

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(Refer Slide Time: 26:29)

So, let’s now move ahead and see what we are getting. So, we see that (𝑅𝑅 ∘ 𝑆𝑆1 ) is this and
this composition of course is again a max-min, here also the composition is max-min. So,
we can quickly get the (𝑅𝑅 ∘ 𝑆𝑆1 ) and (𝑅𝑅 ∘ 𝑆𝑆2 ). So, when we do that then, let us now take
the intersection of these two.

So, when we take the intersection of these 2 fuzzy relation matrices. We since we are
taking intersection. So, we use the min criteria, the basic intersection. So, this can be found
by taking the min of the respective membership values from both the fuzzy relation sets
𝑅𝑅 ∘ 𝑆𝑆1 and 𝑅𝑅 ∘ 𝑆𝑆2 and final outcome is here. And this is nothing but the RHS which is the
intersection of both the fuzzy relation sets.

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(Refer Slide Time: 27:49)

So, this way we have here the LHS and RHS. Now, if we see LHS and RHS both, we see
that both of these fuzzy relation matrices are not same. We can clearly see that the LHS
which is the 𝑅𝑅 ∘ (𝑆𝑆1 ∩ 𝑆𝑆2 ) has the elements 0.5, 0.6, 0.8, 0.5, 0.6, 0.2, 0.5, 0.6, 0.7. And
similarly, when we see the corresponding elements in the case of RHS where we have the
(𝑅𝑅 ∘ 𝑆𝑆1 ) ∩ (𝑅𝑅 ∘ 𝑆𝑆2 ). We see that we have 0.5, 0.6, 0.8, 0.5, 0.6, 0.5, 0.6, 0.6, 0.7.

So, first of all let me make it clear here that, both of the fuzzy relation matrices of LHS
and RHS they are defined in the same universe of discourses. So, here the universe of
discourse is 𝑋𝑋 × 𝑍𝑍 and here also the 𝑋𝑋 × 𝑍𝑍. The universe of discourse is 𝑋𝑋 × 𝑍𝑍. And when
we see the corresponding elements in the fuzzy relation matrices. So, we see that the LHS
the each elements are either equal or lesser than that of the elements in the RHS fuzzy
relation matrix.

So that means, here the we can say that the 𝑅𝑅 ∘ (𝑆𝑆1 ∩ 𝑆𝑆2 ) ⊆ (𝑅𝑅 ∘ 𝑆𝑆1 ) ∩ (𝑅𝑅 ∘ 𝑆𝑆2 ). So, we
see that each elements or every elements of LHS is either less or equal to the corresponding
RHS fuzzy relation matrix. So, this way we can say that the fuzzy relation sets 𝑅𝑅, 𝑆𝑆1 , 𝑆𝑆2
holds good here or we can say satisfy the weak distributivity over intersection criteria.

So, in general, we can say weak distributivity over intersection property is holding good
for the composition of given fuzzy relations.

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(Refer Slide Time: 30:47)

Now, let us discuss the monotonicity as the other property, the fourth property here for
fuzzy relations 𝑅𝑅, 𝑆𝑆1 and 𝑆𝑆2 . So, it is very simple in the sense that if we have 𝑆𝑆1 and 𝑆𝑆2 ,
two fuzzy relations and 𝑆𝑆1 ⊆ 𝑆𝑆2 . So if this is the case, then the 𝑅𝑅 ∘ 𝑆𝑆1 ⊆ 𝑅𝑅 ∘ 𝑆𝑆2 . And this
is called the monotonicity property. So, let us take an example similarly here and let us see
how the monotonicity property also is holding good for fuzzy relations.

(Refer Slide Time: 31:28)

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So, let us take the fuzzy relation set 𝑅𝑅, 𝑆𝑆1 and 𝑆𝑆2 . And here also we see that the universe
of discourse for 𝑅𝑅 is 𝑋𝑋 × 𝑌𝑌. For 𝑆𝑆1 , the universe of discourse is 𝑌𝑌 × 𝑍𝑍 and for 𝑆𝑆2 also the
universe of discourse is 𝑌𝑌 × 𝑍𝑍.

(Refer Slide Time: 32:11)

Now, let us choose 𝑆𝑆1 and 𝑆𝑆2 in such a way that our 𝑆𝑆1 ⊂ 𝑆𝑆2 . So, if we choose 𝑆𝑆1 and 𝑆𝑆2
suitably, so that 𝑆𝑆1 ⊆ 𝑆𝑆2 . We can check here that each and every element of the 𝑆𝑆1 ≤ 𝑆𝑆2 .
So, when we choose the 𝑆𝑆1 , 𝑆𝑆2 fuzzy relation set like this. Then let us now take the
composition of 𝑆𝑆1 and 𝑆𝑆2 separately with 𝑅𝑅. So, here when we take the 𝑅𝑅 ∘ 𝑆𝑆1 , we find
here a new fuzzy relation matrix which is defined in the universe of discourse capital 𝑋𝑋 ×
𝑍𝑍.

Similarly, when the fuzzy composition of 𝑅𝑅 is taken with 𝑆𝑆2 again we have the outcome
and which is fuzzy relation in the universe of discourse 𝑋𝑋 × 𝑍𝑍. Now, let us compare the
𝑅𝑅 ∘ 𝑆𝑆1 and 𝑅𝑅 ∘ 𝑆𝑆2 . So, if we compare this with element wise we see that all the elements
of 𝑅𝑅 ∘ 𝑆𝑆1 or either less or equal to the 𝑅𝑅 ∘ 𝑆𝑆2 . Or in other words we can say that the elements
of 𝑅𝑅 ∘ 𝑆𝑆1 are either less or equal to the corresponding elements of the 𝑅𝑅 ∘ 𝑆𝑆2 which is
clearly shown here.

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(Refer Slide Time: 34:23)

So, this way we see that when we see that all this criteria when we apply all the elements
of 𝑅𝑅 ∘ 𝑆𝑆1 ≤ 𝑅𝑅 ∘ 𝑆𝑆2 , then we can say that the monotonicity property is verified for the
composition of fuzzy relations.

In other words we can say the monotonicity property holds good for fuzzy relations and
especially with this condition that when 𝑆𝑆1 ⊂ 𝑆𝑆2 and when we take a composition either
max-min or max-product like the composition of 𝑅𝑅 and 𝑆𝑆1 and composition of 𝑅𝑅 and 𝑆𝑆2 ,
then the 𝑅𝑅 ∘ 𝑆𝑆1 ⊆ 𝑅𝑅 ∘ 𝑆𝑆2 .

(Refer Slide Time: 35:35)

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So, so far we used max-min compositions of fuzzy relations to verify the properties in all
the examples. However, on the same lines max-product composition of fuzzy relations can
also be taken.

(Refer Slide Time: 35:52)

So, this way we have seen all the four properties with respect to the composition of fuzzy
relations in today’s class and now in the next lecture we will study the fuzzy tolerance and
equivalence relations.

Thank you.

662
 Fuzzy Sets, Logic and Systems and Applications
Prof. Nishchal K. Verma
Department of Electrical Engineering
Indian Institute of Technology, Kanpur

Lecture - 38
Fuzzy Tolerance and Equivalence Relations-I

So, welcome to lecture number 38 of Fuzzy Sets, Logic and Systems and Applications. In
this lecture we will discuss Fuzzy Tolerance and Equivalence Relations.

(Refer Slide Time: 00:39)

So, as I mentioned that in this lecture, we will discuss two relations; fuzzy tolerance and
fuzzy equivalence.

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(Refer Slide Time: 00:43)

So, let us first understand what is a fuzzy tolerance relation. So, here if we have a set, a
relation set 𝑅𝑅. And if it is defined in the universe of discourse 𝑋𝑋 × 𝑋𝑋, such that the fuzzy
relation set 𝑅𝑅 ⊂ 𝑋𝑋 × 𝑋𝑋. You can see here.

Then, the fuzzy relation set 𝑅𝑅 will be a fuzzy tolerance relation if it satisfies the following
two properties. What are these two properties? The first property is the reflectivity, you
can see here. And then, the second property is symmetry. So, reflexivity property, what
does it says is that from the fuzzy relation matrix we have here the strength values or in
other words we can say the membership values of the fuzzy relation set. So, here
𝜇𝜇𝑅𝑅 (𝑥𝑥𝑖𝑖 , 𝑥𝑥𝑖𝑖 ) = 1.0.

So, what does this mean? This means that if we have a fuzzy relation set, if we represent
this fuzzy relation set in the form of a matrix as we have already seen in the previous
lectures. So, all diagonal elements here will be 1. So, in other words we say the
𝜇𝜇𝑅𝑅 (𝑥𝑥𝑖𝑖 , 𝑥𝑥𝑖𝑖 ) = 1.0, ∀𝑥𝑥𝑖𝑖 ∈ 𝑋𝑋. Is very simple to understand that the all the corresponding
elements all the elements corresponding to the same row and same column will be equal
to 1 or will be unity in other words.

So, this is called the reflexivity. Now symmetry property. So, symmetry property is also
very simple to understand here that if we have the membership values corresponding to
𝑥𝑥𝑖𝑖 , 𝑥𝑥𝑗𝑗 ; that means, 𝜇𝜇𝑅𝑅 �𝑥𝑥𝑖𝑖 , 𝑥𝑥𝑗𝑗 � and this should be equal to 𝜇𝜇𝑅𝑅 �𝑥𝑥𝑗𝑗 , 𝑥𝑥𝑖𝑖 �. So, if this satisfies,
then we can say the matrix which is corresponding to the fuzzy relation set 𝑅𝑅 is symmetric.

664
𝑅𝑅 which is which follows the symmetry property. And here of course, this is for every
𝑥𝑥𝑖𝑖 , 𝑥𝑥𝑗𝑗 belonging into 𝑋𝑋 as the universe of discourse.

And here this is needless to say that 𝜇𝜇𝑅𝑅 (𝑥𝑥𝑖𝑖 , 𝑥𝑥𝑖𝑖 ), 𝜇𝜇𝑅𝑅 �𝑥𝑥𝑖𝑖 , 𝑥𝑥𝑗𝑗 � and 𝜇𝜇𝑅𝑅 �𝑥𝑥𝑗𝑗 , 𝑥𝑥𝑖𝑖 � are the
membership values of the elements 𝑥𝑥𝑖𝑖 , 𝑥𝑥𝑖𝑖 , 𝑥𝑥𝑖𝑖 , 𝑥𝑥𝑗𝑗 and 𝑥𝑥𝑗𝑗 , 𝑥𝑥𝑖𝑖 respectively for any fuzzy
relation 𝑅𝑅. So, basically, fuzzy tolerance relation satisfies two properties. First property is
reflexivity and the second property is symmetry. So, if these two properties for any fuzzy
relation set 𝑅𝑅 is satisfied then, we can say fuzzy relation set is a fuzzy tolerance relation.

(Refer Slide Time: 05:58)

So, let us take an example here to understand the fuzzy tolerance relation better. So, here
we are taking a fuzzy relation matrix 𝑅𝑅. You can see here this is the fuzzy relation set 𝑅𝑅
and this fuzzy relation set is given in the form of a matrix. So, we see here the elements as
part of this matrix and these elements are nothing but the membership values.

And these membership values can be represented in terms of 𝜇𝜇 and its corresponding rows
and columns. For example, if we take 0.7 so this 0.7 of fuzzy relation matrix is nothing
but this is 𝜇𝜇𝑅𝑅 and since this 0.7 is corresponding to row number 𝑥𝑥2 . So, we write here 𝑥𝑥2
and then this is corresponding to the column 𝑥𝑥3 . So, we write here 𝑥𝑥3 .

So, 𝜇𝜇𝑅𝑅 (𝑥𝑥2 , 𝑥𝑥3 ) is 0.7. Similarly, if we take another element here let us say 0.6. This is in
terms of its membership value of the fuzzy relation set that we have taken is 𝜇𝜇𝑅𝑅 (𝑥𝑥3 , 𝑥𝑥1 ).
Why (𝑥𝑥3 , 𝑥𝑥1 )? Because, 0.6 is corresponding to the third row and which is designated as

665
𝑥𝑥3 . So, here we have written 𝑥𝑥3 and then separated by comma and we have written 𝑥𝑥1 . So,
𝑥𝑥1 is basically the column corresponding to which 0.6 is.

So, 0.6 can be represented here as 𝜇𝜇𝑅𝑅 . And please understand that 𝑅𝑅 here is nothing but
the name of the fuzzy relation set. So, that is why this 𝑅𝑅 is and so 0.6 here is nothing but
this is corresponding 𝜇𝜇𝑅𝑅 (𝑥𝑥3 , 𝑥𝑥1 ). So, this way we understand that all the elements of the
fuzzy relation matrix can be understood.

(Refer Slide Time: 09:06)

And now here we have all the elements of the fuzzy relation matrix is represented in terms
of its mu membership values. So, you can see here that as to how all these elements are
with respect to its rows and columns and these values are the membership values of fuzzy
relation set 𝑅𝑅.

666
(Refer Slide Time: 09:38)

So, once this is clear, now since we are interested in checking that the given fuzzy relation
set 𝑅𝑅 is a fuzzy tolerance relation or not. So, in order to do that we need to check whether
the two properties that was just mentioned or satisfied or not. So, these two properties are
the reflexivity property and then another property here is the symmetry property.

So, let us now go through the first property and check whether the given fuzzy relation
matrix 𝑅𝑅 satisfies reflexivity property or not. And we know that we need to check this
condition. If this condition is there then we can say that a given fuzzy relation set 𝑅𝑅
satisfies reflexivity property. So, as I have already mentioned that reflexivity property is
satisfied when its diagonal elements are 1. Why diagonal elements are 1? Because you
know in therefore, diagonal elements we have this corresponding elements are having
same row and same column.

So for example, here if we take this element we see that we have this as this element is
nothing but this is 𝜇𝜇𝑅𝑅 (𝑥𝑥2 , 𝑥𝑥2 ) and this is 1. Similarly, we write all these elements where
these elements are corresponding to the same row and same column. So, we see that
𝜇𝜇𝑅𝑅 (𝑥𝑥1 , 𝑥𝑥1 ) is 1, 𝜇𝜇𝑅𝑅 (𝑥𝑥2 , 𝑥𝑥2 ) is 1, 𝜇𝜇𝑅𝑅 (𝑥𝑥3 , 𝑥𝑥3 ) is 1, 𝜇𝜇𝑅𝑅 (𝑥𝑥4 , 𝑥𝑥4 ) is 1 which we can see very
clearly.

667
(Refer Slide Time: 12:09)

So, finally, we see that the diagonal elements here. All these diagonal elements are 1. Just
by looking at the fuzzy relation matrix that is given to us, we can just check the diagonal
elements and if all the diagonal elements are 1 then we can say the reflexivity property for
the fuzzy relation set 𝑅𝑅 is satisfied. So, now we can clearly see that 𝑅𝑅 satisfies the
reflexivity property. So, the first property is satisfied.

(Refer Slide Time: 13:02)

668
Now, let us look for the second property, that is symmetry property. So, in symmetry
property we have 𝜇𝜇𝑅𝑅 �𝑥𝑥𝑖𝑖 , 𝑥𝑥𝑗𝑗 � = 𝜇𝜇𝑅𝑅 (𝑥𝑥𝑗𝑗 , 𝑥𝑥𝑖𝑖 ), which is here. And again this is for every 𝑥𝑥𝑖𝑖 , 𝑥𝑥𝑗𝑗
belonging into 𝑋𝑋.

So, we have the fuzzy relation set 𝑅𝑅 the same fuzzy relation set 𝑅𝑅 that is given here is this
and let us take this and verify this. So, when we check for this condition we find that
𝜇𝜇𝑅𝑅 (𝑥𝑥1 , 𝑥𝑥2 ) = 𝜇𝜇𝑅𝑅 (𝑥𝑥2 , 𝑥𝑥1 ) and here we see that both of these elements is equal to 0.9.

Similarly, when we see for all combinations we find that the symmetry property is very
well satisfied here. So, what does this mean? This means that any element corresponding
to a particular row and column is same if the row and columns are interchanged. So, here
in this case if fuzzy relations set 𝑅𝑅 that has been given satisfies the symmetry property as
well.

So, when both of these properties are satisfied we can clearly say that the fuzzy relation
set 𝑅𝑅 is a fuzzy tolerance relation please understand that a fuzzy relation set 𝑅𝑅 can only be
called as fuzzy tolerance relation when both of these properties are satisfied. So, here in
this case in the example that we have taken. We have taken fuzzy relation set 𝑅𝑅 and for
this fuzzy relation set 𝑅𝑅, both of these properties the reflexivity symmetry are satisfied.
So, this 𝑅𝑅 is qualified to be called as fuzzy tolerance relation.

(Refer Slide Time: 15:48)

669
Now, let us take another example here where we check whether the given fuzzy relation
set 𝑅𝑅 is a fuzzy tolerance relation or not. So, on the same lines we move forward and see
that whether the reflexivity property and symmetry property, both are satisfied for this
fuzzy relation set are not. So, when we try to see for reflexivity property, we apply this
criteria. So, as I mentioned that we can quickly just by looking at the fuzzy relation matrix
we can quickly comment on this property. So, we see that all the diagonal elements are not
1.

So, this is not equal to 1. Only out of these three diagonal elements only 1 diagonal element
is 1, but rest two elements are not 1. So, the condition for the reflexivity property is that
all these diagonal elements should be equal to 1. So, just by looking at it we can say that
the reflexivity property is not satisfied.

So, it is mentioned here you can see that 𝜇𝜇𝑅𝑅 (𝑥𝑥1 , 𝑥𝑥1 ) is equal to 0.8 which is not equal to 1.
𝜇𝜇𝑅𝑅 (𝑥𝑥2 , 𝑥𝑥2 ) is 1, 𝜇𝜇𝑅𝑅 (𝑥𝑥3 , 𝑥𝑥3 ) is 0.5 which is not equal to 1. So, this way we can say the fuzzy
relation set that has been given to us is not satisfying the reflexivity property. Now let us
go ahead and check for the symmetry property. Here I would like to tell you that since we
are finding whether the fuzzy tolerance relation whether these R that is that has been given
to us is a fuzzy tolerance relation or not.

So, please understand that since the reflectivity property is not satisfied, so we need not go
ahead and check for the symmetry property because there is no use of it. It may be
symmetric, but if it is not reflexive then we can say the given for the relation set 𝑅𝑅 is not
a fuzzy tolerance relation. So, the condition is that both the properties needs to be satisfied.
So, here since the reflexivity property is not satisfied. So, which we need not go ahead and
check for the symmetry property because just by checking any one of the properties we
can say that whether we should go forward or not.

So, since reflexivity property is not satisfied we can clearly say that the given fuzzy
relations set is not a fuzzy tolerance relation.

670
(Refer Slide Time: 19:23)

Now, let us take another example here of fuzzy tolerance relation here also let us check
for the given fuzzy relation matrix fuzzy relation set 𝑅𝑅 and let us check for this fuzzy
relation matrix R whether this R is a fuzzy tolerance relation or not.

So, the fuzzy relation matrix that is given to us is here and you can see. And if we are
interested in checking the fuzzy tolerance relationship or whether 𝑅𝑅 is a fuzzy tolerance
relation or not. Again we have to proceed on the similar lines and we need to check for the
two properties first one is reflexivity and the second one is the symmetry.

(Refer Slide Time: 20:33)

671
So, when we see here the given fuzzy relation matrix. So, just by looking at the diagonal
elements here. We can see that all the diagonal elements are 1. So, when as I have already
mentioned when all the diagonal elements are 1, then it is obvious that 𝜇𝜇𝑅𝑅 (𝑥𝑥𝑖𝑖 , 𝑥𝑥𝑖𝑖 ) is 1. And
if this is the case then we can say that the reflexivity property is satisfied.

So, from the fuzzy relation matrix we can write all mu values in 𝜇𝜇(𝑥𝑥𝑖𝑖 , 𝑥𝑥𝑗𝑗 ) format and then
we see that 𝜇𝜇𝑅𝑅 (𝑥𝑥𝑖𝑖 , 𝑥𝑥𝑖𝑖 ) is 1. That means 𝜇𝜇𝑅𝑅 (𝑥𝑥1 , 𝑥𝑥1 ), 1, 𝜇𝜇𝑅𝑅 (𝑥𝑥2 , 𝑥𝑥2 ), 1, 𝜇𝜇𝑅𝑅 (𝑥𝑥3 , 𝑥𝑥3 ), 1,
𝜇𝜇𝑅𝑅 (𝑥𝑥4 , 𝑥𝑥4 ), 1, 𝜇𝜇𝑅𝑅 (𝑥𝑥5 , 𝑥𝑥5 ), 1. So, this way we see that the reflexivity property is satisfied.

Now let us quickly go ahead and check for the symmetry property. And for symmetry
property also if we see that 𝜇𝜇𝑅𝑅 (𝑥𝑥1 , 𝑥𝑥2 ) = 𝜇𝜇𝑅𝑅 (𝑥𝑥2 , 𝑥𝑥1 ) = 0.1. 𝜇𝜇𝑅𝑅 (𝑥𝑥1 , 𝑥𝑥3 ) = 𝜇𝜇𝑅𝑅 (𝑥𝑥3 , 𝑥𝑥1 ) =
0.8. 𝜇𝜇𝑅𝑅 (𝑥𝑥1 , 𝑥𝑥4 ) = 𝜇𝜇𝑅𝑅 (𝑥𝑥4 , 𝑥𝑥1 ) = 0.2 and 𝜇𝜇𝑅𝑅 (𝑥𝑥1 , 𝑥𝑥5 ) = 𝜇𝜇𝑅𝑅 (𝑥𝑥5 , 𝑥𝑥1 ) = 0.3. So, similarly, all
other values can also be written here. So, what does this mean here is that 𝜇𝜇𝑅𝑅 �𝑥𝑥𝑖𝑖 , 𝑥𝑥𝑗𝑗 � =
𝜇𝜇𝑅𝑅 (𝑥𝑥𝑗𝑗 , 𝑥𝑥𝑖𝑖 ). For this fuzzy relation matrix, for this fuzzy relation set which has been given
to us.

So, when this is satisfied we can say that the fuzzy relation set 𝑅𝑅 satisfies the symmetry
property. So, when this property is also satisfied, we can say the reflectivity and symmetry
both of these properties are satisfied for the given fuzzy relation set. So, we can say that 𝑅𝑅
is a fuzzy tolerance relation.

So, this way we can comment on 𝑅𝑅 now and the given fuzzy relations set is fuzzy tolerance
relation. Please note that it may also be possible that any given fuzzy relation set may not
be a fuzzy tolerance relation always.

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(Refer Slide Time: 24:34)

But in this case here, since both the properties for the 𝑅𝑅 given is satisfied. We can say that
the 𝑅𝑅 is a fuzzy tolerance relation.

(Refer Slide Time: 24:53)

So, with this I would like to stop here and in the next lecture, we will discuss the fuzzy
equivalence relations and its properties.

Thank you.

673
 Fuzzy Sets, Logic and Systems and Applications
Prof. Nishchal K. Verma
Department of Electrical Engineering
Indian Institute of Technology, Kanpur

Lecture – 39
Fuzzy Tolerance and Equivalence Relations- II

So, welcome to lecture number 39 of Fuzzy Sets, Logic and Systems and Applications.
And in this lecture we will discuss Fuzzy Tolerance and Equivalence Relations and this
lecture is in continuation to our previous discussions on this topic.

(Refer Slide Time: 00:32)

Now, let us move ahead and talk about fuzzy equivalence relation. So, like fuzzy tolerance
relation we have fuzzy equivalence relation. So, here we have gone one step ahead. So, it
is quite interesting to understand here that any fuzzy tolerance relation can be a fuzzy
equivalence relation.

So, if any fuzzy relation set which is not a fuzzy relation fuzzy tolerance relation cannot
be a fuzzy equivalence relation because for qualifying for any 𝑅𝑅 to qualify to be called as
a fuzzy equivalence relation. First this 𝑅𝑅 has to be a fuzzy tolerance relation. It is because
for fuzzy equivalence relation we have three properties that needs to be satisfied. So, apart
from the reflexivity, symmetry we have an additional property which is transitivity and

674
these three properties need to be satisfied before any fuzzy relations set 𝑅𝑅 can be called as
fuzzy equivalence relation.

So, since we have already discussed in detail the reflexivity, the symmetry. Now, we see
what is transitivity the third property, so transitivity is basically it follows this condition
which I am just going to explain. So, from the fuzzy relation matrix if we have the
membership values corresponding to 𝜇𝜇𝑅𝑅 (𝑥𝑥𝑖𝑖 , 𝑥𝑥𝑗𝑗 ) let us say which is equal to 𝜆𝜆1 , and
𝜇𝜇𝑅𝑅 (𝑥𝑥𝑗𝑗 , 𝑥𝑥𝑘𝑘 ) which is equal to 𝜆𝜆2 , and 𝜇𝜇𝑅𝑅 (𝑥𝑥𝑖𝑖 , 𝑥𝑥𝑘𝑘 ) = 𝜆𝜆. So, if this is the case, then we need to
check whether the 𝜆𝜆 ≥ min[𝜆𝜆1 , 𝜆𝜆2 ].

So, for transitivity condition to be satisfied this 𝜆𝜆 ≥ min[𝜆𝜆1 , 𝜆𝜆2 ] which is here. So, if this
is the case then we can say that the transitivity condition is also satisfied. So, this way if
all the three conditions as mentioned here. The first condition is reflexivity, second
condition is symmetry, third condition is transitivity. If all these three conditions are
satisfied for any fuzzy relation set we can say this fuzzy relation set is a fuzzy equivalence
relation set.

(Refer Slide Time: 04:14)

So, for this also let us take an example here to understand this concept better. Here we
have taken a fuzzy relation set which is represented in the form of matrix here. And as we
have already seen that all its elements are nothing but its membership values corresponding
to its rows and columns. So, if we need to check whether this fuzzy relation is qualified to
be called as fuzzy equivalence relation or not, we need to check for the three conditions.

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First condition is reflexivity, second condition is symmetry and the third condition is the
transitivity.

So, let us quickly go ahead and check for that. So, when we do that when in order to check
for the reflexivity first. So, as I have already mentioned that all its diagonal elements
should be unity, should be equal to 1. So, I am quickly going through the diagonal elements
here. And we see that all its diagonal elements, all the diagonal elements of the fuzzy
relation set are unity or equal to 1 or another words we can say right here all the diagonal
elements are 1. So, if this is the case we can quickly comment on the reflexivity property
because here 𝜇𝜇(𝑥𝑥𝑖𝑖 , 𝑥𝑥𝑖𝑖 ) = 1, ∀𝑥𝑥𝑖𝑖 ∈ 𝑋𝑋.

So, here since this is satisfied because diagonal elements are nothing but the diagonal
elements are the elements are corresponding to the same row and same column. So, we
can say that 𝜇𝜇(𝑥𝑥𝑖𝑖 , 𝑥𝑥𝑖𝑖 ) = 1. And since this is the case all the diagonal elements are 1, so we
can quickly say that the reflexivity property is satisfied. So, first property is satisfied first
property that is reflexivity is satisfied. Now, let us check for the symmetry property. So,
when we see that here if we take up this element, what is this element? This element is
𝜇𝜇(𝑥𝑥4 , 𝑥𝑥1 ) and this is 0.5.

Now, let us interchange the rows and column here. And if you change the rows and
columns, we see that 𝜇𝜇(𝑥𝑥1 , 𝑥𝑥4 ). So, 𝜇𝜇(𝑥𝑥1 , 𝑥𝑥4 ) is here, 𝜇𝜇(𝑥𝑥1 , 𝑥𝑥4 ) and we see that this is also
0.5. So, this way for this element for 𝑥𝑥4 , 𝑥𝑥1 and 𝑥𝑥1 , 𝑥𝑥4 both the elements are same. And this
has to be checked for all the elements in the fuzzy religion set. So, when we do that we see
that all the elements are satisfying this criteria.

So, we can quickly say that the symmetry property is also satisfied. So, this way for the
given fuzzy relation matrix, for the given fuzzy relation set 𝑅𝑅 the reflectivity and symmetry
property both are satisfied. So, here we can also make a comment that 𝑅𝑅 is a fuzzy
tolerance, fuzzy tolerance relation because both the properties are satisfied, reflexivity and
symmetry. But here since we are interested in finding whether fuzzy relation set 𝑅𝑅 is a
fuzzy equivalence relation or not.

So, this is quite interesting to note that a fuzzy relation has to be first a fuzzy tolerance
relation before it can qualify to become fuzzy equivalence relation. So, this step is must.
So, since the given fuzzy relation set capital 𝑅𝑅 is a fuzzy tolerance relation.

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(Refer Slide Time: 10:27)

Now, we can move forward and check for the third property that is transitivity property.
And transitivity property says that the 𝜇𝜇𝑅𝑅 (𝑥𝑥𝑖𝑖 , 𝑥𝑥𝑗𝑗 ) if it is some value that is 𝜆𝜆1 , and
𝜇𝜇𝑅𝑅 (𝑥𝑥𝑗𝑗 , 𝑥𝑥𝑘𝑘 ) here, it is also some value say 𝜆𝜆2 . Then 𝜇𝜇𝑅𝑅 (𝑥𝑥𝑖𝑖 , 𝑥𝑥𝑘𝑘 ) if it is 𝜆𝜆, then this 𝜆𝜆 ≥
min[𝜆𝜆1 , 𝜆𝜆2 ]. So, now, let us quickly check that and in order to do that we have here let’s
say 𝜇𝜇𝑅𝑅 (𝑥𝑥1 , 𝑥𝑥2 ) and this we have found as 0.8 from this given fuzzy relation matrix, this is
here, this is the value 𝜇𝜇𝑅𝑅 (𝑥𝑥1 , 𝑥𝑥2 ).

And now when we find 𝜇𝜇𝑅𝑅 (𝑥𝑥2 , 𝑥𝑥3 ) let us say this value 𝜇𝜇𝑅𝑅 (𝑥𝑥2 , 𝑥𝑥3 ). So, 𝑥𝑥2 is this and then
𝑥𝑥3 is this. So, 𝜇𝜇𝑅𝑅 (𝑥𝑥2 , 𝑥𝑥3 ) this is nothing, but this value. Now, as per the criteria that has
been given for transitivity property we need to take 𝜇𝜇𝑅𝑅 (𝑥𝑥1 , 𝑥𝑥3 ) because you see 𝑥𝑥1 is
coming from here, 𝑥𝑥1 is coming from here and 𝑥𝑥3 is coming from here.

So, this 𝜇𝜇𝑅𝑅 (𝑥𝑥1 , 𝑥𝑥3 ) if we check we are finding you see here 𝜇𝜇𝑅𝑅 (𝑥𝑥1 , 𝑥𝑥3 ) this is 𝜇𝜇𝑅𝑅 (𝑥𝑥1 , 𝑥𝑥3 ).
So, we are finding here this is 0.4, and this 0.4 is when we take min of the previous values
the 0.4, 0.8 here. So, we see that this is equal to the min of 0.8 and 0.4, so this means that
this condition is satisfied. Similarly, we check for 𝜇𝜇𝑅𝑅 (𝑥𝑥2 , 𝑥𝑥4 ) and 𝜇𝜇𝑅𝑅 (𝑥𝑥4 , 𝑥𝑥5 ). So, we see
that this criteria, the transitivity criteria is satisfied.

Similarly, for all the other elements we have checked you can also verify this. And we find
here 𝜇𝜇𝑅𝑅 (𝑥𝑥𝑖𝑖 , 𝑥𝑥𝑗𝑗 ) which is 𝜆𝜆1 and 𝜇𝜇𝑅𝑅 (𝑥𝑥𝑗𝑗 , 𝑥𝑥𝑘𝑘 ) which is 𝜆𝜆2 . And with this if we find 𝜇𝜇𝑅𝑅 (𝑥𝑥𝑖𝑖 , 𝑥𝑥𝑘𝑘 ).
Then in that case this value the 𝜆𝜆 ≥ min[𝜆𝜆1 , 𝜆𝜆2 ], for this given fuzzy relation matrix. So,

677
since we have checked for all the elements the transitivity condition, the transitivity
property is satisfied we can say here that the transitivity property for the fuzzy relation set
is satisfied here.

(Refer Slide Time: 14:42)

So, this way the fuzzy relation that has been given fuzzy relation 𝑅𝑅 that has been given is
a fuzzy equivalence relation. And we all know why we are making this comment because
we have already checked all the three properties. The first one is the reflexivity, second
one is the symmetry and the third one is the transitivity. All these three conditions are all
these three properties are satisfied. So, the given 𝑅𝑅 is qualified to be called as a fuzzy
equivalence relation.

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(Refer Slide Time: 15:36)

Now, let us take another example and see whether in this example the fuzzy relation matrix
R is a fuzzy equivalence relation are not.

So, this is the fuzzy relation 𝑅𝑅 that has been given and again we have to check for the three
properties all the three properties must be holding good before we can say that 𝑅𝑅 is a fuzzy
equivalence relation. So, let us quickly go through the reflexivity property, symmetry
property, transitivity property for the given fuzzy relations set and comment on it.

(Refer Slide Time: 16:26)

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So, we see that the given fuzzy relation set 𝑅𝑅 has its all its diagonal elements as unity. So,
all the diagonal elements are unity.

So, this way we can say that 𝜇𝜇𝑅𝑅 (𝑥𝑥𝑖𝑖 , 𝑥𝑥𝑖𝑖 ) is 1 and you can see here all the elements all the x
𝜇𝜇𝑅𝑅 (𝑥𝑥1 , 𝑥𝑥1 ), 𝜇𝜇𝑅𝑅 (𝑥𝑥2 , 𝑥𝑥2 ), 𝜇𝜇𝑅𝑅 (𝑥𝑥3 , 𝑥𝑥3 ), 𝜇𝜇𝑅𝑅 (𝑥𝑥4 , 𝑥𝑥4 ), 𝜇𝜇𝑅𝑅 (𝑥𝑥5 , 𝑥𝑥5 ) all the elements are 1. So, we
can say that 𝑅𝑅 is satisfying the reflexivity property. So, the first property is satisfied.

(Refer Slide Time: 17:31)

Now, let us check for the symmetry property. Symmetry says that 𝜇𝜇𝑅𝑅 �𝑥𝑥𝑖𝑖 , 𝑥𝑥𝑗𝑗 � =
𝜇𝜇𝑅𝑅 �𝑥𝑥𝑗𝑗 , 𝑥𝑥𝑖𝑖 �, ∀𝑥𝑥𝑖𝑖 , 𝑥𝑥𝑗𝑗 ∈ 𝑋𝑋. So, when we have checked for this here we see that this condition
is also satisfied for fuzzy relation set that has given to us. And this way we can say that
the 𝑅𝑅 is satisfying the symmetry property.

So, now since we have checked for so far we have checked for two properties. First one is
reflexivity and the second one is symmetry. So, at this juncture we can say that fuzzy
relation matrix 𝑅𝑅 is a fuzzy tolerance relation. So, here I would like to make a comment
for fuzzy relation 𝑅𝑅. So, I can write here that the given fuzzy relation set 𝑅𝑅 is a fuzzy
tolerance relation. So, up to this is very clear. Now, this 𝑅𝑅 before it can be called as fuzzy
equivalence relation should satisfy the third property, that is a transitivity.

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(Refer Slide Time: 19:36)

So, when we take the 𝑅𝑅 here and when we check for the transitivity condition. So,
transitivity condition says that 𝜇𝜇𝑅𝑅 �𝑥𝑥𝑖𝑖 , 𝑥𝑥𝑗𝑗 � is equal to 𝜆𝜆1 and 𝜇𝜇𝑅𝑅 (𝑥𝑥𝑗𝑗 , 𝑥𝑥𝑘𝑘 ) is equal to 𝜆𝜆2 . Then
the 𝜇𝜇𝑅𝑅 (𝑥𝑥𝑖𝑖 , 𝑥𝑥𝑘𝑘 ) should be the value of this should be say 𝜆𝜆 ≥ min[𝜆𝜆1 , 𝜆𝜆2 ].

So, when we take 𝜇𝜇𝑅𝑅 (𝑥𝑥1 , 𝑥𝑥2 ) which is nothing but this element you can see this is the
element. And this is nothing, but 0.8 and when we take 𝜇𝜇𝑅𝑅 (𝑥𝑥2 , 𝑥𝑥5 ) is this, this is 𝜇𝜇𝑅𝑅 (𝑥𝑥2 , 𝑥𝑥5 ).
So, this means we have 𝜆𝜆1 to 0.8. In this case and 𝜆𝜆2 is equal to 0.9. Now, if we take 𝜇𝜇𝑅𝑅 of
𝑥𝑥1 which is coming from here and 𝑥𝑥5 which is coming from here.

So, 𝜇𝜇𝑅𝑅 of 𝑥𝑥1 and 𝑥𝑥5 if we see 𝑥𝑥1 , 𝑥𝑥5 which is nothing but 0.2, so I can just mention this it
is here. So, when we see this this is coming out to be 0.2 and this is let’s say is 𝜆𝜆, so 𝜆𝜆 is
0.2. So, now, let us put the condition of transitivity. So, 𝜆𝜆 ≥ min[𝜆𝜆1 , 𝜆𝜆2 ]. So, we have all
the values. So, let us see whether this holds the condition. So, if we take the min here min
of 0.8, 0.9 and then here we have lambda is 0.2. So, this is not holding means this is not
true.

So, when this is not true then we can stop here and we can say that the transitivity property
is not satisfied this for this fuzzy relation set that is given to us that is 𝑅𝑅. So, 𝑅𝑅 does not, I
can write here that 𝑅𝑅 does not satisfy the transitivity property. So, now, in this case when
we have the checked that the third condition is not satisfied we can quickly say that
although the given fuzzy relation set is a fuzzy tolerance relation, but it is not fuzzy

681
equivalence relation. So, we can now, quickly say that the given fuzzy relation set is not a
fuzzy equivalence relation.

(Refer Slide Time: 23:52)

So, with this I would like to stop here. And in the next lecture we will discuss that as to
how we can fuzzy tolerance relation transform or reformed into a fuzzy equivalence
relation and we will discuss other properties as well.

Thank you.

682
 Fuzzy Sets, Logic and Systems and Applications
Prof. Nishchal K. Verma
Department of Electrical Engineering
Indian Institute of Technology, Kanpur

Lecture – 40
Fuzzy Tolerance and Equivalence Relations –III

So, welcome to lecture number 40 of Fuzzy sets, Logic and Systems and Applications, and
this lecture is in continuation to our previous lectures and here we will discuss a Fuzzy
Tolerance and Equivalence Relations.

(Refer Slide Time: 00:34)

So, this way we have understood as to, when we can say that particular fuzzy relation set
is a fuzzy equivalence relation are not.

One thing we have understood that a fuzzy equivalence relation is always a fuzzy tolerance
relation that is an important outcome of this discussion. And there are fuzzy relation sets
which many a times they are not fuzzy equivalence relations, but we can make them fuzzy
equivalence relation.

So, here is a very important concept that I am going to mention is that a fuzzy relation set
which is a fuzzy tolerance relation let us say fuzzy tolerance relation 𝑅𝑅, if it is defined in
this space 𝑋𝑋 × 𝑋𝑋 so; that means, these 𝑅𝑅 is already a reflexive and symmetric, means this

683
fuzzy relation set 𝑅𝑅 is satisfying the reflectivity and symmetry property. So, if we have a
fuzzy tolerance relation and this fuzzy tolerance relation if it is not fuzzy equivalence
relation it can be converted into fuzzy equivalence relation by at most n minus 1
compositions.

So, this is very important concept that we need to know here. So, if we can make a
composition, at most 𝑛𝑛 − 1 composition with itself. So, within this any fuzzy tolerance
relation 𝑅𝑅 can be converted into a fuzzy is equivalence relation. So, as I mentioned a fuzzy
tolerance relation 𝑅𝑅 which is defined in the space 𝑋𝑋 × 𝑋𝑋 that has already satisfied the
properties of reflexivity and symmetry can be reformed into a fuzzy equivalence relation
by at most 𝑛𝑛 − 1 composition with itself. Where n is the cardinal number of the set defining
𝑅𝑅. So, this can be written as this expression.

So, 𝑅𝑅𝑛𝑛−1 this shows that we are making 𝑛𝑛 − 1 compositions. So, this is the maximum
number of composition that it can go through and within this the fuzzy relation set which
is not a fuzzy equivalence relation can be transformed into fuzzy equivalence relation.

So, I would like to just amend here that a fuzzy tolerance relation 𝑅𝑅 only can get
transformed into fuzzy equivalence relation in this process. And the process is at most n
minus 1 compositions with itself, where n is the cardinal number of the set defining 𝑅𝑅.

(Refer Slide Time: 04:30)

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So, let us take an example here to understand this concept also. So, here we have taken
fuzzy relation set in form of matrix we can call this as the fuzzy relation matrix 𝑅𝑅. And if
we look at the fuzzy relation matrix we can quickly comment on the reflexivity property,
all the diagonal elements are 1. And not only this the symmetry property is also satisfied
because we have already done this in the previous example. Example number 5 where we
had a fuzzy relation 𝑅𝑅 which is here and we have already proved that 𝑅𝑅 is a fuzzy tolerance
relation.

So, this is a fuzzy tolerance relation. So, 𝑅𝑅 is a fuzzy tolerance relation, why because 𝑅𝑅
satisfies the reflexivity property and symmetry property, now if we are interested in
checking whether this fuzzy tolerance relation is qualified to be called as fuzzy equivalence
relation or not. So, the third property that is transitivity property it needs to be satisfied.

Now, in the example number 5 we have checked this that the transitivity property is not
satisfied. So, this fuzzy relation set 𝑅𝑅 is not a fuzzy equivalence relation. So, here I am just
mentioning that I am just writing that the what are the properties that are satisfied. So,
reflexivity, then symmetry, then transitivity.

So, transitivity criteria transitivity property is not satisfied for this 𝑅𝑅, however the
reflexivity and symmetry both the properties are satisfied. So, that is why the given fuzzy
tolerance relation are given fuzzy relation matrix 𝑅𝑅 is not qualified to be called as fuzzy
equivalence relation. So, now, the question is how to make this a fuzzy equivalence
relation by having the compositions as we have just discussed. So, let us proceed for that.

So we apply now the composition of 𝑅𝑅 on 𝑅𝑅; so that is 𝑅𝑅2 . So, 𝑅𝑅2 is nothing, but the 𝑅𝑅 ∘
𝑅𝑅. So, we have the fuzzy tolerance relation. Now, 𝑅𝑅 and we are now having the 𝑅𝑅 ∘ 𝑅𝑅 and
let us see what happens.

685
(Refer Slide Time: 08:20)

So, we have the 𝑅𝑅 that is given again and the same 𝑅𝑅 is here. And when we take the
composition of these two and here we have ∘ sign, ∘ is for the composition and here we
are taking the max-min composition. So, when we do that we see that this is the outcome.

So, after taking the max min composition of 𝑅𝑅 ∘ 𝑅𝑅, we get a new fuzzy relation which is
represented by 𝑅𝑅2 . So, this is the first composition, this is you know the first stage of the
composition. So, the fuzzy relation matrix if we see if we look at we see that we have
𝜇𝜇𝑅𝑅2 (𝑥𝑥1 , 𝑥𝑥2 ) which is equal to 0.8 and 𝜇𝜇𝑅𝑅2 (𝑥𝑥2 , 𝑥𝑥4 ) is 0.5.

And we see that here also the transitivity condition is not satisfied because the 𝜇𝜇𝑅𝑅2 (𝑥𝑥1 , 𝑥𝑥4 )
equal to 0.2 which is 𝜆𝜆. And this is not either equal to or greater than the mean of the 0.8
and 0.5. So, this way we can clearly say that the transitivity condition is still not satisfied.

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(Refer Slide Time: 10:01)

Now, let us go one step further and let us have one more composition. So, we already have
𝑅𝑅2 and now let us have a composition of 𝑅𝑅2 ∘ 𝑅𝑅 which is 𝑅𝑅3 here. So, when we have the
max-min composition of 𝑅𝑅2 on 𝑅𝑅 we have this as the outcome, we have a new fuzzy
relation here that is 𝑅𝑅3 .

New fuzzy relation matrix and of course, this fuzzy relation matrix is a fuzzy tolerance
relation because we have already seen that this satisfies the reflexivity and symmetry. We
can again once again we can check this and here also the condition of reflexivity and
symmetry both are satisfied.

Now, when we check for transitivity we see that a 𝜇𝜇𝑅𝑅2 here that is 𝜇𝜇𝑅𝑅3 (𝑥𝑥1 , 𝑥𝑥2 ) and
𝜇𝜇𝑅𝑅3 (𝑥𝑥2 , 𝑥𝑥4 ) values are 0.8 and 0.5. So, 𝜇𝜇𝑅𝑅3 (𝑥𝑥1 , 𝑥𝑥2 ); 𝑥𝑥1 , 𝑥𝑥2 is this, and 𝜇𝜇𝑅𝑅3 (𝑥𝑥2 , 𝑥𝑥4 )value is
0.5 here.

Now, when we check for 𝜇𝜇𝑅𝑅3 (𝑥𝑥1 , 𝑥𝑥4 ), which is 0.5 again. And if we see that this 0.5 is
either greater than or equal to the mean of these two the 0.8 and 0.5. So, here for this case
the transitivity condition is satisfied, but we have to go further and we have to check this
condition for all its elements.

So, when we have checked for all its elements all the membership values of the fuzzy
tolerance relation 𝑅𝑅3 . Then we see that all the elements are satisfying the transitivity

687
condition and this way we can comment here that the 𝑅𝑅3 is satisfies the transitivity
condition.

So, now when the 𝑅𝑅3 is satisfying the transitivity condition and we have already seen that
𝑅𝑅3 is reflexive and symmetric. So, this means all the three conditions of fuzzy equivalence
relations satisfied. So, we can now say that the fuzzy relation matrix 𝑅𝑅 is converted into
𝑅𝑅3 . And 𝑅𝑅3 is a fuzzy equivalence relation.

So, what does this mean? This means that if we have any fuzzy tolerance relation which is
not a fuzzy equivalence relation. So, with this with the help of this fuzzy tolerance relation
we can use the fuzzy tolerance relation and we can convert this into fuzzy equivalence
relations by taking the suitable compositions.

(Refer Slide Time: 14:09)

Now, the anti-reflexivity and anti-symmetry these are two other properties, we can just
understand. So, at this stage the fuzzy relations for fuzzy relations we have anti-reflexivity
anti-symmetry and these are nothing but the antonyms of the properties that we have
already discussed. So, anti-reflexivity is the property where the reflexivity is not satisfied;
reflexivity condition is not satisfied.

688
(Refer Slide Time: 14:50)

And similarly anti-symmetry is the property where the symmetry condition is not satisfied.
So, let us now formally understand what is anti-reflexivity. So, if we have any fuzzy
relation 𝑅𝑅 which is defined in the universe of discourse 𝑋𝑋 × 𝑋𝑋 such that 𝑅𝑅 ⊂ 𝑋𝑋 × 𝑋𝑋. Then
𝑅𝑅 will satisfy the anti-reflexivity property.

If 𝜇𝜇𝑅𝑅 (𝑥𝑥𝑖𝑖 , 𝑥𝑥𝑖𝑖 ) is equal to 0 instead of 1 in the case of reflexivity. So, here 𝜇𝜇𝑅𝑅 (𝑥𝑥𝑖𝑖 , 𝑥𝑥𝑖𝑖 ) means,
the diagonal elements so all the diagonal elements of the fuzzy relation set need to be 0.
So, this condition if this is the case then we can say that the fuzzy relation has anti-
reflexivity condition.

689
(Refer Slide Time: 15:47)

And similarly, now have an example to understand this better. So, if we have a fuzzy
relation matrix 𝑅𝑅 as shown here. And we see that the all the diagonal elements of it are
zero. So, then in that case we can say that the anti-reflexivity criteria is satisfied. So, we
can say that the fuzzy relation 𝑅𝑅 is anti-reflexive.

(Refer Slide Time: 16:26)

Now, anti-symmetry; so, for any fuzzy relation set 𝑅𝑅 if we have this condition satisfied
which is 𝜇𝜇𝑅𝑅 (𝑥𝑥𝑖𝑖 , 𝑥𝑥𝑗𝑗 ) > 0, then 𝜇𝜇𝑅𝑅 �𝑥𝑥𝑗𝑗 , 𝑥𝑥𝑖𝑖 � = 0, ∀𝑥𝑥𝑖𝑖 , 𝑥𝑥𝑗𝑗 ∈ 𝑋𝑋, 𝑥𝑥𝑖𝑖 ≠ 𝑥𝑥𝑗𝑗 .

690
(Refer Slide Time: 16:55)

So, let us now understand this property the anti-symmetry property by taking an example.
So, we have any fuzzy relation set 𝑅𝑅 which is represented in the form of matrix here.

(Refer Slide Time: 17:12)

So, we see that the condition that is necessary for anti-symmetry is satisfied here. So, this
means that if we have 𝜇𝜇𝑅𝑅 (𝑥𝑥1 , 𝑥𝑥2 ). So, 𝜇𝜇𝑅𝑅 (𝑥𝑥1 , 𝑥𝑥2 ) is 0.9 here. And this is greater than 0 of
course, and then 𝜇𝜇𝑅𝑅 when we interchange the row and columns we see that 𝜇𝜇𝑅𝑅 (𝑥𝑥2 , 𝑥𝑥1 ) is
equal to here 0.

691
So, this way we can say that the anti-symmetry for this element is satisfied. Similarly,
when we check or for all the elements we see that the 𝑅𝑅 satisfies the anti-symmetry
property. And this way we can say that the fuzzy relation 𝑅𝑅 that has been given satisfies
the anti-symmetry property.

So, this way we have understood that a fuzzy relation can go through multiple tests and
based on that we can comment on its whether it is the fuzzy tolerance relation or a fuzzy
equivalence relation. And if a fuzzy tolerance relation is not a fuzzy equivalence relation
let’s say, then we can make a fuzzy equivalence relation by using a fuzzy tolerance relation
with proper composition of it.

(Refer Slide Time: 18:59)

So, this way we will finish the lecture here. And in the next lecture we will study the
Linguistic Hedges ahead.

Thank you.

692
 Fuzzy Sets, Logic and Systems and Applications
Prof. Nishchal K. Verma
Department of Electrical Engineering
Indian Institute of Technology, Kanpur

Lecture - 41
Linguistic Hedges

(Refer Slide Time: 00:19)

Welcome to lecture number 41 of Fuzzy Sets Logic and Systems and Applications. In this
lecture, we will discuss Linguistic Hedges. So, before we finally discuss linguistic hedges
let us go through the term linguistic variables.

693
(Refer Slide Time: 00:42)

So, a linguistic variable is characterised by a quintuple and this quintuple basically has
five variables and here these five variables are 𝑥, 𝑇(𝑥). So, 𝑥 is generic variable and 𝑇(𝑥)
here is the term set, 𝑋 is the universe of discourse 𝐺 here is this syntactic rule which
generates the term set 𝑇(𝑥). And we have 𝑀 which is nothing but, the semantic rule which
associates with each linguistic value.

So, that is how we have these five variables and these five variables are put together to be
called as quintuple. So, if here let us say we take an example of a linguistic variable.

(Refer Slide Time: 02:00)

694
Let say a linguistic variable here is age. So, if we have age as the linguistic variable. So,
let’s see how these five variables look like. So, we see that if we have 𝑎𝑔𝑒, 𝑎 𝑔 𝑒, 𝑎𝑔𝑒.
So, what is the generic variable in this case? So, here the linguistic variable or the generic
variable basically, the linguistic variable is the 𝑎𝑔𝑒, but we have 𝑥 as the generic variable,
generic variable. And then here we have the term set 𝑇(𝑥).

So, for 𝑎𝑔𝑒, we can have term set and this term set can be a collection of multiple linguistic
values. When we say linguistic values it means fuzzy sets. So, every linguistic value is
represented by a particular fuzzy set and here the term set 𝑇(𝑥) and since here 𝑎𝑔𝑒 is a
linguistic variable. So our 𝑇(𝑎𝑔𝑒) or I would say the term set for 𝑎𝑔𝑒 can be here in this
case 𝑦𝑜𝑢𝑛𝑔, 𝑚𝑖𝑑𝑑𝑙𝑒 𝑎𝑔𝑒𝑑, 𝑜𝑙𝑑. So, this is just for example, and we can have multiples
segregations multiple fuzzy sets for 𝑎𝑔𝑒. So, term set can be basically a collection of the
linguistic values for a particular linguistic variable.

So, if we have a linguistic term like 𝑎𝑔𝑒 is 𝑦𝑜𝑢𝑛𝑔, this will denote the assignment of the
linguistic value young to the linguistic variable 𝑎𝑔𝑒. So, the linguistic variable as I have
mentioned here if we have 𝑎𝑔𝑒 as the linguistic variable. So, linguistic variable is normally
represented in terms of the generic variable 𝑥.

So, as I mentioned that this linguistic variable can be segregated can be divided into
multiple linguistic values. Here in this case, we have divided the whole region of 𝑎𝑔𝑒 in
3. So, these three regions these three regions basically are 𝑦𝑜𝑢𝑛𝑔, 𝑚𝑖𝑑𝑑𝑙𝑒 𝑎𝑔𝑒𝑑, 𝑜𝑙𝑑 like
that, but we can have multiple such regions which could be represented by the linguistic
values and these linguistic values are normally represented by fuzzy sets.

So, here we have this fuzzy set, this linguistic value 𝑦𝑜𝑢𝑛𝑔 and this 𝑦𝑜𝑢𝑛𝑔 is nothing but,
the lower side of the 𝑎𝑔𝑒 representation. So, 𝑦𝑜𝑢𝑛𝑔 is a fuzzy set here is a left open fuzzy
set and we have this the middle one is 𝑚𝑖𝑑𝑑𝑙𝑒 𝑎𝑔𝑒 and here this 𝑚𝑖𝑑𝑑𝑙𝑒 𝑎𝑔𝑒 is
represented by another fuzzy set. Similarly, we have the 𝑜𝑙𝑑 𝑎𝑔𝑒 here which is represented
by a fuzzy set which is right open.

So, as I mentioned that when we are dealing with the term set we can suitably divide a
particular linguistic variable into multiple fuzzy regions and every fuzzy region is
represented by a particular linguistic value which is represented by a fuzzy set. So this way
a linguistic variable is normally represented and as in this case we have seen that 𝑎𝑔𝑒 if

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we have taken age as the linguistic variable 𝑎𝑔𝑒 has been divided into various fuzzy
regions. And all these regions are nothing but, is the collection of all these regions is term
set.

So, here we have a generic variable 𝑥 which is the measure of age, for example we have 1
year, 2 years, 3 year like that and it can further go up to here in this case we are we have
90. So, these values are actually the values of the generic variable 𝑥 and 𝑥 here is the 𝑎𝑔𝑒.

(Refer Slide Time: 07:55)

Now, what is the universe of discourse? So, the universe of discourse can be all you know
the possible limit of 𝑎𝑔𝑒 where the all possible values of the generic variable age, linguistic
variable age could be settle. So, within the total a space basically is here termed as universe
of discourse 𝑋. So, in the term set 𝑇(𝑎𝑔𝑒) each term can be characterized by a fuzzy set
of a universe of discourse.

So, the 𝑋 here is the universe of discourse and this universe of discourse basically says
that whatever value that 𝑥 can take or 𝑎𝑔𝑒 can take will be belonging into the limit of the
universe of discourse. Then comes the syntactic rule here, so, the syntactic rule is 𝐺 and it
is symbolically defined by 𝐺. So, the syntactic rule refers to the way the linguistic values
in the term set 𝑇 are generated.

So, in the figure that we have just discussed for 𝑎𝑔𝑒 where we have created multiple fuzzy
regions for 𝑎𝑔𝑒; 𝑦𝑜𝑢𝑛𝑔, 𝑚𝑖𝑑𝑑𝑙𝑒 𝑎𝑔𝑒 etcetera are the syntactic levels. So, based on this

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syntactic rule, we create 𝑦𝑜𝑢𝑛𝑔, 𝑚𝑖𝑑𝑑𝑙𝑒 𝑎𝑔𝑒𝑑, 𝑜𝑙𝑑 etcetera. Similarly, semantic rule 𝑀.
So, 𝑀 basically helps us in specifying the procedure for computing the meaning of any
linguistic value through specified membership function. So, we will have couple of
examples and with this we will be able to understand all these parameters.

(Refer Slide Time: 10:00)

Now, comes the composite linguistic term. So, here for linguistic variables we use word
as values of linguistic variables in cases of linguistics we often use more than one word to
describe a variable. So, for example, here the intensity of light, if we choose the intensity
of light as a linguistic variable. So, let us now see how these five parameters as we have
seen those parameters which were as a quintuple. How actually are these coming up for
this linguistic variable.

So, intensity of the light basically here, if we are taking this as the linguistic variable. So,
what is the generic variable? How we are going to measure the intensity of light? How we
are going to represent the intensity of light, let us say it is 𝑥, so, 𝑥 is the generic variable
and then what is the universe of discourse 𝑋. So, that the limit the range within which we
are supposed to take the generic variable values and then the term set.

So, the term set here for this case could be if it is the intensity of light, it could be simply
either 𝑙𝑜𝑤 or ℎ𝑖𝑔ℎ or 𝑚𝑒𝑑𝑖𝑢𝑚 or like that or may be 𝑑𝑖𝑚, 𝑏𝑟𝑖𝑔ℎ𝑡. So, these kinds of
fuzzy regions or the linguistic values can be included in the term set. And similarly, the
syntactic and semantic rules could be created for these the linguistic variable the intensity

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of light. Here we have written 𝑣𝑒𝑟𝑦 𝑏𝑟𝑖𝑔ℎ𝑡, 𝑠𝑙𝑖𝑔ℎ𝑡𝑙𝑦 𝑑𝑖𝑚, 𝑚𝑜𝑟𝑒 𝑜𝑟 𝑙𝑒𝑠𝑠 𝑏𝑟𝑖𝑔ℎ𝑡, all these
are little further if we discussed the hedges. So, these comes under that.

So, we will understand these terms when we move little ahead. So, the linguistic variables
we see here the term set which we have for linguistic variables. So, this can further be
generated and let us now understand first that the linguistic variable may be a composite
term and can be classified into three groups. So, as a whole linguistic variable can have
primary terms and then linguistic hedges and then we have the negation and complement
or and connectives. For example, if we take intensity of light so, what are the primary
terms for the intensity of light?

So, primary terms could be for intensity of light, it could be as I already mentioned that
could be low intensity, medium intensity or may be high intensity. Similarly, the linguistic
hedges for the intensity of light could be maybe we if we add an adjective here, so we
could simply write 𝑣𝑒𝑟𝑦 𝑙𝑜𝑤 intensity or may be 𝑚𝑜𝑟𝑒 𝑜𝑟 𝑙𝑒𝑠𝑠, 𝑚𝑒𝑑𝑖𝑢𝑚 or whatever.

So, similar linguistic values could be included. So, the linguistic hedges basically are with
the adjectives in along with the primary terms. And then comes the negation class here we
have you know the linguistic fuzzy regions where the primary terms are taken as either the
complement or negation or with some other connectives like and or, or. So, for example,
here we can write we can use a term 𝑛𝑜𝑡 𝑙𝑜𝑤 or may be 𝑛𝑜𝑡 𝑚𝑒𝑑𝑖𝑢𝑚. Similarly, we can
have another fuzzy value which could be like this 𝑙𝑜𝑤, but not very 𝑙𝑜𝑤.

So, here we see that, but is the, but is a connective. So, this way we see that we have some
composite terms and these are basically divided into three groups; first group is primary
terms, second group is the linguistic hedges and the third group is the complement and
connectives. So, let us further understand this by taking some examples and these primary
terms and then the linguistic hedges negation complements and connectives can be
understood further.

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(Refer Slide Time: 15:48)

So, if we talk of the primary terms let us say the 𝑎𝑔𝑒 which we have already taken in the
beginning as the linguistic variable, then it is primary term could be the 𝑦𝑜𝑢𝑛𝑔 as I have
already mentioned and in this primary terms, the 𝑚𝑖𝑑𝑑𝑙𝑒 𝑎𝑔𝑒𝑑 can also be one of the
linguistic value, similarly 𝑜𝑙𝑑. So, we have already seen that we have three fuzzy regions.
This could be like this.

So, we have let’s say this is the universe of discourse here and within this we have the
generic variable 𝑥 which is nothing but the 𝑎𝑔𝑒 the measure of 𝑎𝑔𝑒 let’s say years. So, let
us say we have created three primary terms and these three primary terms here are
𝑚𝑖𝑑𝑑𝑙𝑒 𝑎𝑔𝑒 or 𝑚𝑖𝑑𝑑𝑙𝑒 𝑎𝑔𝑒𝑑 and then we have the 𝑜𝑙𝑑 and here we have let’s say 𝑦𝑜𝑢𝑛𝑔.
So, these three are the primary terms which have been crated here and further these primary
terms can be used with some adjectives are connectives to get the linguistic hedges.

So, we will discuss this in the coming slides. So, here the primary terms basically are the
fuzzy regions, primary fuzzy regions which are being created just to represent the basic
building regions basic regions. So, for example, here the age as the generic variable, 𝑎𝑔𝑒
as the linguistic variable here is divided into three region, three basic regions and hence
these 𝑦𝑜𝑢𝑛𝑔, 𝑚𝑖𝑑𝑑𝑙𝑒 𝑎𝑔𝑒𝑑, 𝑜𝑙𝑑 are called as basically the primary terms.

Now, these primary terms as I have already mentioned can be used as hedges by using the
adjectives along with the primary terms.

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(Refer Slide Time: 18:47)

So, here as I mentioned linguistic hedges. So, in linguistic, fundamental atomic terms.
Basically, atomic terms are often modified with adjectives. So, here with adjectives and
adverbs such as 𝑣𝑒𝑟𝑦, 𝑠𝑙𝑖𝑔ℎ𝑡, 𝑚𝑜𝑟𝑒 𝑜𝑟 𝑙𝑒𝑠𝑠, 𝑓𝑎𝑖𝑟𝑙𝑦, 𝑠𝑙𝑖𝑔ℎ𝑡𝑙𝑦, 𝑎𝑙𝑚𝑜𝑠𝑡, 𝑏𝑎𝑟𝑒𝑙𝑦, 𝑚𝑜𝑠𝑡𝑙𝑦,
𝑟𝑜𝑢𝑔ℎ𝑙𝑦, 𝑎𝑝𝑝𝑟𝑜𝑥𝑖𝑚𝑎𝑡𝑒𝑙𝑦 etcetera. So, we see that everywhere we have some adjectives
or adverbs and when this is being used along with the primary terms or atomic terms then
this becomes linguistic hedges.

(Refer Slide Time: 19:46)

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So, let us understand this further with some mathematical expressions of linguistic hedges.
So, if we have here a fuzzy set let say 𝐴 and if it is continuous fuzzy set then we represent
a fuzzy set, continuous fuzzy set like this. And similarly, we represent discrete fuzzy set
like this. So, the membership values which we use here for representing any fuzzy set let
say 𝐴 whether it is a continuous fuzzy set or discrete fuzzy set.

So, if we have a membership function 𝜇(𝑥) which has been used in the basic fuzzy set in
the primary term or the atomic term that the symbol that has been used. So, if we have
𝜇(𝑥), then how can we represent a linguistic hedge using the membership function of the
primary term. So, here if we have the 𝜇(𝑥) which is here. The 𝜇(𝑥) is nothing but the
membership function of primary fuzzy value.

So, if we have 𝜇(𝑥) as the primary fuzzy value when we say fuzzy set, primary fuzzy set
it means it is the primary term like in case of 𝑎𝑔𝑒 we have seen the 𝑦𝑜𝑢𝑛𝑔. So, 𝑦𝑜𝑢𝑛𝑔 is
represented by a primary set, this is a primary fuzzy set these also sometimes termed as a
fuzzy value or the linguistic value because all these the linguistic value, fuzzy value
etcetera are represented by a suitable fuzzy set

So, now let us make use of 𝜇(𝑥) which is used in the primary fuzzy set and let us make
linguistic hedge out of it. So, if we have 𝜇(𝑥) for a primary fuzzy set then if we have to let
say use very for example, if we have let say this 𝜇(𝑥), I will write mu x here and this 𝜇(𝑥)
has been used for let say 𝑦𝑜𝑢𝑛𝑔. This 𝜇(𝑥) has been used for 𝑦𝑜𝑢𝑛𝑔. It means, we have
a fuzzy region or fuzzy value which is represented by a fuzzy set 𝐴 and this 𝐴 is nothing
but, this is for 𝑦𝑜𝑢𝑛𝑔. So, this 𝜇(𝑥) is for 𝜇(𝑥) here is a membership function and this is
for the primary fuzzy set primary region primary term.

So, if let say we would like to say it like this. We would like to modify the fuzzy set like
this like 𝑣𝑒𝑟𝑦 𝑦𝑜𝑢𝑛𝑔. So, how can we make use of this 𝜇(𝑥) and we convert it into very
𝑦𝑜𝑢𝑛𝑔. So, very young we have used very before 𝑦𝑜𝑢𝑛𝑔, 𝑦𝑜𝑢𝑛𝑔 we already know. So,
how can we get the a fuzzy set let say which is 𝐴 and this is for 𝑣𝑒𝑟𝑦 𝑦𝑜𝑢𝑛𝑔. It is very
easy and we simply make use of the 𝜇(𝑥) that was given to us we have to is square the
𝜇(𝑥). So, the 𝜇(𝑥) that was given to us means simply take it and then we will square it
and that is it.

701
So, 𝜇𝐴 can be converted into 𝜇𝑣𝑒𝑟𝑦 𝐴 very easily by just taking square of the membership
functions. So, this way we are able to convert this into a linguistic hedge.

(Refer Slide Time: 24:52)

Similarly, a membership value if it is given let’s say again this is the basic membership
value 𝜇𝐴 and we are interested in more or less as the linguistic hedge, we have to simply
dilate it. When we say dilate means we are reducing the power, means we are taking these
square root of it here.

So, more or less can be represented by 𝜇, the membership function for more or less can be
simply the 𝜇𝑚𝑜𝑟𝑒 𝑜𝑟 𝑙𝑒𝑠𝑠 𝐴 (𝑥) = [𝜇𝐴 (𝑥)]1/2.

702
(Refer Slide Time: 25:46)

Similarly, when we want to find the membership function of the extremely something
extremely if we want to use the hedge is 𝑒𝑥𝑡𝑟𝑒𝑚𝑒𝑙𝑦, 𝜇𝑒𝑥𝑡𝑟𝑒𝑚𝑒𝑙𝑦 𝐴 (𝑥) =
𝜇𝑣𝑒𝑟𝑦 𝑣𝑒𝑟𝑦 𝑣𝑒𝑟𝑦𝐴 (𝑋) = [𝜇𝐴 (𝑥)]8 . This means we have applied very 3 times. If we take very
one time it is squares the membership function means we write 𝜇𝐴 (𝑥) raise to the power
2.

So, this way we are able to make use of the hedge and we can applied the adjective over
the primary term to make the linguistic hedge. So, with this I would like to stop here.

(Refer Slide Time: 26:48)

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And in the next lecture we will continue with some examples on linguistic hedges and
negations complements and connectives.

Thank you.

704
 Fuzzy Sets, Logic and Systems and Applications
Prof. Nishchal K. Verma
Department of Electrical Engineering
Indian Institute of Technology, Kanpur

Lecture – 42
Linguistic Hedges and Negation/ Complement and Connectives

(Refer Slide Time: 00:18)

Welcome to lecture number 42 of Fuzzy Sets, Logic and Systems and Applications. In this
lecture we will discuss some examples on Linguistic Hedges and then we will discuss the
Negation, Complement and Connectives.

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(Refer Slide Time: 00:32)

So, let us take an example here to understand this better. So, if we have a linguistic variable
bright. So, we have a bright and as I already mentioned that this bright is termed as a
primary term, this bright basically is a fuzzy value which is represented by a fuzzy set.

So, if we have bright, simply bright here and bright here is a discrete fuzzy set, this is a
discrete fuzzy set. So, just to make you understand we have taken this example. So, let us
now convert this fuzzy set here which is characterized by a membership values along with
the corresponding generic variable values 1, 2, 3, 4, 5. So, let us now using the given
𝑏𝑟𝑖𝑔ℎ𝑡 a discrete fuzzy set let us now find 𝑣𝑒𝑟𝑦 𝑏𝑟𝑖𝑔ℎ𝑡 and then 𝑣𝑒𝑟𝑦 𝑣𝑒𝑟𝑦 𝑏𝑟𝑖𝑔ℎ𝑡 and
then 𝑚𝑜𝑟𝑒 𝑜𝑟 𝑙𝑒𝑠𝑠 𝑏𝑟𝑖𝑔ℎ𝑡.

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(Refer Slide Time: 02:00)

So, since we have been given here the primary discrete set we can write primary discrete
fuzzy value or fuzzy set all these names can be used interchangeably. So, a 𝑏𝑟𝑖𝑔ℎ𝑡 fuzzy
set has been given, now let us convert this fuzzy set into 𝑣𝑒𝑟𝑦 𝑏𝑟𝑖𝑔ℎ𝑡. So, when let us say
we are supposed to use 𝑣𝑒𝑟𝑦 𝑏𝑟𝑖𝑔ℎ𝑡 word in order to communicate something.

So, how can we make use of this 𝑏𝑟𝑖𝑔ℎ𝑡 fuzzy set and we can generate a new fuzzy set
using the fuzzy set 𝑏𝑟𝑖𝑔ℎ𝑡 for 𝑣𝑒𝑟𝑦 𝑏𝑟𝑖𝑔ℎ𝑡. So, 𝑣𝑒𝑟𝑦 𝑏𝑟𝑖𝑔ℎ𝑡 as I have already mentioned
𝑣𝑒𝑟𝑦 when 𝑣𝑒𝑟𝑦 word comes here this means we have to raise the power here by 2 on the
membership function and if it is a discrete fuzzy set it is all the membership values are
squared. So, 𝑣𝑒𝑟𝑦 𝑏𝑟𝑖𝑔ℎ𝑡 is very simple to get here.

So, since we already have 1/1 + 0.8/2 + 0.6/3 + 0.4/4 + 0.2/5. So, here what we have
to do for 𝑣𝑒𝑟𝑦 𝑏𝑟𝑖𝑔ℎ𝑡 is because we are only adding the adjective 𝑣𝑒𝑟𝑦 here on the
𝑏𝑟𝑖𝑔ℎ𝑡, 𝑏𝑟𝑖𝑔ℎ𝑡 is already given. So, this 𝑣𝑒𝑟𝑦 term has to simply increase the power of
increase the membership value, that means the we have to square the values of the
membership. And let us see what we are getting.

So, here we see that we have a squared the values we do not have to touch we do not have
to change its corresponding membership, its corresponding generic variable 1, 2, 3, 4 and
5. We do not have to touch that we do not have to change these values we only have to
square the values of the membership. So, here we are squaring 1.

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So, 1 will become 1 only and then 0.8 when we square it we are getting 0.64 and here is
square of 0.6 will become 0.36, similarly square of 0.4 will become 0.16 and then square
of 0.2 will become 0.04. So, this way we see that we are getting a new discrete fuzzy set.

So, this fuzzy set is a fuzzy set which is for 𝑣𝑒𝑟𝑦 𝑏𝑟𝑖𝑔ℎ𝑡 I can write here the this as the
discrete fuzzy set for 𝑣𝑒𝑟𝑦 𝑏𝑟𝑖𝑔ℎ𝑡. So, what is interesting here is to note is that we had
𝑏𝑟𝑖𝑔ℎ𝑡 only we were given the discrete fuzzy set for 𝑏𝑟𝑖𝑔ℎ𝑡 and here we are converting
the 𝑏𝑟𝑖𝑔ℎ𝑡 into 𝑣𝑒𝑟𝑦 𝑏𝑟𝑖𝑔ℎ𝑡. And as I have already discussed when we add 𝑣𝑒𝑟𝑦, before
any linguistic value before any primary fuzzy set.

So, then this 𝑏𝑟𝑖𝑔ℎ𝑡 becomes 𝑣𝑒𝑟𝑦 𝑏𝑟𝑖𝑔ℎ𝑡 here by adding 𝑣𝑒𝑟𝑦 and since we are adding
𝑣𝑒𝑟𝑦 and we have already seen that 𝑣𝑒𝑟𝑦 comes with squaring of the membership values
or membership function in case of continuous fuzzy set. So, in case the fuzzy set is a
discrete fuzzy set then simply we have to square the respective membership values, but if
it is a continuous fuzzy set, then we will have to simply square the continuous membership
function that has been given for the primary set.

(Refer Slide Time: 06:45)

Let us now move to the second example which is for 𝑣𝑒𝑟𝑦 𝑣𝑒𝑟𝑦 𝑏𝑟𝑖𝑔ℎ𝑡. So, 𝑏𝑟𝑖𝑔ℎ𝑡 has
been given to us now we have to convert the given 𝑏𝑟𝑖𝑔ℎ𝑡 discrete fuzzy set into
𝑣𝑒𝑟𝑦 𝑣𝑒𝑟𝑦 𝑏𝑟𝑖𝑔ℎ𝑡. So obviously here we have two times 𝑣𝑒𝑟𝑦 𝑣𝑒𝑟𝑦, so we have to square
it two times, we have to square the respective membership values in twice. So, when we
square the membership values twice this the respective membership values basically

708
becomes the original membership values raised to the power 4 which you can see here,
this is for 𝑣𝑒𝑟𝑦 𝑣𝑒𝑟𝑦.

So, when we use the 𝑏𝑟𝑖𝑔ℎ𝑡 and we want to have 𝑣𝑒𝑟𝑦 𝑣𝑒𝑟𝑦 𝑏𝑟𝑖𝑔ℎ𝑡 we this way by taking
the powers increased by 4, the powers of the respective membership values by 4 we are
getting its membership values like this. And please note that here no change will happen
to its corresponding generic variable values which are 1, 2, 3, 4, and 5. So, we do not have
to change these values only the change will happen to the corresponding membership
values in case of the discrete fuzzy set simply we take the membership value and we use
the membership value raised to the power 4 and then whatever value comes we will write.
But if it is a continuous fuzzy set, then the membership function will become the twice
square it means we will write the 𝜇(𝑥)4 .

(Refer Slide Time: 08:47)

Now, let us quickly go to the third part of the example here and here the bright is given to
us and we have to find the 𝑚𝑜𝑟𝑒 𝑜𝑟 𝑙𝑒𝑠𝑠 𝑏𝑟𝑖𝑔ℎ𝑡. So, 𝑚𝑜𝑟𝑒 𝑜𝑟 𝑙𝑒𝑠𝑠 is a hedge as I have
already mentioned. So, like we had 𝑣𝑒𝑟𝑦 and then 𝑣𝑒𝑟𝑦, 𝑣𝑒𝑟𝑦, 𝑣𝑒𝑟𝑦 and then extremely
like that. Now, let us take 𝑚𝑜𝑟𝑒 𝑜𝑟 𝑙𝑒𝑠𝑠 as hedge and let us see what happens with
𝑚𝑜𝑟𝑒 𝑜𝑟 𝑙𝑒𝑠𝑠. 𝑀𝑜𝑟𝑒 𝑜𝑟 𝑙𝑒𝑠𝑠 basically we get when we change its membership value we
dilate its membership value in other words I would say.

So, 𝑚𝑜𝑟𝑒 𝑜𝑟 𝑙𝑒𝑠𝑠 means we rather than squaring or raising the power we are decreasing
the power here. So, decreasing the power means we are taking the square root of the

709
original membership value or membership function. So, in case of the continuous fuzzy
set we simply take the 𝜇(𝑥)1/2 , whereas if it is a discrete fuzzy set we simply take the
square root of all the respective membership values and here also we will not touch any of
the generic variable values.

So, so all the generic variable values will remain unchanged. So, let us find
𝑚𝑜𝑟𝑒 𝑜𝑟 𝑙𝑒𝑠𝑠 𝑏𝑟𝑖𝑔ℎ𝑡. So, 𝑚𝑜𝑟𝑒 𝑜𝑟 𝑙𝑒𝑠𝑠 𝑏𝑟𝑖𝑔ℎ𝑡 is here 𝑚𝑜𝑟𝑒 𝑜𝑟 𝑙𝑒𝑠𝑠,
𝑚𝑜𝑟𝑒 𝑜𝑟 𝑙𝑒𝑠𝑠 𝑏𝑟𝑖𝑔ℎ𝑡 fuzzy set is here. So, what we are doing here is we had 1. So, we are
taking the square root of 1 and then we are taking the square root of 0.8, we are taking the
square root of 0.6, we are taking the square root of 0.4, we are taking the square root of
0.2. And this way we are getting when we are taking square root of 1 we are getting 1, we
are getting here in this case when we are taking 0.8 and then we are taking square root of
0.8 we are getting 0.8944.

So, similarly we are getting all these values when we are taking the square root and this
way we are forming a new set and here we have formed a linguistic hedge. So, hedges are
coming out of the modifications of the original or the primary fuzzy sets by adding the
adjectives or adverbs. So, this way the bright fuzzy set is converted into the
𝑚𝑜𝑟𝑒 𝑜𝑟 𝑙𝑒𝑠𝑠 𝑏𝑟𝑖𝑔ℎ𝑡. So, when we say 𝑚𝑜𝑟𝑒 𝑜𝑟 𝑙𝑒𝑠𝑠 𝑏𝑟𝑖𝑔ℎ𝑡 it means we simply take the
square roots of all the respective membership values which are characterizing the fuzzy
set, this is discrete fuzzy set.

(Refer Slide Time: 12:00)

710
So, that way we have understood as to how we managed to get the, a primary fuzzy set
converted into the linguistic hedges. So, now let us understand here move to the 3rd class
and 3rd class here is the negation, complement and connectives. So, when have been given
a primary fuzzy set, a primary term all these names can be interchangeably used and when
we try to find a negation of it let’s say we have a primary set say 𝑚𝑖𝑑𝑑𝑙𝑒 𝑎𝑔𝑒𝑑 and then
we say 𝑛𝑜𝑡 𝑚𝑖𝑑𝑑𝑙𝑒 𝑎𝑔𝑒𝑠, then how to get the 𝑛𝑜𝑡 𝑚𝑖𝑑𝑑𝑙𝑒 𝑎𝑔𝑒𝑠 fuzzy set out of the given
fuzzy set 𝑚𝑖𝑑𝑑𝑙𝑒 𝑎𝑔𝑒𝑠.

So, here if I have been given any fuzzy set 𝐴, let us say 𝐴, the given fuzzy set, given fuzzy
set and this fuzzy set represents; this represents basically a primary set which is part of the
term set. So, if we are interested in finding 𝑁𝑂𝑇(𝐴), means as I mentioned if 𝐴 is a
𝑚𝑖𝑑𝑑𝑙𝑒 𝑎𝑔𝑒𝑑, then if you are interested in 𝑁𝑂𝑇(𝑚𝑖𝑑𝑑𝑙𝑒 𝑎𝑔𝑒𝑑) then we simply take
the 𝑁𝑂𝑇 of it and 𝑁𝑂𝑇 of it is represented by this sign here if 𝑁𝑂𝑇 is represented by this
sign.

So, 𝑁𝑂𝑇 is here and 𝑁𝑂𝑇(𝐴) can be symbolically written like this and what is done here
to get 𝑁𝑂𝑇(𝐴) we have already learned this when we have studied, when we have
discussed in the one of the previous lectures, the negation the complement. So, what we
do here is we take we subtract the corresponding membership values from 1 which you
can see here.

So, simply when we have been given a fuzzy set let us say if I write the given fuzzy set 𝐴
like this, if it is a continuous fuzzy set we will be representing it like this the integration
sign and then the universe of discourse just below it and then 𝜇(𝑥) and then 𝑥 here. And
this is here what is done for 𝑁𝑂𝑇(𝐴) is simply we take the complement of it and when we
take compliment of it the membership function is subtracted from 1 which you can see
here rest other things remain the same.

So, this way we can get 𝑁𝑂𝑇 is as the negation or the complement and here in other case
we can have the connectives like 𝐴 𝑎𝑛𝑑 𝐵 when we have let us say a fuzzy set 𝐴 and fuzzy
set 𝐵 both the fuzzy set have been given to you, then how to connect both the fuzzy sets
together? For example, I can say a 𝑚𝑖𝑑𝑑𝑙𝑒 𝑎𝑔𝑒𝑑 and 𝑦𝑜𝑢𝑛𝑔. So, we have two fuzzy sets
and both the primary terms are getting connected by 𝐴𝑁𝐷.

711
So, 𝐴𝑁𝐷 is here a connective. So, the connective can be either 𝐴𝑁𝐷, 𝑂𝑅, so these two are
the connectives. So, here there could be 𝐴 𝐴𝑁𝐷 𝐵 or 𝐴 𝑂𝑅 𝐵. So, like 𝑚𝑖𝑑𝑑𝑙𝑒 𝑎𝑔𝑒𝑑 and
𝑦𝑜𝑢𝑛𝑔 or 𝑚𝑖𝑑𝑑𝑙𝑒 𝑎𝑔𝑒𝑑 or 𝑦𝑜𝑢𝑛𝑔. So, whatever way any two fuzzy sets can be
connected. So, this and can also be replaced by but, so if we use, but or and both are same.

So, let us first take 𝐴 𝐴𝑁𝐷 𝐵. So, when we take 𝐴 𝐴𝑁𝐷 𝐵, so this means what? When we
have already done this exercise in one of the lectures previous lectures. So, when we have
two fuzzy sets let us say and they are being connected by 𝐴𝑁𝐷 connective. So, we simply
use intersection. So, 𝐴 𝐴𝑁𝐷 𝐵 will become 𝐴 intersection 𝐵 means both the fuzzy sets are
being intersected means we have we will have to take the intersection of the primary term
set 𝐴 and the primary term set 𝐵.

So, this way when we do what is happening with the corresponding membership function
is this we take the min of min[𝜇𝐴 (𝑥), 𝜇𝐵 (𝑥)]. Similarly, when we connect 𝐴 𝐴𝑁𝐷 𝐵 by
and we take the union of 𝐴 and 𝐵 and similarly we use the max sign in place of the min
sign. So, here we connect both the membership function, functions of 𝐴 𝐴𝑁𝐷 𝐵 like this
like 𝜇𝐴 (𝑥) and then we take max 𝜇𝐵 (𝑥), means we take max[𝜇𝐴 (𝑥), 𝜇𝐵 (𝑥)] which you
can see here.

And please note again that the generic variable values 𝑥 will remain the same, will remain
unchanged. Here these connectives are very interesting and even negation also. So,
negation and connectives both are basic connectives shown here, but since we have already
done in the previous lectures that we have multiple kinds of negations, multiple kinds of
connectives. So, like we could use for connectives various kinds of t-norms and s-norms.

So, as an when it is required we can use that also, but here if nothing is mentioned then we
simply use the basic s-norm and t-norms. So, s-norm is used for 𝑂𝑅 and t-norm is used for
𝐴𝑁𝐷. So, this way the negation complement and connectives can be managed and this is
quite interesting to note here that any primary fuzzy set any primary linguistic value, when
we say linguistic value linguistic value is nothing but a fuzzy set, linguistic value is
represented by a primary fuzzy set.

So, any primary term set, any primary fuzzy set, any linguistic value can be converted into
its hedges or in other fuzzy sets with the negation or you know with connectives you can
manage to get a new fuzzy set.

712
(Refer Slide Time: 19:50)

So, let us take an example here to understand the negation complement and connectives.
Here we have a two fuzzy sets both the fuzzy sets are discrete fuzzy sets 𝐴 and 𝐵 and let
us using 𝐴 and 𝐵 let us find 𝑁𝑂𝑇 𝐴 and then let us find 𝐴 𝐴𝑁𝐷 𝐵. Here this is a connective.
And in the third case also 𝑂𝑅 is connective, so 𝐴𝑁𝐷, 𝑂𝑅 both are connectives. So, let us
now go one by one and try to find 𝑁𝑂𝑇 𝐴 first.

(Refer Slide Time: 20:48)

So, 𝐴 has already been given. So, we represent 𝑁𝑂𝑇(𝐴) by a negation 𝐴 and as I have
already mentioned that simply when we have 𝜇𝐴 the membership function given. And then

713
when it comes to negation of it we subtract the membership values from 1 or if it is a
membership function we subtract this also from 1.

So, when we do that we get 𝑁𝑂𝑇(𝐴) like this means here we are negating all these
corresponding membership values and this comes out to be this. So, 𝑁𝑂𝑇(𝐴) is this, this
is represented by 𝑁𝑂𝑇(𝐴). So, a new fuzzy set 𝑁𝑂𝑇(𝐴) given 𝐴 is this. So, this way we
find 𝑁𝑂𝑇(𝐴) very quickly very easily.

(Refer Slide Time: 21:57)

Now, let us find 𝐴 𝐴𝑁𝐷 𝐵. So, 𝐴 𝐴𝑁𝐷 𝐵 as I have already mentioned that 𝐴𝑁𝐷 is a
connective; 𝐴𝑁𝐷 is a connective. We have been given fuzzy set 𝐴 this is fuzzy set 𝐴 first
fuzzy set and this is second fuzzy set. So, both the fuzzy sets have been given now we have
to connect these two fuzzy sets together and as I have already mentioned that when we
have 𝐴𝑁𝐷 we have to take the intersection of it.

So, when we take intersection, the basic intersection uses the min. So, when we use this
we find the new fuzzy set like this. So, in the new fuzzy set is with min of both the
membership values. So, when we use this the new fuzzy set is coming out to be this. So,
we can say this fuzzy set is a new fuzzy set; a new fuzzy set after connective 𝐴𝑁𝐷 so this
is for 𝐴𝑁𝐷, now if we use 𝑂𝑅.

714
(Refer Slide Time: 23:23)

So, 𝑂𝑅 can also be very quickly managed to get we see that here when it comes to 𝑂𝑅
means when we have two fuzzy sets and when we have to make a new fuzzy set by 𝑂𝑅ing
both the sets. So, 𝐴 ∪ 𝐵 gives us 𝐴 𝑂𝑅 𝐵 and here we use the max sign as a basic union.
So, you see here the max sign the inverted open triangle and this way we have 𝐴 𝑂𝑅 𝐵 and
this 𝐴 𝑂𝑅 𝐵 = ∫𝑥∈𝑋[𝜇𝐴 (𝑥) ∨ 𝜇𝐵 (𝑥)]/𝑥 .

So, the new fuzzy set, the new fuzzy set which is coming after connecting 𝐴 𝐴𝑁𝐷 𝐵 as 𝑂𝑅
by 𝑂𝑅 we are getting a new fuzzy set here the by 𝑂𝑅. So, 𝑂𝑅 is the connective. So, this
way we see that we have been able to manage to get new fuzzy sets. So, either taking the
negation of the primary set primary fuzzy set or by connecting the two primary sets or
maybe even further we can connect two or more fuzzy sets by 𝐴𝑁𝐷 or 𝑂𝑅 or any other
connectives and we can get the expression for the fuzzy set. So, this way we are able to
manage to get the, a new fuzzy set. So, with this I would like to stop here in this lecture.

715
(Refer Slide Time: 25:30)

And in the next lecture we will discuss the concentration and dilation of linguistic values.

Thank you.

716
 Fuzzy Sets, Logic and Systems and Applications
Prof. Nishchal K. Verma
Department of Electrical Engineering
Indian Institute of Technology, Kanpur

Lecture – 43
Concentration and Dilation & Composite Linguistic Term and Some Examples

(Refer Slide Time: 00:19)

So, welcome to the Lecture Number 43 of Fuzzy Sets, Logic and Systems and
Applications. In this lecture, we will discuss the Concentration and Dilation of fuzzy sets
and also we will discuss a composite linguistic terms and the related examples.

717
(Refer Slide Time: 00:41)

So, here let us first have the concentration and then the dilation.

(Refer Slide Time: 00:46)

Concentration basically is nothing but its a actually the concentration of a fuzzy set. And
when we say fuzzy set it means that we have some linguistic value and which is
characterized by a fuzzy set or which is represented by a fuzzy set.

Let us say 𝐴, then of course, we will have its membership value or membership function
in case of continuous linguistic value. So, we’ll have 𝜇𝐴 (𝑥). In case of 𝑥, we have the

718
generic variable. So if let us say we have 𝐴 which is a linguistic value, and when we say
linguistic value it means in fuzzy system, we represents a linguistic value by a suitable
fuzzy set.

So we can say that it is nothing but a suitable fuzzy set. So, if we have a linguistic value
which is represented by a suitable fuzzy set 𝐴 and if we are interested in having the
concentration of a fuzzy set 𝐴, so this concentration of fuzzy set is nothing but 𝐴(𝑘) . So
concentration of a fuzzy set 𝐴 is normally represented by here, 𝐴(𝑘) .

And if it is a continuous fuzzy set which is being concentrated, then of course, this will be
represented by the integral sign and then we have the universe of discourse 𝑋 and then we
have 𝜇𝐴 (𝑥) and since we have the 𝐴(𝑘) so this 𝑘 will be here as well. So, this way the
concentration of 𝐴 is represented by the expression 𝐴(𝑘) .

And similarly, if we have a discrete fuzzy set which is being concentrated, we will use
sigma and then we use again the suitable universe of discourse. So, in this case we have
the universe of discourse as 𝑋. So, for continuous and discrete fuzzy sets, we have the
concentrated versions of these, the concentrated fuzzy sets. So in summary, a
concentration of any fuzzy set basically is nothing, but we get another set which has its
membership values or its membership functions raised to the power 𝑘.

So, rest other things remains the same. Which we can see here that we simply we raise the
power of membership function in case of continuous fuzzy set and we raise the power of
membership values in case of discrete fuzzy sets. And as I mentioned, rest other things
will remain the same.

So here as I mentioned, the power of membership function is raised by 𝑘. So, 𝑘 is here

some number. In general, we write 𝑘 for concentration. But for normal concentration are
mentioned here that when we do not mention any value of k then normally, the for simpler
concentration the value of 𝑘 can be 2 and the value of 𝑘 can be any value more than 1.

So if nothing has been mentioned, then we can simply write the value we can take the
value of 𝑘 as 2. So when we take a normal concentration, we have here we have taken 𝑘 =
2 and when we substitute the 𝑘 = 2, what we are getting is here for continuous
membership concentration.

719
So, if we have a fuzzy set 𝐴 which is a continuous fuzzy set and if we are interested in
finding the concentration of fuzzy set 𝐴, then we simply write here 𝐴(2) . Why within
bracket? Because we are not exactly squaring the fuzzy set 𝐴, we are only squaring here
the membership values or membership functions of the fuzzy set 𝐴.

So here, this way this 𝐴(2) = ∫𝑋[𝜇𝐴 (𝑥)]2 /𝑥. So, this is going to be our concentrated fuzzy
set and when we have a discrete fuzzy set. So for discrete fuzzy set, everything remain the
same except the summation in place of the integration sign. So, this way we have
understood that what is the concentration of a fuzzy set.

So once again, I would like to tell you that if we are interested in finding the concentration
of any fuzzy set, we will simply write the fuzzy set and we will try to find the value of 𝑘
if the value of 𝑘 is given here, the when we are interested in concentration, the value of 𝑘
will always be greater than 1. This has to be noted, for concentration.

So, we will first look for the value of 𝑘 and if the value of 𝑘 has been given, then we will
use the value of 𝑘 which will be more than 1. So we will use that, but if the value of 𝑘 is
not mentioned, then we will go for the normal concentration. Means, we will take the value
of 𝑘 as 2. So, as we have done here.

So, concentration is very simple and we simply use the value of 𝑘 which is more than 1
and this value of 𝑘 basically increases the power of the membership function for
continuous fuzzy set or the power of membership value for discrete fuzzy sets.

And here, the notion of this concentration is nothing but, when we concentrate any fuzzy
set whatever fuzzy set that we have here let us say we have a fuzzy set 𝐴, let us say this is
a fuzzy set 𝐴, and if we are interested in concentrating this fuzzy set so this the
concentrated fuzzy set will be something like this, depending upon the value of 𝑘. So, I
can say this is my concentrated discrete fuzzy set.

So, I can write here that this is a concentrated fuzzy set 𝐴. So what do we see here? What
basically do we see here is that, when we go for the resulting fuzzy set that is a concentrated
fuzzy set gets a squeezed. Means, they spread gets reduced. So that needs to be understood
here, that whenever we concentrate any fuzzy set, its spread is going to get reduced.

720
(Refer Slide Time: 11:18)

Let us take couple examples on the concentration of fuzzy sets to understand the concept
better. So here we have a simple discrete fuzzy set 𝐴, a discrete fuzzy set 𝐴 with a universe
of discourse 1 to 5 and here we are interested in the concentration of 𝐴 and 𝑘 has not been
given to us. So, obviously, we will have to go for the value of 𝑘 = 2. So, I am writing here
since the value of 𝑘 is not given, so we will go for the normal concentration and this means
we would take the value of 𝑘 = 2.

(Refer Slide Time: 13:00)

721
So, when we do this, here the concentration of a discrete fuzzy set 𝐴, this can be written
as this equation. So, 𝐴(2) = ∑𝑋[𝜇𝐴 (𝑥)]2 /𝑥. So the 𝑘 comes here. Here, 𝑘 = 2. So, the
membership values of this discrete fuzzy set will get squared.

So when we go ahead, the we write the symbol of the concentration as 𝐶𝑂𝑁, so 𝐶𝑂𝑁(𝐴) =
∑𝑋[𝜇𝐴 (𝑥)]2 /𝑥 = (0.1)^2/2 + (0.7)^2/3 + (0.8)^2/4 + (1.0)2 /5.

So we are doing nothing except we are squaring the membership values of the
corresponding generic values of 𝑥. So, when we are squaring this what we are getting we
see here that, we are getting 0.01 here. When we square 0.1 and we are getting 0.49 when
we square 0.7, we are getting 0.64 when we square 0.8 and we are getting 1 when we are
square 1. So this way, we are getting a new expression of the discrete fuzzy set which is
here, corresponding to the generic variable values, we are getting the modified values of
the membership.

So, we get the concentration of the fuzzy set which we have taken as a discrete fuzzy set.
So, we can write here the concentration of 𝐴 is nothing but 0.01/2, then 0.49/3, then
0.64/4 then 1/5. So this is what is the new fuzzy set as a result of the concentration of the
discrete fuzzy set that was given to us. All right, so this was the example for the discrete
fuzzy set or I would say the concentration of discrete fuzzy set.

(Refer Slide Time: 16:32)

722
Now, let us take an another example for the concentration of a continuous fuzzy set. So, if
we have a linguistic term which is defined by a suitable fuzzy set let us say it is named as
a 𝐵𝑟𝑖𝑔ℎ𝑡. So, fuzzy set for 𝐵𝑟𝑖𝑔ℎ𝑡. So, fuzzy so, 𝐵𝑟𝑖𝑔ℎ𝑡 is a fuzzy set, 𝐵𝑟𝑖𝑔ℎ𝑡 is a
represented by a suitable continuous fuzzy set and the universe of discourse here is from
0 to 50. So, I can write here linguistic term 𝐵𝑟𝑖𝑔ℎ𝑡 is represented by a suitable fuzzy set
here.

Where, the membership of this 𝐵𝑟𝑖𝑔ℎ𝑡 fuzzy set is represented by 𝜇𝐵𝑟𝑖𝑔ℎ𝑡 (𝑥) here and
this is nothing, but a gaussian function. So, this is represented by 𝑔𝑎𝑢𝑠𝑠𝑖𝑎𝑛( 𝑥; 20,5) =
1 𝑥−20 2
𝑒𝑥𝑝 (− 2 ( ) ). Where, this 5 is nothing but the standard deviation. And this 20 is
5

nothing, but the mean.

So, this is the standard deviation and this 20 is the mean. Now, here we have been asked
to concentrate this fuzzy set 𝐵𝑟𝑖𝑔ℎ𝑡 and 𝐵𝑟𝑖𝑔ℎ𝑡 is a continuous fuzzy set. So, let us do
that. So, we’ll write the concentration of 𝐵𝑟𝑖𝑔ℎ𝑡 fuzzy set by the 𝐶𝑂𝑁s of 𝐵𝑟𝑖𝑔ℎ𝑡, the
concentration write concentration and then we will write 𝐵𝑟𝑖𝑔ℎ𝑡 like this.

(Refer Slide Time: 19:13)

So, when we do this, let us first take this fuzzy set which has been given to us, let us first
understand the continuous fuzzy set that is for the linguistic term 𝐵𝑟𝑖𝑔ℎ𝑡. So, when we
plot the fuzzy set here, the 𝐵𝑟𝑖𝑔ℎ𝑡 fuzzy set, this looks like this and this is nothing, but
the gaussian membership function.

723
(Refer Slide Time: 19:53)

So, we have the fuzzy set which is for 𝐵𝑟𝑖𝑔ℎ𝑡. And this 𝐵𝑟𝑖𝑔ℎ𝑡 has the membership
function as the gaussian with its mean 20 and its standard deviation 5. We can see here, is
20. So, we can represent the 𝐵𝑟𝑖𝑔ℎ𝑡 fuzzy set like this.

(Refer Slide Time: 20:22)

Now, let us concentrate this fuzzy set 𝐵𝑟𝑖𝑔ℎ𝑡 and as I already mentioned that we can write
the 𝐶𝑂𝑁(𝐵𝑟𝑖𝑔ℎ𝑡) means, the concentration of 𝐵𝑟𝑖𝑔ℎ𝑡 like this. And this can also be
written as 𝐵𝑟𝑖𝑔ℎ𝑡 (2) and then this is again going to be equal to since this is continuous
fuzzy set so, we will use the integral sign to represent this fuzzy set.

724
So, integral sign over 𝑋 which is the universe of discourse, and then we will write the
𝜇𝐵𝑟𝑖𝑔ℎ𝑡 (𝑥)2 /𝑥. So initially, for 𝐵𝑟𝑖𝑔ℎ𝑡 we had simply the 𝜇𝐵𝑟𝑖𝑔ℎ𝑡 (𝑥). But when we are
concentrating the fuzzy set 𝐵𝑟𝑖𝑔ℎ𝑡, then we will have to raise the power of the membership
function by 𝑘. And 𝑘 since is not given in this example so, we can take 𝑘 = 2.

If 𝑘 has been given, then we will use the same value of 𝑘 and this 𝑘 since we are
concentrating the fuzzy set so the value of 𝑘 is going to be always more than 1. So here
we are taking 𝑘 = 2, we are squaring the membership function. So, when we do that the
whole fuzzy set can be written

2
1 𝑥−20 2
([𝑒𝑥𝑝 (− 2 ( ) )] )⁄
5
𝐶𝑂𝑁(𝐵𝑟𝑖𝑔ℎ𝑡) = 𝑥.
∫
𝑋

And then again whatever is here, as the membership function is squared, means raised to
the power 2 and then we have the oblique 𝑥. So, this is 𝐶𝑂𝑁(𝐵𝑟𝑖𝑔ℎ𝑡), again.

(Refer Slide Time: 22:56)

Or in other words, we can write here the 𝑏𝑟𝑖𝑔ℎ𝑡 and then within bracket 2. And here please
note that we have used 𝑘 = 2 only. We have used the 𝑘 = 2. So, let us not get confused
with the value of 𝑘, as I have already mentioned if no value of 𝑘 has been given, then we
will simply use the value of 𝑘 as 2 for concentration. So when we plot the concentration

725
of 𝐵𝑟𝑖𝑔ℎ𝑡, means the fuzzy set which has come out of the concentration of 𝐵𝑟𝑖𝑔ℎ𝑡, we
get this fuzzy set.

So, which is represented by the red color here. So, this fuzzy set is the concentration of
𝐵𝑟𝑖𝑔ℎ𝑡.

(Refer Slide Time: 24:09)

I can write it either this way or the 𝐶𝑂𝑁(𝐵𝑟𝑖𝑔ℎ𝑡). And this blue plot is for the 𝐵𝑟𝑖𝑔ℎ𝑡
fuzzy set. So, we can clearly see that when we concentrate any fuzzy set, we get its spread
reduced or squeezed. So, here if we once again go for further concentration of the
concentrated 𝐵𝑟𝑖𝑔ℎ𝑡 fuzzy set, we will further get this spread reduced. So, here the
concentration basically helps us in reducing the fuzziness the uncertainties that is involved
in the fuzzy representation.

So, concentration of any fuzzy set basically gives us reduces spread than that of the original
spread of the fuzzy set that was taken for concentration. So, with this the discussion on the
concentration of a fuzzy set is over.

726
(Refer Slide Time: 25:28)

And in the next lecture we will discuss the dilation and the composite linguistic terms with
some suitable examples.

Thank you.

727
 Fuzzy Sets, Logic and Systems and Applications
Prof. Nishchal K. Verma
Department of Electrical Engineering
Indian Institute of Technology, Kanpur

Lecture - 44
Dilation and Composite Linguistic Term and Some Examples

So, welcome to lecture number 44 of Fuzzy Sets Logic and Systems and Applications. In
this lecture we will discuss a Dilation of fuzzy set and Composite Linguistic Term and
also we will discuss some of the examples.

(Refer Slide Time: 00:40)

So, let us first take the dilation of any fuzzy set, so as we already know that any linguistic
value is always represented by a suitable fuzzy set. So, let us say if we have linguistic
value which is represented by a fuzzy set 𝐴 and we are interested in the dilation of this
fuzzy set.

So, the dilation basically here is expressed by or I would say the represented by 𝐷𝐼𝐿. So,
dilation in short symbolically represented by 𝐷𝐼𝐿 and if the fuzzy set 𝐴 which is being
dilated is represented by 𝐷𝐼𝐿(𝐴). So, dilation of any fuzzy set here is

𝐷𝐼𝐿(𝐴) = 𝐴(𝑘) = ∫ [𝜇𝐴 (𝑥)]𝑘 /𝑥

𝑋

728
So, what we are doing here in dilation of any fuzzy set is we are basically taking the we
are raising the power of the membership function, but this here this raise of 𝑘 is the
opposite and in other way; in other words we can say we are decreasing the power. So,
here [𝜇𝐴 (𝑥)]𝑘 /𝑥 and here please understand that this 𝑘 is always less than 1.

So, the value of 𝑘 here is for dilation is always less than 1 and similarly here if we have
any discrete fuzzy set if we have any discrete fuzzy set 𝐴. So, on the same lines we go for
dilation and here also we write

𝐷𝐼𝐿(𝐴) = 𝐴𝑘 = ∑[𝜇𝐴 (𝑥)]𝑘 /𝑥

𝑋

And here this 𝑘 is again is the value of 𝑘 is less than 1, but normally the value of 𝑘 here is
0.5.

So, if we do not mention any value of k the value of 𝑘 here is understood as 0.5 for dilation,
so then if we are let us say dilating a fuzzy set 𝐴 continuous fuzzy set 𝐴. So, we can
represent the dilation of 𝐴, as

𝐴(0.5) = ∫ [𝜇𝐴 (𝑥)]0.5 /𝑥

𝑋

Similarly, we have the discrete fuzzy set, then we can use this expression for dilating fuzzy
set 𝐴, dilating a discrete fuzzy set 𝐴. And, this fuzzy set will be 𝐴(0.5)

𝐴(0.5) = ∑[𝜇𝐴 (𝑥)]0.5 /𝑥

𝑋

So, this is for the discrete fuzzy set the first one is for the discrete fuzzy set and then second
one is sorry, first one is for the continuous fuzzy set and the second one is for the discrete
fuzzy set here.

So, this way we have understood the formulation remain the same, formulations remain
the same for the dilation same as the concentration except the value of 𝑘. So, in
concentration the value of 𝑘 remained always more than 1, whereas for dilation the value
of 𝑘 is always less than 1.

729
(Refer Slide Time: 06:12)

Now, let us take an example here to understand the dilation of any discrete fuzzy set here
in our case it is 𝐴. So, 𝐴 is given to us and it is defined by the elements here the 0.1/2 +
0.7/3 + 0.8/4 + 1/5 see here. So, this is a discrete fuzzy set within the universe of
discourse 𝑋 that is 1 to 5 and you see now we have to find 𝐷𝐼𝐿 means dilation of 𝐴. So,
we see here that for dilation of any fuzzy set we need the value of 𝑘 as well, but here since
the value of 𝑘 is not mentioned in this example.

So, we can take we will take value of 𝑘 as default here 0.5, so the value of 𝑘 is
automatically coming as 0.5, for dilation for dilation if nothing is mentioned, similarly if
it was a concentration it was for concentration we could have taken 𝑘 = 2.

730
(Refer Slide Time: 07:52)

So, now, let us take the dilation of 𝐴 let us do the dilation of 𝐴. So, we have 𝐴 as a discrete
fuzzy set here discrete fuzzy set. Now, when we do the dilation we need to increase the
power we need to raise the power of the respective membership values to 0.5 which is here
corresponding its generic variable values. So, when we do that what we are getting here is
this see 𝐷𝐼𝐿(𝐴) is coming out to be 0.316/2 + 0.836/3 + 0.894/4 + 1/5.

So, we get a new fuzzy set which is a dilated fuzzy set, which is the outcome of the dilation
of a fuzzy set 𝐴. And as we know that we have got this by taking the square roots of the
respective membership values corresponding to all generic variable values. So, this way
we have found the dilation of a fuzzy set, discrete fuzzy set 𝐴 here and similarly other
fuzzy sets can be taken and the same on the same lines that dilation of the fuzzy set 𝐴 can
be found out.

731
(Refer Slide Time: 09:48)

Now, if we take another example which is on the continuous fuzzy set. So, here we have
a continuous fuzzy set, let us say this fuzzy set is a 𝑏𝑟𝑖𝑔ℎ𝑡 this fuzzy set is used this fuzzy
set is to represent the linguistic term 𝑏𝑟𝑖𝑔ℎ𝑡. So, we have this example where we are taking
a fuzzy set continuous fuzzy set here to represent the linguistic term bright. Now, if we are
interested in finding the dilation of this fuzzy set again we have to do the same which we
have done in the previous example.

But here we have since we have the continuous fuzzy set, since 𝑏𝑟𝑖𝑔ℎ𝑡 is a continuous
fuzzy set. It means the membership function is a continuous fuzzy set to give this
continuous fuzzy set. So, in our case in this example 𝜇𝑏𝑟𝑖𝑔ℎ𝑡 (𝑥) is a Gaussian membership
function which is

1 𝑥 − 20 2
𝜇𝑏𝑟𝑖𝑔ℎ𝑡 (𝑥) = 𝑔𝑎𝑢𝑠𝑠𝑖𝑎𝑛(𝑥; 20,5) = 𝑒𝑥𝑝 (− ( ) )⁄𝑥
2 5

So, this is the continuous membership function and this continuous membership function
is used to represent the continuous fuzzy set.

732
(Refer Slide Time: 11:46)

Now, if we are interested in finding the dilation of this fuzzy set, the 𝑏𝑟𝑖𝑔ℎ𝑡 fuzzy set we
can write the dilation of 𝑏𝑟𝑖𝑔ℎ𝑡. And then in the in in this way the this can be written by
the 𝑏𝑟𝑖𝑔ℎ𝑡 (0.5) . So, since we are going for dilation so we have to have the value of 𝑘 less
than 1 and since the value of 𝑘 has not been given in this problem as well.

So, we have to take the standard value of 𝑘 that is 0.5 for dilation for dilation. And when
we do that we simply raise the power by raise the power of the membership function by 𝑘
1 𝑥−20 0.5
and here 𝑘 is 0.5. So, we simply write here 𝑒𝑥𝑝 (− 2 ( ) ) /𝑥
5

So, this way when we plot this when we represent this here so this blue coloured fuzzy set
basically is for this is for 𝑏𝑟𝑖𝑔ℎ𝑡 and then here the green coloured fuzzy set is nothing but
the dilation of 𝑏𝑟𝑖𝑔ℎ𝑡. So, we can clearly see here that what is happening here is the
original fuzzy set original continuous fuzzy set which after taking the dilation of it has
wider spread.

So, we see that the spread of the fuzzy set is increased, so dilation of any fuzzy set always
increases its spread. So, in this case in this example we have a Gaussian membership
function for the fuzzy set which is used to represent the linguistic term 𝑏𝑟𝑖𝑔ℎ𝑡 and when
we have dilated the 𝑏𝑟𝑖𝑔ℎ𝑡 linguistic term. So, the dilation the normal dilation is giving
us a new fuzzy set which has more spread and it is shown by a green colour here this
picture.

733
(Refer Slide Time: 14:42)

So, this way we have understood the dilation here and now let us go further to talk about
the composite linguistic terms. So, composite linguistic terms can be formed from one or
more combinations of primary terms here logical connectives and linguistic hedges. So,
let us use all of these words all of these combinations together to get or to build the better
linguistic variables or in other words I would say that is use all of these to modify to get
the modified linguistic variables.

So, if we can use primary terms logical connectives and linguistic hedges, so we see that
when we use all of these one of the composite linguistic terms can be like this like we have
𝑛𝑜𝑡 𝑣𝑒𝑟𝑦 𝑦𝑜𝑢𝑛𝑔 and 𝑛𝑜𝑡 𝑣𝑒𝑟𝑦 𝑜𝑙𝑑. So, here we have and as the connective and we see
that what is the primary term primary term is basically 𝑦𝑜𝑢𝑛𝑔 and 𝑣𝑒𝑟𝑦 𝑜𝑙𝑑. So, primary
term is this primary term, so like that when we use all of these together we can form a new
linguistic variable this is also called composite linguistic term.

And, similarly we can have another composite linguistic term which is here as
𝑦𝑜𝑢𝑛𝑔 𝑏𝑢𝑡 𝑛𝑜𝑡 𝑡𝑜𝑜 𝑦𝑜𝑢𝑛𝑔. So obviously, here we see that we have but is the another
connective which is used for and so here but is connective. And then in other case in the
third one we have 𝑚𝑖𝑑𝑑𝑙𝑒 𝑎𝑔𝑒𝑑 or 𝑜𝑙𝑑 so we have or as the connective in this case.

Similarly, in the fourth case we have and as the connective so this way in all the cases we
are getting a new linguistic term and we would call these as a composite linguistic terms.

734
And this we are getting out of the out of all the primary terms logical connectives and in
some cases the linguistic hedges.

(Refer Slide Time: 18:13)

So, let us first understand that the how the linguistic edges we have we can obtain we have
already discussed these linguistic hedges and these linguistic hedges basically we obtain
by either dilation dilating the original fuzzy set or concentrating the fuzzy sets. So, let us
take an example here to understand the composite linguistic term out of the primary term
and the connectives and hedges.

So, let us take this example, so in this example here we have the linguistic term 𝑙𝑖𝑔ℎ𝑡 and
ℎ𝑒𝑎𝑣𝑦 and of course, these 𝑙𝑖𝑔ℎ𝑡 and ℎ𝑒𝑎𝑣𝑦 are since these are the linguistic terms,
linguistic values. So, these will be defined by some fuzzy set and here 𝑙𝑖𝑔ℎ𝑡 is defined by
this fuzzy set heavy is defined by this fuzzy set and these two are the continuous fuzzy
sets. And these membership values membership functions 𝜇𝑙𝑖𝑔ℎ𝑡 is defined by a bell
shaped fuzzy set.

Here this mu light is 𝜇𝑙𝑖𝑔ℎ𝑡 is this and then 𝜇ℎ𝑒𝑎𝑣𝑦 is this, so this way both the fuzzy sets
are continuous fuzzy sets and both the fuzzy sets are different fuzzy sets. Now our job is
to find the composite fuzzy sets out of the given fuzzy sets as primary terms. So, we see
that we can have the composite linguistic terms like this like we have we can have 𝑙𝑖𝑔ℎ𝑡
𝑏𝑢𝑡 𝑛𝑜𝑡 𝑡𝑜𝑜 𝑙𝑖𝑔ℎ𝑡 and then we can have 𝑠𝑙𝑖𝑔ℎ𝑡𝑙𝑦 𝑙𝑖𝑔ℎ𝑡.

735
We can have 𝑒𝑥𝑡𝑟𝑒𝑚𝑒𝑙𝑦 𝑙𝑖𝑔ℎ𝑡, we can have ℎ𝑒𝑎𝑣𝑦 𝑏𝑢𝑡 𝑛𝑜𝑡 𝑡𝑜𝑜 ℎ𝑒𝑎𝑣𝑦 we can have
𝑠𝑙𝑖𝑔ℎ𝑡𝑙𝑦 ℎ𝑒𝑎𝑣𝑦, but slightly heavy and then finally we can have 𝑒𝑥𝑡𝑟𝑒𝑚𝑒𝑙𝑦 ℎ𝑒𝑎𝑣𝑦. So,
like that all these composite linguistic terms we can manage to get by using the primary
terms, primary linguistic terms for 𝑙𝑖𝑔ℎ𝑡 and ℎ𝑒𝑎𝑣𝑦 and then its hedges and connectives.

(Refer Slide Time: 21:02)

So, let us go 1 by 1 so here since the primary terms, primary linguistic terms are defined
by or characterized by suitable continuous fuzzy sets. And here we have these fuzzy sets
its membership functions as well bell functions for 𝑙𝑖𝑔ℎ𝑡 and for ℎ𝑒𝑎𝑣𝑦 as well. So, it
means that we have 𝜇𝑙𝑖𝑔ℎ𝑡 (𝑥) = 𝑏𝑒𝑙𝑙(𝑥; 10,2,30) and then we have 𝜇ℎ𝑒𝑎𝑣𝑦 (𝑥) =
𝑏𝑒𝑙𝑙(𝑥; 15,3,70). So, this means that we have two bell functions which are the membership
functions for the 𝑙𝑖𝑔ℎ𝑡 fuzzy set, the fuzzy set for 𝑙𝑖𝑔ℎ𝑡 and the fuzzy set for ℎ𝑒𝑎𝑣𝑦.

So, let us first define the fuzzy set for 𝑙𝑖𝑔ℎ𝑡 so fuzzy set for 𝑙𝑖𝑔ℎ𝑡 is this here you can see.
And since the 𝑙𝑖𝑔ℎ𝑡 has its 𝜇(𝑥) its membership function and this membership function is
coming from here, this membership function is coming from here. So, the light fuzzy set
can be defined by

𝑤
𝐿𝑖𝑔ℎ𝑡 = ∫ ( )⁄𝑥
𝑋 𝑥 − 30 4
1 + | 10 |

736
So, we see that we have a bell shaped membership function for 𝑙𝑖𝑔ℎ𝑡 and for ℎ𝑒𝑎𝑣𝑦 both
and we all know that the parameters that are given here for the bell shaped are 10 to 30 in
place of a, b, c respectively. So, we have basically 𝑎 is equal to 10 and 𝑏 is equal to 2, 𝑐 is
equal to 30 for membership function for light.

And similarly, the membership function for heavy set ℎ𝑒𝑎𝑣𝑦 fuzzy set we have 𝑎 is equal
to 15, 𝑏 is equal to 3, 𝑐 is equal to 70 you can see here. And on the same lines we can have
a fuzzy set for fuzzy set representation for heavy.

𝑤
𝐻𝑒𝑎𝑣𝑦 = ∫ ( )⁄𝑥
𝑋 𝑥 − 70 6
1+| |
15

So, these both the fuzzy sets light and heavy are different because of its membership
functions. So, membership functions are different the nature of membership function
remains the same both are bell shaped membership functions, but the parameters are
different and we can represent these fuzzy sets here.

So, here we have the fuzzy set for 𝑙𝑖𝑔ℎ𝑡 which is represented by blue colour and then we
have the fuzzy set for heavy which is represented by the green colour here. So, this way
we have two fuzzy sets as the primary terms or in other word we can say the primary
linguistic terms 𝑙𝑖𝑔ℎ𝑡 and ℎ𝑒𝑎𝑣𝑦, so we can say this is as 𝑙𝑖𝑔ℎ𝑡 and this is as say ℎ𝑒𝑎𝑣𝑦.
So, these both the fuzzy sets are there so these are already given. So, now, we are interested
in the composite linguistic term versus here which is 𝑙𝑖𝑔ℎ𝑡 𝑏𝑢𝑡 𝑛𝑜𝑡 𝑡𝑜𝑜 𝑙𝑖𝑔ℎ𝑡.

737
(Refer Slide Time: 26:02)

So, let us try to get 𝑙𝑖𝑔ℎ𝑡 𝑏𝑢𝑡 𝑛𝑜𝑡 𝑡𝑜𝑜 𝑙𝑖𝑔ℎ𝑡 I am writing this here
𝑙𝑖𝑔ℎ𝑡 𝑏𝑢𝑡 𝑛𝑜𝑡 𝑡𝑜𝑜 𝑙𝑖𝑔ℎ𝑡. So, please understand that the fuzzy set for 𝑙𝑖𝑔ℎ𝑡 is already there
so I am writing here 𝑙𝑖𝑔ℎ𝑡 this is fuzzy set for 𝑙𝑖𝑔ℎ𝑡. Now, we have to make the composite
fuzzy set means we need to find first the 𝑡𝑜𝑜 𝑙𝑖𝑔ℎ𝑡. So, when we say 𝑡𝑜𝑜 𝑙𝑖𝑔ℎ𝑡 it means
you know for getting 𝑡𝑜𝑜 𝑙𝑖𝑔ℎ𝑡 we need to concentrate the fuzzy set 𝑙𝑖𝑔ℎ𝑡.

(Refer Slide Time: 26:49)

So, let us concentrate the fuzzy set here so when we concentrate the fuzzy set here this is
the fuzzy set for 𝑡𝑜𝑜 𝑙𝑖𝑔ℎ𝑡 this is fuzzy set for 𝑡𝑜𝑜 𝑙𝑖𝑔ℎ𝑡. So, the fuzzy set for 𝑙𝑖𝑔ℎ𝑡 is

738
already there and here what we are doing is we are concentrating the 𝑙𝑖𝑔ℎ𝑡 fuzzy set to get
the 𝑡𝑜𝑜 𝑙𝑖𝑔ℎ𝑡.

So, when we concentrate the fuzzy set 𝑙𝑖𝑔ℎ𝑡 we are getting the fuzzy set as 𝑡𝑜𝑜 𝑙𝑖𝑔ℎ𝑡
which you can now very easily understand. So, this way we have got the fuzzy set for
𝑡𝑜𝑜 𝑙𝑖𝑔ℎ𝑡 I can write it like this the 𝑡𝑜𝑜 𝑙𝑖𝑔ℎ𝑡. Now, since we have obtained the fuzzy set
for 𝑡𝑜𝑜 𝑙𝑖𝑔ℎ𝑡 and we need 𝑛𝑜𝑡 𝑡𝑜𝑜 𝑙𝑖𝑔ℎ𝑡 also 𝑛𝑜𝑡 𝑡𝑜𝑜 𝑙𝑖𝑔ℎ𝑡 the fuzzy set for
𝑛𝑜𝑡 𝑡𝑜𝑜 𝑙𝑖𝑔ℎ𝑡, so now let us take the complement of 𝑡𝑜𝑜 𝑙𝑖𝑔ℎ𝑡 to get the 𝑛𝑜𝑡 𝑡𝑜𝑜 𝑙𝑖𝑔ℎ𝑡
fuzzy set.

(Refer Slide Time: 28:11)

So, here we are taking the complement of 𝑡𝑜𝑜 𝑙𝑖𝑔ℎ𝑡 and this is going to give us the fuzzy
set representation for 𝑛𝑜𝑡 𝑡𝑜𝑜 𝑙𝑖𝑔ℎ𝑡, 𝑡𝑜𝑜 𝑙𝑖𝑔ℎ𝑡 and this way we have getting here the fuzzy
set for 𝑛𝑜𝑡 𝑡𝑜𝑜 𝑙𝑖𝑔ℎ𝑡. Now we are interested in 𝑙𝑖𝑔ℎ𝑡, but 𝑛𝑜𝑡 𝑡𝑜𝑜 𝑙𝑖𝑔ℎ𝑡 so first we have
here the fuzzy set for 𝑙𝑖𝑔ℎ𝑡 then we have the fuzzy set for 𝑡𝑜𝑜 𝑙𝑖𝑔ℎ𝑡 and then here we have
the fuzzy set for 𝑛𝑜𝑡 𝑡𝑜𝑜 𝑙𝑖𝑔ℎ𝑡.

Now, we have to find a composite fuzzy set which is out of the 𝑙𝑖𝑔ℎ𝑡, but not here a
connective but, and but is nothing, but and it is equivalent to and so we will basically
compose these fuzzy set. So, we have a composition of 𝑙𝑖𝑔ℎ𝑡 fuzzy set here because we
are interested in 𝑙𝑖𝑔ℎ𝑡 𝑏𝑢𝑡 𝑛𝑜𝑡 𝑡𝑜𝑜 𝑙𝑖𝑔ℎ𝑡.

739
So, we should take 𝑙𝑖𝑔ℎ𝑡 and then 𝑛𝑜𝑡 𝑡𝑜𝑜 𝑙𝑖𝑔ℎ𝑡 and then we connect this by 𝑏𝑢𝑡, so but
is as I already said it is and so let us now find the composite fuzzy set here, the modified
fuzzy set here out of 𝑙𝑖𝑔ℎ𝑡 and 𝑛𝑜𝑡 𝑡𝑜𝑜 𝑙𝑖𝑔ℎ𝑡 and both the fuzzy sets are composed by
composed with and connective.

(Refer Slide Time: 30:15)

So, here when both the fuzzy sets are superimposed on each other so the
𝑙𝑖𝑔ℎ𝑡 𝑏𝑢𝑡 𝑛𝑜𝑡 𝑡𝑜𝑜 𝑙𝑖𝑔ℎ𝑡 is going to give us the intersection of 𝑙𝑖𝑔ℎ𝑡 𝑎𝑛𝑑 𝑛𝑜𝑡 𝑡𝑜𝑜 𝑙𝑖𝑔ℎ𝑡.
So, this way we are getting the this portion when we take the intersection this I will again
mention as 𝑛𝑜𝑡 𝑡𝑜𝑜 𝑙𝑖𝑔ℎ𝑡 and this is for 𝑙𝑖𝑔ℎ𝑡 this is for 𝑡𝑜𝑜 𝑙𝑖𝑔ℎ𝑡.

740
(Refer Slide Time: 31:37)

So, this way we have you see composite fuzzy set which is the outcome of
𝑙𝑖𝑔ℎ𝑡 𝑏𝑢𝑡 𝑛𝑜𝑡 𝑡𝑜𝑜 𝑙𝑖𝑔ℎ𝑡 and of course, here this fuzzy set is for 𝑛𝑜𝑡 𝑡𝑜𝑜 𝑙𝑖𝑔ℎ𝑡 and this
fuzzy set is for too light, this fuzzy set is for 𝑙𝑖𝑔ℎ𝑡. So, this way we have obtained the
composite linguistic term which is represented by the fuzzy set 𝑙𝑖𝑔ℎ𝑡 𝑏𝑢𝑡 𝑛𝑜𝑡 𝑡𝑜𝑜 𝑙𝑖𝑔ℎ𝑡
which is here.

(Refer Slide Time: 32:41)

So, with this I would like to stop here and in the next lecture we will continue with few
more examples on composite linguistic terms.

741
Thank you.

742
 Fuzzy Sets, Logic and Systems and Applications
Prof. Nishchal K. Verma
Department of Electrical Engineering
Indian Institute of Technology, Kanpur

Lecture – 45
Some Examples on Composite Linguistic Terms

So, welcome to lecture number 45 of Fuzzy Sets, Logic and Systems and Applications,
here in this lecture we will discuss Some Examples on Composite Linguistic Terms.

(Refer Slide Time: 00:28)

Now, let us take an example here where we have to find a composite linguistic term for
slightly light. So, for light linguistic term we have a fuzzy set here which you can see is
for light which is represented by a continuous membership function 𝜇𝑙𝑖𝑔ℎ𝑡 by bell shaped
members function here.

And now, since we are interested in 𝑠𝑙𝑖𝑔ℎ𝑡𝑙𝑦 𝑙𝑖𝑔ℎ𝑡, so slightly, so I will write here for
slightly we need to dilate the fuzzy set. So, this way when we dilate the fuzzy set we get a
new fuzzy set which is here, here this is slightly for slightly the value of 𝑘 because we are
dilating. So, the value of 𝑘 here I am writing 𝑘 is equal to 0.5 for slightly. Similarly, if we
have more or less this is same as 𝑠𝑙𝑖𝑔ℎ𝑡𝑙𝑦. So, either we say 𝑠𝑙𝑖𝑔ℎ𝑡𝑙𝑦 or we say more or
less we need to dilate the fuzzy set.

743
So, here in our case if we say 𝑙𝑖𝑔ℎ𝑡 and then we say 𝑠𝑙𝑖𝑔ℎ𝑡𝑙𝑦 𝑙𝑖𝑔ℎ𝑡. So, then for
𝑠𝑙𝑖𝑔ℎ𝑡𝑙𝑦 𝑙𝑖𝑔ℎ𝑡 we have to dilate the original fuzzy set 𝑙𝑖𝑔ℎ𝑡 to get 𝑠𝑙𝑖𝑔ℎ𝑡𝑙𝑦 𝑙𝑖𝑔ℎ𝑡 which
is mentioned here which is the composite fuzzy set for 𝑠𝑙𝑖𝑔ℎ𝑡𝑙𝑦 𝑙𝑖𝑔ℎ𝑡 is shown here by
the red color.

(Refer Slide Time: 02:46)

Now, in case we are interested in 𝑒𝑥𝑡𝑟𝑒𝑚𝑒𝑙𝑦 𝑙𝑖𝑔ℎ𝑡. So, for 𝑒𝑥𝑡𝑟𝑒𝑚𝑒𝑙𝑦 𝑙𝑖𝑔ℎ𝑡 the value of
𝑘 becomes 8. So, the value of 𝑘 for 𝑒𝑥𝑡𝑟𝑒𝑚𝑒𝑙𝑦 this for 𝑒𝑥𝑡𝑟𝑒𝑚𝑒𝑙𝑦, for 𝑒𝑥𝑡𝑟𝑒𝑚𝑒𝑙𝑦 and
this 𝑒𝑥𝑡𝑟𝑒𝑚𝑒𝑙𝑦 is same as which is which is same as very very very, 3 times very.

So, we have already have the fuzzy set for 𝑙𝑖𝑔ℎ𝑡 which is here we have fuzzy set for 𝑙𝑖𝑔ℎ𝑡
and then if we are interested in 𝑒𝑥𝑡𝑟𝑒𝑚𝑒𝑙𝑦 𝑙𝑖𝑔ℎ𝑡 it means we have to concentrate the 𝑙𝑖𝑔ℎ𝑡
fuzzy set 3 times. So, this we have we this fuzzy set which is represented by blue color is
for 𝑙𝑖𝑔ℎ𝑡 fuzzy set and here you see the red fuzzy set the red color fuzzy set is for
𝑒𝑥𝑡𝑟𝑒𝑚𝑒𝑙𝑦 𝑙𝑖𝑔ℎ𝑡. And here this 𝑒𝑥𝑡𝑟𝑒𝑚𝑒𝑙𝑦 𝑙𝑖𝑔ℎ𝑡 is basically we have got by taking 𝑘 is
equal to 8 means we have, means this has been concentrated, means the fuzzy set has been
concentrated.

So, this way the composite fuzzy sets can be obtained here this is called the hedge
𝑒𝑥𝑡𝑟𝑒𝑚𝑒𝑙𝑦 𝑙𝑖𝑔ℎ𝑡 means we are obtaining the linguistic hedge and thereby we are finding
the resulting fuzzy set. So, we see that this red color fuzzy set is for the 𝑒𝑥𝑡𝑟𝑒𝑚𝑒𝑙𝑦 𝑙𝑖𝑔ℎ𝑡
fuzzy set. And, this we have found as the concentration of, concentration of 𝑙𝑖𝑔ℎ𝑡. So,
𝑒𝑥𝑡𝑟𝑒𝑚𝑒𝑙𝑦 𝑙𝑖𝑔ℎ𝑡 we have got out of concentrating the fuzzy set thrice.

744
(Refer Slide Time: 05:55)

Now, let us take another example another composite fuzzy set where we need to find a
ℎ𝑒𝑎𝑣𝑦 𝑏𝑢𝑡 𝑛𝑜𝑡 𝑡𝑜𝑜 ℎ𝑒𝑎𝑣𝑦. So, we have the fuzzy set here, fuzzy set representation for
the linguistic term ℎ𝑒𝑎𝑣𝑦 and here also the 𝜇ℎ𝑒𝑎𝑣𝑦 is nothing, but bell shaped membership
function.

And, if a ℎ𝑒𝑎𝑣𝑦 fuzzy set and I mean the fuzzy set for ℎ𝑒𝑎𝑣𝑦 linguistic term is shown
here. And, then let us first find the fuzzy set for 𝑡𝑜𝑜 ℎ𝑒𝑎𝑣𝑦 and then 𝑛𝑜𝑡 𝑡𝑜𝑜 ℎ𝑒𝑎𝑣𝑦 and
then let us connect ℎ𝑒𝑎𝑣𝑦 𝑎𝑛𝑑 𝑛𝑜𝑡 𝑡𝑜𝑜 ℎ𝑒𝑎𝑣𝑦 together to get the composite fuzzy set. So,
this fuzzy set is for ℎ𝑒𝑎𝑣𝑦, then this fuzzy set is for 𝑡𝑜𝑜 ℎ𝑒𝑎𝑣𝑦, this is for 𝑡𝑜𝑜 ℎ𝑒𝑎𝑣𝑦, this
is for 𝑡𝑜𝑜 ℎ𝑒𝑎𝑣𝑦; 𝑡𝑜𝑜 ℎ𝑒𝑎𝑣𝑦 you know how did we get we simply concentrated this and
we have got the fuzzy set for 𝑡𝑜𝑜 ℎ𝑒𝑎𝑣𝑦.

Now, let us quickly find the 𝑛𝑜𝑡 𝑡𝑜𝑜 ℎ𝑒𝑎𝑣𝑦. So, the fuzzy set for ℎ𝑒𝑎𝑣𝑦 is here 𝑡𝑜𝑜 ℎ𝑒𝑎𝑣𝑦
is here and 𝑛𝑜𝑡 𝑡𝑜𝑜 ℎ𝑒𝑎𝑣𝑦 is here, 𝑛𝑜𝑡 𝑡𝑜𝑜 ℎ𝑒𝑎𝑣𝑦. Now since we have
ℎ𝑒𝑎𝑣𝑦 𝑎𝑛𝑑 𝑛𝑜𝑡 𝑡𝑜𝑜 ℎ𝑒𝑎𝑣𝑦 fuzzy sets let us now use the connective but in between and
get the modified fuzzy set for, ℎ𝑒𝑎𝑣𝑦 𝑏𝑢𝑡 𝑛𝑜𝑡 𝑡𝑜𝑜 ℎ𝑒𝑎𝑣𝑦.

So, when we do this we see that since we have ℎ𝑒𝑎𝑣𝑦 here and then we have 𝑡𝑜𝑜 ℎ𝑒𝑎𝑣𝑦
here, we have 𝑛𝑜𝑡 𝑡𝑜𝑜 ℎ𝑒𝑎𝑣𝑦 here. So, now, for finding ℎ𝑒𝑎𝑣𝑦 𝑏𝑢𝑡 𝑛𝑜𝑡 𝑡𝑜𝑜 ℎ𝑒𝑎𝑣𝑦 we
have to use the connective, but and we have to use and for but, means we are taking the
intersection. So, we will take the intersection of these two. So, here we this is
𝑛𝑜𝑡 𝑡𝑜𝑜 ℎ𝑒𝑎𝑣𝑦, this is 𝑛𝑜𝑡 𝑡𝑜𝑜 ℎ𝑒𝑎𝑣𝑦 and this fuzzy set is for, this fuzzy set is for. So,

745
means we have two fuzzy sets: one is for ℎ𝑒𝑎𝑣𝑦 and other one is for 𝑛𝑜𝑡 𝑡𝑜𝑜 ℎ𝑒𝑎𝑣𝑦 and
when we super impose both the fuzzy sets on each other and when we take the intersection
of this we are getting the resultant as ℎ𝑒𝑎𝑣𝑦 𝑏𝑢𝑡 𝑛𝑜𝑡 𝑡𝑜𝑜 ℎ𝑒𝑎𝑣𝑦.

So, let us now look at this. So, this portion is giving us the intersection the intersection
here of ℎ𝑒𝑎𝑣𝑦, but ℎ𝑒𝑎𝑣𝑦 𝑎𝑛𝑑 𝑛𝑜𝑡 𝑡𝑜𝑜 ℎ𝑒𝑎𝑣𝑦, intersection of ℎ𝑒𝑎𝑣𝑦 and
𝑛𝑜𝑡 𝑡𝑜𝑜 ℎ𝑒𝑎𝑣𝑦. And we already know that this is for ℎ𝑒𝑎𝑣𝑦, this fuzzy set is for ℎ𝑒𝑎𝑣𝑦
this for 𝑡𝑜𝑜 ℎ𝑒𝑎𝑣𝑦 and this is for 𝑛𝑜𝑡 𝑡𝑜𝑜 ℎ𝑒𝑎𝑣𝑦.

(Refer Slide Time: 11:15)

So, this way we have found this outcome after taking the intersection of the
ℎ𝑒𝑎𝑣𝑦 𝑎𝑛𝑑 𝑛𝑜𝑡 𝑡𝑜𝑜 ℎ𝑒𝑎𝑣𝑦 and this is what is the fuzzy representation of the linguistic
term ℎ𝑒𝑎𝑣𝑦 𝑏𝑢𝑡 𝑛𝑜𝑡 ℎ𝑒𝑎𝑣𝑦. And we already know that this is for ℎ𝑒𝑎𝑣𝑦 this fuzzy set is
for 𝑡𝑜𝑜 ℎ𝑒𝑎𝑣𝑦 and this fuzzy set is for 𝑛𝑜𝑡 𝑡𝑜𝑜 ℎ𝑒𝑎𝑣𝑦. So, this way we have seen that we
have found the new composite fuzzy set which is the outcome of
ℎ𝑒𝑎𝑣𝑦 𝑏𝑢𝑡 𝑛𝑜𝑡 𝑡𝑜𝑜 ℎ𝑒𝑎𝑣𝑦.

746
(Refer Slide Time: 12:12)

Let us quickly go ahead and find the fuzzy set for slightly ℎ𝑒𝑎𝑣𝑦 here this 𝑠𝑙𝑖𝑔ℎ𝑡𝑙𝑦 is a
hedge and for 𝑠𝑙𝑖𝑔ℎ𝑡𝑙𝑦 for getting the 𝑠𝑙𝑖𝑔ℎ𝑡𝑙𝑦 ℎ𝑒𝑎𝑣𝑦 from ℎ𝑒𝑎𝑣𝑦 we need to dilate. So,
for slightly 𝑘 is going to be equal 0.5. So, for 𝑠𝑙𝑖𝑔ℎ𝑡𝑙𝑦 ℎ𝑒𝑎𝑣𝑦 what we have to do is we
need to dilate the primary fuzzy set that is the heavy fuzzy set the fuzzy set for ℎ𝑒𝑎𝑣𝑦.

So, we have here the fuzzy set for ℎ𝑒𝑎𝑣𝑦 and this red color fuzzy set is the fuzzy set which
is represented by red color is basically the basically for this 𝑠𝑙𝑖𝑔ℎ𝑡𝑙𝑦 ℎ𝑒𝑎𝑣𝑦 here. So, how
did we get this 𝑠𝑙𝑖𝑔ℎ𝑡𝑙𝑦 ℎ𝑒𝑎𝑣𝑦 is just fine in just dilating the ℎ𝑒𝑎𝑣𝑦 fuzzy set. So, this is
very simple. So, when we have any primary fuzzy set and we are interested in the 𝑠𝑙𝑖𝑔ℎ𝑡𝑙𝑦
of this fuzzy set this primary fuzzy set we dilate.

So, this is going to give us the fuzzy set like this. So, 𝑠𝑙𝑖𝑔ℎ𝑡𝑙𝑦, so dilation is going to give
us the 𝑠𝑙𝑖𝑔ℎ𝑡𝑙𝑦 ℎ𝑒𝑎𝑣𝑦. So, dilation of ℎ𝑒𝑎𝑣𝑦 is going to give us the 𝑠𝑙𝑖𝑔ℎ𝑡𝑙𝑦 ℎ𝑒𝑎𝑣𝑦.

747
(Refer Slide Time: 14:16)

Now, this way on the same lines we can find 𝑒𝑥𝑡𝑟𝑒𝑚𝑒𝑙𝑦 ℎ𝑒𝑎𝑣𝑦. So, for 𝑒𝑥𝑡𝑟𝑒𝑚𝑒𝑙𝑦 as I
have already mentioned this 𝑒𝑥𝑡𝑟𝑒𝑚𝑒𝑙𝑦 word this 𝑒𝑥𝑡𝑟𝑒𝑚𝑒𝑙𝑦 we can get by concentrating
the primary term thrice. So, the 𝑒𝑥𝑡𝑟𝑒𝑚𝑒𝑙𝑦 can be obtained like 𝑒𝑥𝑡𝑟𝑒𝑚𝑒𝑙𝑦 ℎ𝑒𝑎𝑣𝑦 ,
𝑒𝑥𝑡𝑟𝑒𝑚𝑒𝑙𝑦 ℎ𝑒𝑎𝑣𝑦, 𝑒𝑥𝑡𝑟𝑒𝑚𝑒𝑙𝑦 ℎ𝑒𝑎𝑣𝑦 can be simply the concentration and then
concentration and then concentration of ℎ𝑒𝑎𝑣𝑦. So, this means that 3 times concentration
of heavy and is a normal concentration it means the final value of 𝑘 is going to be 8.

So, in this case when we are 𝑒𝑥𝑡𝑟𝑒𝑚𝑒 when we are finding the fuzzy set for 𝑒𝑥𝑡𝑟𝑒𝑚𝑒𝑙𝑦,
𝑒𝑥𝑡𝑟𝑒𝑚𝑒𝑙𝑦 ℎ𝑒𝑎𝑣𝑦. So, 𝑒𝑥𝑡𝑟𝑒𝑚𝑒𝑙𝑦 ℎ𝑒𝑎𝑣𝑦 basically is the concentration with 𝑘 is equal
to 8 and when we do that we are going to get this fuzzy set which is represented by a red
color and this is the original the primary term for ℎ𝑒𝑎𝑣𝑦. This is how we can find the
composite linguistic terms the fuzzy sets for composite linguistic terms

748
(Refer Slide Time: 15:58)

Now, here the same can be represented on the represented together. So, we see that when
we have a bell shaped membership function for light linguistic value and a bell shaped
membership function for ℎ𝑒𝑎𝑣𝑦 linguistic value. But, here the parameters of this bell
shaped function membership function are different for 𝑙𝑖𝑔ℎ𝑡 and ℎ𝑒𝑎𝑣𝑦, you can see here
in case of 𝑙𝑖𝑔ℎ𝑡 we have 20, 3, 50, in case of ℎ𝑒𝑎𝑣𝑦 we have 20, 3, 130.

So, this membership this fuzzy set is for heavy this for ℎ𝑒𝑎𝑣𝑦 and this fuzzy set is for
𝑙𝑖𝑔ℎ𝑡. So, when we have these two primary terms and now you can see when we obtain
the linguistic hedges and thereby the composite linguistic terms and representation as
fuzzy sets, we see that here when we had 𝑙𝑖𝑔ℎ𝑡 fuzzy set this blue color is for 𝑙𝑖𝑔ℎ𝑡 I can
just tick here.

So, you can see blue color for the set is for 𝑙𝑖𝑔ℎ𝑡 and then here we have green colored
fuzzy set for ℎ𝑒𝑎𝑣𝑦 and when we are interested in 𝑠𝑙𝑖𝑔ℎ𝑡𝑙𝑦 ℎ𝑒𝑎𝑣𝑦 which we have
obtained by dilating ℎ𝑒𝑎𝑣𝑦. So, this we have obtained by dilating ℎ𝑒𝑎𝑣𝑦 means we have
simply taken 𝑘 is equal to 0.5. So, this you can see here this is the dilation of ℎ𝑒𝑎𝑣𝑦 and
you see the 𝑒𝑥𝑡𝑟𝑒𝑚𝑒𝑙𝑦 ℎ𝑒𝑎𝑣𝑦 here which is here.

So, 𝑒𝑥𝑡𝑟𝑒𝑚𝑒𝑙𝑦 ℎ𝑒𝑎𝑣𝑦 which is which we can simply we can concentrate thrice a ℎ𝑒𝑎𝑣𝑦
membership heavy fuzzy set and so we basically concentrate here thrice for getting the
𝑒𝑥𝑡𝑟𝑒𝑚𝑒𝑙𝑦 ℎ𝑒𝑎𝑣𝑦 this is 𝑒𝑥𝑡𝑟𝑒𝑚𝑒𝑙𝑦 ℎ𝑒𝑎𝑣𝑦. And, similarly when we are interested in
ℎ𝑒𝑎𝑣𝑦 𝑏𝑢𝑡 𝑛𝑜𝑡 𝑡𝑜𝑜 ℎ𝑒𝑎𝑣𝑦.

749
So, we use the connective, but and this we are getting by simply getting the intersection of
ℎ𝑒𝑎𝑣𝑦 and 𝑛𝑜𝑡 𝑡𝑜𝑜 ℎ𝑒𝑎𝑣𝑦 and then we when we are interested in 𝑠𝑙𝑖𝑔ℎ𝑡𝑙𝑦 𝑙𝑖𝑔ℎ𝑡, we take
the value of 𝑘 is equal to 0.5 as I mentioned is here we dilate and similarly for
𝑒𝑥𝑡𝑟𝑒𝑚𝑒𝑙𝑦 𝑙𝑖𝑔ℎ𝑡 we are taking the concentration three times means 𝑘 is equal to 8
similarly here we use 𝑙𝑖𝑔ℎ𝑡 𝑏𝑢𝑡 𝑛𝑜𝑡 𝑡𝑜𝑜 𝑙𝑖𝑔ℎ𝑡, similarly we have 𝑙𝑖𝑔ℎ𝑡 and then we take
the intersection of 𝑙𝑖𝑔ℎ𝑡 𝑤𝑖𝑡ℎ 𝑛𝑜𝑡 𝑡𝑜 𝑙𝑖𝑔ℎ𝑡, means we get first the complement of
𝑡𝑜𝑜 𝑙𝑖𝑔ℎ𝑡 which becomes 𝑛𝑜𝑡 𝑡𝑜𝑜 𝑙𝑖𝑔ℎ𝑡 and then we take the intersection of this with
𝑙𝑖𝑔ℎ𝑡. Similarly, 𝑛𝑜𝑡 𝑙𝑖𝑔ℎ𝑡 𝑎𝑛𝑑 𝑛𝑜𝑡 ℎ𝑒𝑎𝑣𝑦 we take the complement of 𝑙𝑖𝑔ℎ𝑡 and not we
take the complement of 𝑙𝑖𝑔ℎ𝑡 and ℎ𝑒𝑎𝑣𝑦 and then we take the intersection of these two to
get the composite terms.

So, this way we see that when we have primary terms primary fuzzy sets, then we can
obtain suitable composite linguistic terms and thereby you know its representations as
fuzzy sets very easily. So, 𝑙𝑖𝑔ℎ𝑡, ℎ𝑒𝑎𝑣𝑦 in this case in our case our primary terms and then
we have found the 𝑠𝑙𝑖𝑔ℎ𝑡𝑙𝑦 ℎ𝑒𝑎𝑣𝑦, 𝑒𝑥𝑡𝑟𝑒𝑚𝑒𝑙𝑦 ℎ𝑒𝑎𝑣𝑦, ℎ𝑒𝑎𝑣𝑦 𝑏𝑢𝑡 𝑛𝑜𝑡 𝑡𝑜𝑜 ℎ𝑒𝑎𝑣𝑦,
𝑠𝑙𝑖𝑔ℎ𝑡𝑙𝑦 𝑙𝑖𝑔ℎ𝑡, 𝑒𝑥𝑡𝑟𝑒𝑚𝑒𝑙𝑦 𝑙𝑖𝑔ℎ𝑡, 𝑙𝑖𝑔ℎ𝑡 𝑏𝑢𝑡 𝑛𝑜𝑡 𝑡𝑜𝑜 𝑙𝑖𝑔ℎ𝑡, 𝑛𝑜𝑡 𝑙𝑖𝑔ℎ𝑡 𝑎𝑛𝑑 𝑛𝑜𝑡 ℎ𝑒𝑎𝑣𝑦.

So, like that we can find any suitable composite linguistic terms by using the primary terms
hedges and connectives and this way we can manage to get lot many composite linguistic
terms and that’s understand as to how we get all these composite linguistic terms using the
primary linguistic terms hedges and connectives.

(Refer Slide Time: 21:44)

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So, with this I would like to stop here and in the next lecture we will study the contrast
intensification and orthogonality of fuzzy sets.

Thank you.

751
 Fuzzy Sets, Logic and Systems and Applications
Prof. Nishchal K. Verma
Department of Electrical Engineering
Indian Institute of Technology, Kanpur

Lecture - 46
Contrast Intensification and Orthogonality

Welcome to lecture number 46 of Fuzzy Sets, Logic and Systems and Applications. In this
lecture, we are going to discuss a Contrast Intensification and Orthogonality. And this
lecture is in continuation to our previous lecture which was on linguistic hedges where we
have learnt the concentration and dilation of fuzzy sets.

(Refer Slide Time: 00:41)

So, since we are going to discuss the contrast intensification in this lecture and we have
already the idea of the concentration of a fuzzy set. So here, in contrast intensification, we
use the concentration of fuzzy sets.

So let us assume that we have a fuzzy set 𝐴 and this is nothing but this fuzzy set 𝐴 is
linguistic, this fuzzy set is for a linguistic value. So in other words, we can say we have a
linguistic value and which is characterized by a fuzzy set and if it is a continuous fuzzy
set, then we have 𝜇(𝑥) here as the membership function 𝜇(𝑥) as the membership function.

And if we have a discrete fuzzy set, which is just the representation of linguistic value
then the 𝜇𝐴 (𝑥) is the membership value. And of course, this membership value is basically

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the corresponding to the generic variable 𝑥. So the contrast intensification is basically
represented by 𝐼𝑁𝑇 which is you can see here 𝐼 − 𝑁 − 𝑇 is the symbol for the
intensification, the contrast intensification.

So we have 𝐴 if we have 𝐴 as a fuzzy set which whose intensification is needed or whose

contrast intensification is needed to be done, then we write this as the 𝐼𝑁𝑇(𝐴). So we
normally write this 𝐼𝑁𝑇 of the normal bracket the small bracket, here. And this is
represented by, this is expressed by the 𝐼𝑁𝑇(𝐴) = 2𝐴(2) .

This means that we are doing the concentration and this concentration is a normal
concentration because for normal concentration, we have 𝑘 is equal to 2. So here we have
the intensification of 𝐴, this will be equal to 2𝐴(2) . It means we are concentrating the fuzzy
set 𝐴 or in other words I can say the 𝐴 is being normally concentrated so that you know
we have 𝑘 is equal to 2. So 𝐴(2) and then whatever comes here as we multiply it by 2.

So this is for the membership value, this computation is for the membership value. So let
us now understand this, this is true for 0 ≤ 𝜇𝐴 (𝑥) ≤ 0.5. And this of course, is for all 𝑥
for every 𝑥 belonging into the universe of discourse 𝑋. And then we have another
expression, this is valid for 0.5 ≤ 𝜇𝐴 ≤ 1, ∀𝑥 ∈ 𝑋.

And if this is the case, then we have another expression. What is this expression? We have
the ¬2(¬𝐴)(2) . So this needs to be completely understood that we apply this
intensification in 2 zones.

So, first zone starts with it starts below 0.5 and the second zone is starts above 0.5. So
above 0.5 as the membership value, we have 2𝐴(2) and below 0.5 as the membership value
of 𝐴 holds the value of the intensified membership ¬2(¬𝐴)(2) .

which is clearly written here you can understand by this expression. So the intensification
is very important concept and this is used for you know, the reducing the for reducing the,
I am writing here for reducing for reducing the fuzziness. So whenever, we intensify we
take the we do contrast intensification of any linguistic value which is characterized by a
fuzzy set, we essentially do the reduction of fuzziness which is present in the fuzzy set.

So you can just see here this paragraph mentions the same. That means, that the contrast
intensification 𝐼𝑁𝑇 increases the value of 𝜇𝐴 (𝑥) which are above 0.5 and diminishes

753
means reduces those which are below this point means the 0.5 as the membership value of
fuzzy set 𝐴 or the linguistic value 𝐴. Thus, the contrast intensification has the effect of
reducing the fuzziness of the linguistic value 𝐴, the inverse operator of the contrast
intensifier is also available is also used and this is called the contrast diminisher, which is
represented by 𝐷𝐼𝑀, 𝐷 − 𝐼 − 𝑀 which is as already mentioned that it is the opposite of
the contrast intensification.

(Refer Slide Time: 08:53)

So let us have 1 example of a discrete fuzzy set to understand the contrast intensification
better. So here we are taking this example where we have a fuzzy set 𝐴 which is of course,
a linguistic value and this is with the universe of discourse which is defined here and the
universe of discourse here we have 1, 2, 3, 4 and 5.

So if we see here that we have a fuzzy set which is a discrete fuzzy set which is available
with us and this represents the fuzzy linguistic value and if we are interested in finding the
contrast intensification or the 𝐼𝑁𝑇(𝐴), then let us see how we can manage to get. So as I
have already mentioned that we first look for the point where we have to apply the first
expression that I mentioned that above 0.5 as the membership value and below 0.5 as the
membership value.

So if we look at this fuzzy set where we have these 2 you see here, these 2 terms so 0.7 / 1,
0.6 / 2 both are above 0.5. And so here the expression for contrast for contrast
intensification that we will be using is nothing but the 2(𝐴)(2) . Means we are going to

754
concentrating the fuzzy set and then we will the then the membership value that we have
for this fuzzy set we will multiply this membership value by 2.

Similarly, we have 1 term here which is with 0.5. So this 0.5 now the question comes
which form which expression we should apply because both the expression are applicable.
So whatever what whatever expression that we apply both are going to give the same result
because both are applicable for the 0.5 as the membership value. So now, here we see the
here we see that here we see that we have these two terms which are below 0.5. So, these
are below 0.5, these are below 0.5 as the membership value and here these are above 0.5
as the membership value.

So this way, we see that we can clearly apply the expression which are valid for these
membership for computing these membership value for intensification of 𝐴.

(Refer Slide Time: 13:21)

So let us quickly go ahead and see how we proceed. So for 𝑥 is equal to 1, we have 𝜇(𝐴)
of 1 which is 0.7 and this is within this range. So this is above 0.7. So this expression
applies. What is this expression? Expression here that applies is the ¬2(¬𝐴)(2) . So when
we compute, we are going to get the new membership value which is point which is 0.82
for the generic variable 1. Similarly, for 𝑥 is equal to 2 that is 𝜇𝐴 (2) which was given as
0.6.

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So here and now since this is 0.6, so this is very clear that the 0.6 is lying within this range
where 𝜇𝐴 is more than or equal to 0.5 or less than or equal to 1. So this way, when we
apply the expression of 𝐼𝑁𝑇 contrast intensification, we get 0.68 / 2. We here 2 is the
generic variable value that is, 𝑥 is equal to 2. Similarly, for 𝑥 is equal to 2, the same can
be computed and this please remember that this membership value for the generic variable
3 for the generic variable value 3 which is 0.1.

So it is clearly understood that this membership values since it is below 0.5. So the contrast
intensification expression will change and the contrast intensification basically here will
be 2(𝐴)(2) is equal to 2 × (0.1)2 . Means, the new value that comes out to be here is 0.02
for 3.

So similarly for 𝑥 is equal to 3, we can compute and we see that we are getting for generic
variable value 0.5; 4 0.5 and similarly for generic variable value 5, we are getting the
membership value of the contrast intensified fuzzy set as 0.185. So, this is how we get a
new fuzzy set.

So please remember that this is how we get a new fuzzy set which is the intensified fuzzy
set, this is called contrast intensification of fuzzy set. And when we apply these expressions
as I just discussed and for you know these 2 conditions and when we when we compute
these new membership values, new membership grades for respective membership for
respective generic variable values.

(Refer Slide Time: 17:21)

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So this way we get the all these computed and then when we write the new fuzzy set which
is nothing but the intensified fuzzy set which is the contrast intensification of 𝐴 and this is
nothing but when we since we have already computed all these new membership values
of respective fuzzy respective generic variable values 1, 2, 3 and 4. So we get this new
fuzzy set. So this is the intensified fuzzy set. 𝐼𝑁𝑇 or in other words, we can say the
intensified or we can say the intensification of a fuzzy set 𝐴. So that is how we managed
to get the contrast intensification of a fuzzy set that was given to us.

(Refer Slide Time: 18:48)

Now let us take another example and this example is for the continuous fuzzy set. So if we
have a fuzzy set A which is a continuous fuzzy set. If we have a fuzzy set 𝐴 which is a
continuous fuzzy set you see here, we have a fuzzy set which is a continuous fuzzy set
continuous fuzzy set. So and if we look at this fuzzy set, we this fuzzy set is characterized
by membership function and this membership function is 𝜇𝐴 (𝑥) and 𝜇𝐴 (𝑥) here is
triangular membership function.

So I can write here a Triangular Membership Function. This is represented by the

𝑡𝑟𝑖𝑎𝑛𝑔𝑙𝑒(𝑥; 1,3,7) are the vertices of the triangle that we are intending to know, intending
to use for to represent the membership function.

So here we have a triangular membership function which is a triangle triangular

membership function and the vertices are 1, 3, 7. We can see here the same membership

757
function and this membership function is for fuzzy set 𝐴. So this is for this is a membership
function which is characterizing a fuzzy set 𝐴.

(Refer Slide Time: 21:01)

Now let us use the contrast intensification function 𝐼𝑁𝑇 for this fuzzy set and we can use
this formula this expression for contrast, intensification and this is going to give us a new
fuzzy set which is the intensified fuzzy set which is contrast intensified fuzzy set and we
will look as to how this fuzzy set is, we will see as to how this fuzzy set looks like.

So as this intensification function the contrast intensification function is applicable and

these are the, this intensification function has 2 expressions and the first expression is
applicable in the first zone and the second intensification function expression is applicable
for the second zone. So first zone is basically the zone which is below the 0.5 membership
value.

So this is the first zone, I write here the first zone and then we have the second zone which
is more than which is applicable to more than 0.5 as the membership value. So now, and
please understand that any membership value which is exactly equal to 0.5, so please
understand that both of them means either of these expressions can be applied so and of
course, these any of these if you apply both of these are going to give the same value of
the intensified value of the membership function of the intensified fuzzy set.

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(Refer Slide Time: 23:23)

So let us now see what is happening. So when we apply this for the first time so for the
first time means we apply the intensification function here, the contrast intensification
function. So when we do that we see that below 0.5 line this is below 0.5 line is 𝜇 is equal
to I can write here is 𝜇𝐴 is equal to 0.5. So below this this in Zone 1, the fuzziness is getting
reduced because you see the membership values of the corresponding generic variables
are getting reduced and when we talk of the Zone 2, here the Zone 2. So in Zone 2, we see
that the membership values of the intensified fuzzy set 𝐴 is increased as compared to the
original membership value of the fuzzy set 𝐴 for corresponding generic variable values.

So this way we have when we are when we apply this contrast intensification for
continuous fuzzy set, we and when we plot the intensified the contrast intensified I can
write here intensified fuzzy set. So the fuzzy set that was given to us was A and when we
intensify this will look like this. So this is shown by a red colour fuzzy set. So now, when
we have found another fuzzy set which is the intensified fuzzy set, I can write here
intensified fuzzy set let us say 𝐵.

If it is 𝐵, if it is represented by let us say 𝐵 fuzzy set. So this is 𝐵 and now we can further
intensify this fuzzy set and let us see what happens.

So now, when we further go for the contrast intensification of fuzzy set 𝐵, let us see what
it how it looks like.

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(Refer Slide Time: 26:04)

So intensification once again is going to give us if this is 𝐵 then we let us represent this by
𝐶 the new fuzzy set. So, intensification is going to give us, you see here the new fuzzy set
is represented by fuzzy set 𝐶 and which is shown by a yellow colour see here. So this was
the, so this new fuzzy set 𝐶 is compared to the fuzzy set 𝐵, this is 𝐵 fuzzy set the red colour
fuzzy set 𝐵 as I have already mentioned. So this is 𝐵.

So I can write here that 𝐵 and 𝐶 both are both 𝐵 and 𝐶 both have reduced fuzziness as
compared to the fuzzy set 𝐴. So fuzziness can be you know can by taking the
intensification by taking the contrast intensification the fuzzy set gets reduced.

760
(Refer Slide Time: 27:52)

When we go further, let us say this fuzzy set is 𝐷 fuzzy set intensification once again. So
if we represent this fuzzy set by a violet colour which is 𝐷. So this is further intensified
and we can clearly see that this the you know the 0.5 line is the separator. Below this, the
behaviour is different and the above this line the behaviour is different. Means, below this
line whatever membership values are there, it is getting reduced above 0.5 line, the
membership values are getting increased.

(Refer Slide Time: 28:58)

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So this way, we are able to intensify the membership value. With this, I would like to stop
here and in the next lecture, we will continue with the orthogonality of fuzzy sets.

Thank you.

762
 Fuzzy Sets, Logic and Systems and Applications
Prof. Nishchal K. Verma
Department of Electrical Engineering
Indian Institute of Technology, Kanpur

Lecture – 47
Orthogonality of Fuzzy Sets

Welcome to lecture number 47 of Fuzzy Sets, Logic and Systems and Applications. In this
lecture, we will be discussing the Orthogonality of Fuzzy Sets.

(Refer Slide Time: 00:20)

Now, let us discuss another term, let us understand the orthogonality of fuzzy sets. So, if
we have a fuzzy if we have a term set 𝑇(𝑥). So, we have already seen that the 𝑇(𝑥) includes
many linguistic values and these linguistic values if these are represented by fuzzy sets.

So let us say the term sets 𝑇(𝑥) has multiple linguistic values and which are represented
by fuzzy sets 𝑡1 (𝑥), 𝑡2 (𝑥), 𝑡3 (𝑥), … . . , 𝑡𝑛 (𝑥). This means that term set has n number of
fuzzy sets. So this means what this means that if we have any variable any linguistic
variable and if these linguistic variable if this linguistic variable is within the universe of
discourse, let us say this is the universe of discourse, within this, we have multiple fuzzy
sets. Means this the reason within the universe of discourse is covered by multiple fuzzy
sets and let us say we have here n fuzzy sets, small n number of fuzzy sets. So here what
is the orthogonality? Is that the if we have a term set 𝑇(𝑥) and this term set has

763
𝑡1 (𝑥), 𝑡2 (𝑥), 𝑡3 (𝑥), … . . , 𝑡𝑛 (𝑥) of a linguistic variable x on the universe of discourse 𝑋. So
this is called the orthogonal if it is satisfies the following property.

So this means that the we take summation of all these fuzzy summation of all these
membership values which are there for a particular generic variable value 𝑥. So here, this

∑ 𝜇𝑡𝑖 (𝑥) = 1, ∀𝑥 ∈ 𝑋
𝑖=1

So this is very simple to compute by taking an example, I will be able to understand you
better I hope.

And so the 𝑡𝑖 (𝑥) we can say, the 𝑖 𝑡ℎ fuzzy set in the term set in the complete term set are
the convex and normal fuzzy sets that is very interesting to know that all these fuzzy set
which are present in the term set, must be a convex and normal fuzzy sets.

(Refer Slide Time: 03:42)

So let us take an example here, and understand. So, first we take a discrete fuzzy set and
first we take discrete fuzzy sets 𝐴, 𝐵 and 𝐶 to understand the orthogonality betters. So very
simple, so by taking the example by going through these examples 2, 3 examples, we are
taking will be able to understand the orthogonality of fuzzy sets better.

764
So here, we are taking 3 fuzzy sets, 3 discrete fuzzy sets and this these 3 fuzzy sets are
within the universe of discourse 1, 2, 3, 4, 5. So means from 1 to 5. So now, we see we
have a fuzzy set discrete fuzzy set 𝐴 0.2/2 + 0.1/3 + 0.2/4 + 0.3/5.

We have another fuzzy set 𝐵 which is 0.6/1 + 0.6/2 + 0.7/4 + 0.7/5. The C fuzzy set
that we have is 0.4/1 + 0.2/2 + 0.9/3 + 0.1/4. So this way we have 4. So, we have the
3 fuzzy sets and now let us check whether the fuzzy sets, the orthogonality condition is
satisfied for these fuzzy sets or not, for the term set or not.

So, basically the orthogonality when we talk of orthogonality, it’s for the whole term set.
So, we simply check for a particular generic variable value if we sum all the membership
values if it is equal to 1, we say the, and it should be there for all the generic variable
values. Means, throughout all the generic variable values at any generic variable value we
should have the summation of the membership values is equal to 1. So this is the condition.

(Refer Slide Time: 06:11)

So, let us now quickly check. So since we have the 3 discrete fuzzy sets, and if we take a
particular a generic variable let us say 𝑥 is equal to 1, so see here that if we take the generic
variable 1 the summation here is going to give us the 1, the membership value 1. So we
can say that at generic variable 1 generic variable value 1, the summation the total of the
membership values of all the coming from all the fuzzy sets is equal to 1.

765
So this way we can say that this is 1, I mean for corresponding the generic variable value
1, we are getting the total of the membership values is equal to 1.

(Refer Slide Time: 07:14)

Similarly now, we check for the other. So here, we can write it like this and similarly here
also we see that summation of membership values summation of membership values for
all the 3 membership for all the 3 fuzzy sets. So, this is also equal to 1.

(Refer Slide Time: 08:04)

Similarly, we see that this is for 2. This is for the generic variable value 2. Here we see for
generic variable value 3. If we sum all the membership values. So, this is for 3. So, 𝑖 is

766
equal to 1 to 3, all the 3 membership values corresponding to the 3 fuzzy sets if we are
adding we are getting here one.

Similarly, here for this also for the generic variable value 4. When we sum this, we see
that 𝑖 is equal to 1 to 3, 𝜇 this I can write here 𝑖. So 𝜇𝑖 and now if I write the 𝜇𝑖 (4) of 4,
this again, you see 0.2, 0.7 and 0.1 both are going to give us 1. Similarly, here also the
summation of the 𝑖 is equal to 𝑖 1 to 3 and then 𝜇𝑖 (5) this is again is going to give us 1.

So we see that for all the generic variable values in the universe of discourse, we are getting
the summation of all the membership values in the corresponding fuzzy sets in the term
set is equal to 1. So this way, we can say for these 3 fuzzy sets these 3 discrete fuzzy sets,
the orthogonality the orthogonality condition holds for the term set containing 𝐴, 𝐵 and 𝐶.

(Refer Slide Time: 10:33)

So this way we can say that the both the all the 3 discrete fuzzy sets are basically helping
the term set to be the orthogonal. Now you can see the solution in more detail here, on the
same lines it has been explained and we can have one more example here.

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(Refer Slide Time: 10:56)

So we are assuming that we have a term set we have a term set 𝑇(𝑥) which is nothing but
which has 𝐴, 𝐵 and 𝐶.

(Refer Slide Time: 11:21)

So, we assume that these term set 𝑇(𝑥) is orthogonal. So here, we assume that this term
set. So here we are assuming. So assuming that term set 𝑇(𝑥) which has the linguistic
values 𝐴, 𝐵, 𝐶 and 𝐴, 𝐵, 𝐶 are nothing but the fuzzy sets.

768
So out of these 𝐴, 𝐵, 𝐶, 2 fuzzy sets 𝐴 and 𝐵 are which is again in the universe of discourse
here as mentioned 1 to 6 and they are these 𝐴 and 𝐵 are known to us. Now the question is
if the term set 𝑇(𝑥) is orthogonal then what is 𝐶? What is the fuzzy set 𝐶? So its very
simple, we apply the same condition here for orthogonality and since we already know
that we have 2 fuzzy sets 𝐴 and 𝐵 and all of the membership values corresponding the
generic variable values which are known.

Now we can have a fuzzy set 𝐶 and since it is already said that the term set is orthogonal,
so we can quickly manage to find the third fuzzy set applying this condition. The 𝜇𝑡𝑖 (𝑥)
and the summation i is equal to 1 to n. So, here we have 3 fuzzy sets. So, n is equal to 3
here A, A, B and C.

So, let us now quickly go through we have this fuzzy sets this 3 fuzzy sets, I can again
write the 𝑥 is nothing, but this 𝐴, 𝐵, 𝐶.

(Refer Slide Time: 13:30)

So these 3 fuzzy sets we have and 𝐴 is known, 𝐵 is known. So 𝐶 can be known by subtract
by having 1 minus these membership values for the corresponding membership for the
corresponding generic variable value.

So, we can assume here the 𝐶 is going to be the 𝜇𝐶 (𝑥)/1 and then we can have similarly
the 𝜇𝐶 , we can write like this. There is let us say 𝐶. So 𝐶 can be 𝜇𝐶 (1) and then we can

769
have 𝜇𝐶 (2)/2 then we can have 𝜇𝐶 (3)/3 then we can have 𝜇𝐶 (4)/4 then we can have
𝜇𝐶 (5)/5 then we can have 𝜇𝐶 (6)/6.

So now here, we are we can compute all these respective membership values 𝜇𝐶 (1), 𝜇𝐶 (2),
𝜇𝐶 (3), 𝜇𝐶 (4), 𝜇𝐶 (5), 𝜇𝐶 (6) and this way we can represent it like 𝐶 is equal to

𝐶 = ∑ 𝜇𝐶 (𝑥)/𝑥
𝑋

So this can be written like this here and when we use the condition, this condition here we
can find the 𝐶 is equal to 0.9/1 + 0.4/2 + 0/3 + 0.2/4 + 0.3/5 + 0/6. So the condition
here is that the sum of all the values of memberships corresponding to the generic variable
1 here.

So, for 1, for 2, for 3, for 4, for 5, for 6 all the generic variable values the corresponding
membership values are known now and when you sum these values. So at any point at any
point of the generic variable value, the summation of the membership values are 1. So this
way, given the condition that a 𝑇(𝑥) which is comprising of 𝐴, 𝐵 and 𝐶 fuzzy sets of the
linguistic values and if 𝐴 and 𝐵 are given 𝐶 can be computed if 𝑇(𝑥) is orthogonal.

(Refer Slide Time: 17:33)

So, it’s very simple and the solution is here, you can further extend it. So now we have an
other example here. We have.

770
(Refer Slide Time: 17:55)

So before this, let me just quickly go through this and see that the with the orthogonality
we are getting the 𝐶 very easily.

(Refer Slide Time: 18:05)

Okay so let us now take the continuous fuzzy sets or means a term set which has multiple
continuous fuzzy sets. So here the same formula applies means at any point of time or any
point of generic variable value, the summation of membership values given the universe
of discourse are within the universe of discourse this going to be 1.

771
So we can clearly see that if we have a term set let us say 𝑇(𝑎𝑔𝑒) and this term set has 3
linguistic values which is which are 𝑦𝑜𝑢𝑛𝑔, 𝑚𝑖𝑑𝑑𝑙𝑒 − 𝑎𝑔𝑒𝑑 and 𝑜𝑙𝑑. And these 3
linguistic values are represented by or characterized by fuzzy sets. So if we have this as
𝑦𝑜𝑢𝑛𝑔 and this as 𝑚𝑖𝑑𝑑𝑙𝑒 𝑎𝑔𝑒𝑠, 𝑚𝑖𝑑𝑑𝑙𝑒 𝑎𝑔𝑒𝑑 and this as the 𝑜𝑙𝑑.

So we have 3 fuzzy sets. So now we see within the universe of discourse 0 to 90 which is
given. Within this, at any point of the generic variable we see that when we sum the
membership value corresponding to a particular membership value we get its summation
always 1. So the young fuzzy set has the trapezoidal membership function and 𝑚𝑖𝑑𝑑𝑙𝑒 −
𝑎𝑔𝑒𝑑 also has the membership function trapezoidal and for 𝑜𝑙𝑑 also, we have trapezoidal
as the membership function.

But 𝑦𝑜𝑢𝑛𝑔 and 𝑜𝑙𝑑 are open trapezoidal means left open, 𝑦𝑜𝑢𝑛𝑔 has left open trapezoidal,
𝑜𝑙𝑑 is right open trapezoidal. So this way, we when we see that we at any point of time.
So if we let us say if we draw a line here and let us say here for 𝑥 is equal to let us say 43.
So, for 43 we see that we have the membership value. So, let us say 𝜇(43), I am just
writing here.

For 𝑥 is equal to 43 the 𝜇𝑦𝑜𝑢𝑛𝑔 (43) is 0. 𝜇𝑚𝑖𝑑𝑑𝑙𝑒 − 𝑎𝑔𝑒𝑑(43) is 1. You can see here the
membership value corresponding to 𝑥 is equal to 43.

And then we have 𝜇𝑜𝑙𝑑 this is 𝜇𝑜𝑙𝑑 (43) is equal to 0. So now, when we sum this for 𝑥 is
equal to point for 𝑥 is equal to 43, we see that we are getting all 𝜇’s for 43, let us say this
is 1 to 3. So this is here and then we are getting 1. So likewise, if we move this line
throughout within the universe of discourse, we are getting at each and every place the
summation of all the membership values unity.

772
(Refer Slide Time: 22:19)

So this way here you can see every place we are getting the summation of its membership
values 1. Similarly, if we take this value which is 𝑥 is equal to 30, the generic variable
value is equal to 30. So we see that for generic variable value that is 𝑥 is equal to 30. What
is happening? So what is happening here is that 𝜇𝑦𝑜𝑢𝑛𝑔 (30) is.

You see here, just look at the membership function characterizing the fuzzy set young
which is represented by the red colour. So no which is represented by the yellow colour.
This is sorry, blue colour. So this is characterized by this is represented by this mu this is
the fuzzy set young. This represented by blue colour. So if we look at this at 𝑥 is equal to
30 we are getting its membership value 0.5.

Similarly now so at 𝑥 is equal to 30 as the membership as the generic variable value, we

see that we are coming across 2 membership functions. So first is the young and the second
one is middle aged. So we see that which is represented by red colour and then when we
see that 𝜇𝑚𝑖𝑑𝑑𝑙𝑒 𝑎𝑔𝑒𝑑 (30), we are getting this also as 0.5.

So no other membership is available here for 𝑥 is equal to 30. So we can write here 𝜇𝑜𝑙𝑑 ,
so this is for 30 or at 30 this is at 30, 0. So when we sum these 3, when we take this
summation of these mu all 𝜇’s at 30, 𝑖 equal to 1 to 𝑖 is equal to 1 to 3, we see that this is
1.

773
So similarly, for this these terms for this term set having the fuzzy sets 𝑦𝑜𝑢𝑛𝑔, 𝑚𝑖𝑑𝑑𝑙𝑒 −
𝑎𝑔𝑒𝑑, 𝑜𝑙𝑑, we see that within the universe of discourse no matter what value you take for
𝑥 as the generic variable value for each and every value of 𝑥, we are getting the summation
of its corresponding membership values 1. And this way we can say the term set here given
is the orthogonal term set. So orthogonality condition for this term set is satisfied.

(Refer Slide Time: 26:01)

Similarly, here if we take 𝑥 is equal to 60. So for this also we see that we are coming across
2 membership functions. For 2 membership functions of 𝑚𝑖𝑑𝑑𝑙𝑒 − 𝑎𝑔𝑒𝑑 and 𝑜𝑙𝑑,
respectively. So we see that both of these we have 0.5 as the membership values and when
we see that for 𝑥 is equal to 60 when we sum these sum all the membership values.

We find 0.5 for 𝑚𝑖𝑑𝑑𝑙𝑒 − 𝑎𝑔𝑒𝑑 and 0.5 for 𝑜𝑙𝑑 and the summation is 1. And please
understand that at 𝑥 is equal to, at 𝑥 is equal to 60, 𝜇𝑦𝑜𝑢𝑛𝑔 is 0. I can write here 𝜇𝑦𝑜𝑢𝑛𝑔 (60),
𝜇𝑚𝑖𝑑𝑑𝑙𝑒 𝑎𝑔𝑒𝑑 (60) and then 𝜇𝑜𝑙𝑑 (60). So this is going to give us the summation 1.

774
(Refer Slide Time: 27:52)

So this way, we can say that the sum of the membership values for linguistic values
𝑦𝑜𝑢𝑛𝑔, 𝑚𝑖𝑑𝑑𝑙𝑒 − 𝑎𝑔𝑒𝑑 and 𝑜𝑙𝑑 are 1 for every generic variable value belonging into the
universe of discourse 𝑋. And therefore, the term set which is for age is orthogonal.

(Refer Slide Time: 28:19)

So that is how we test the orthogonality of the term set, now at this point we will stop and
in the next lecture, we will discuss the fuzzy rules and fuzzy reasoning.

Thank you very much.

775
 Fuzzy Sets, Logic and Systems and Applications
Prof. Nishchal K. Verma
Department of Electrical Engineering
Indian Institute of Technology, Kanpur

Lecture - 48
Fuzzy Rules and Fuzzy Reasoning

Welcome to lecture number 48th of Fuzzy Sets, Logic and Systems and Applications. In
this lecture we are going to discuss Fuzzy Rules and Fuzzy Reasoning.

(Refer Slide Time: 00:18)

So, in this lecture we are going to first take up the fuzzy if-then rule. So, let me before
going ahead tell you that fuzzy if-then rule is an essential component of a fuzzy system.
So, without fuzzy if and then rules there is no fuzzy system.

So, this means that in any fuzzy system we must have a set of fuzzy 𝑖𝑓 − 𝑡ℎ𝑒𝑛 rules. A
fuzzy 𝑖𝑓 − 𝑡ℎ𝑒𝑛 rule is also known as fuzzy rule, fuzzy implication or fuzzy conditional
statement. So, fuzzy 𝑖𝑓 and 𝑡ℎ𝑒𝑛 rule has a form of 𝐼𝐹 𝑥 𝑖𝑠 𝐴, 𝑇𝐻𝐸𝑁 𝑦 𝑖𝑠 𝐵. So, this is the
syntax of syntax of any fuzzy 𝑖𝑓 and 𝑡ℎ𝑒𝑛 rules. So, here we have two components first
component of this fuzzy 𝑖𝑓 − 𝑡ℎ𝑒𝑛 rule is the antecedent or premise.

So, this part is called the antecedent or premise and then the second part which is just after
then is called the consequence or conclusion. So, in any fuzzy 𝑖𝑓 − 𝑡ℎ𝑒𝑛 rule will have
antecedent part or premise part and consequent part or conclusion part? And it is very

776
important here to note that antecedent part in fuzzy 𝑖𝑓 and 𝑡ℎ𝑒𝑛 rule is always fuzzy, this
means that 𝐴 is always fuzzy, 𝐴 is a linguistic value and which is always a fuzzy quantity,
a fuzzy set.

Whereas we have the consequent part or conclusion part and this consequent and or
conclusion part has 𝐵 and this 𝐵 can be either can be either a fuzzy set or a crisp value.
And this crisp value can be expressed in terms of some function which is actually the
function of the generic variable. So, we can say that the 𝐴 that has been taken here in this
fuzzy rule fuzzy 𝑖𝑓 − 𝑡ℎ𝑒𝑛 rule.

𝐴 is a linguistic value 𝐴 is a linguistic value characterized by a fuzzy set with the universe
of discourse 𝑋 and 𝐵 here 𝐵 here can be either 𝐵 here can be either a linguistic value and
of course we all know that linguistic value can always be represented by a fuzzy set. So,
𝐴 is a linguistic value 𝐴 is a linguistic value characterized by a fuzzy set, whereas 𝐵 can
be either a linguistic value that means the fuzzy set or a crisp value expressed in terms of
a function of linguistic variables used in this for example, here the linguistic variable is 𝑥,
so this is the linguistic variable all right.

So, now, often 𝑥 is A is called antecedent or premise as I have already mentioned here,
while 𝑦 is 𝐵 is called the consequence or the conclusion. So, this must be understood very
clearly and as I have already mentioned the fuzzy if and then rule is very important
component of any fuzzy system and without a set of fuzzy if and then rule the fuzzy system
cannot be cannot exist.

777
(Refer Slide Time: 06:31)

So, let us now move ahead and we discuss a fuzzy 𝑖𝑓 − 𝑡ℎ𝑒𝑛 rule where we have let us
say the fuzzy 𝑖𝑓 and 𝑡ℎ𝑒𝑛 rule like this like if 𝑥 𝑖𝑠 𝐴 𝑡ℎ𝑒𝑛 𝑦 𝑖𝑠 𝑏. So, let us first understand
that this can also be represented by 𝐴 𝑒𝑛𝑡𝑎𝑖𝑙𝑠 𝐵 𝑜𝑟 𝐴 𝑐𝑜𝑢𝑝𝑙𝑒𝑑 𝑤𝑖𝑡ℎ 𝐵. So, please
understand here as it is written here that there are two ways to interpret a fuzzy 𝑖𝑓 − 𝑡ℎ𝑒𝑛
rule. First way is 𝐴 𝑐𝑜𝑢𝑝𝑙𝑒𝑑 𝑤𝑖𝑡ℎ 𝐵 or the other way I mean the second way is
𝐴 𝑒𝑛𝑡𝑎𝑖𝑙𝑠 𝐵.

So, fuzzy 𝑖𝑓 − 𝑡ℎ𝑒𝑛 rule when it is written in that syntax that I have already mentioned
like if A x A is A, 𝑥 is 𝐴 then 𝑦 is 𝐵. So, this can also be interpreted like this like 𝐴 coupled
with 𝐵 or 𝐴 entails with 𝐵. So, assuming 𝐴 and 𝐵 both are fuzzy sets. So, let us please
understand let us assume here that 𝐴 and 𝐵 both are the fuzzy sets. So, when we assume
𝐴 and 𝐵 means for 𝐴 and 𝐵 for 𝐴 and 𝐵 both fuzzy sets all right.

So, please understand why are we saying this that for 𝐴 and 𝐵 both fuzzy set because 𝐵
can be here in fuzzy 𝑖𝑓 and 𝑡ℎ𝑒𝑛 rule 𝐵 can be either crisp or fuzzy, but here we are
assuming for this discussion that we have both 𝐴 and 𝐵 fuzzy sets. So, assuming 𝐴 and 𝐵
both fuzzy sets if and then rule can be defined as a fuzzy relation. So, this is very important
to note when we have 𝐴 and 𝐵 both fuzzy sets. So, in this case this 𝐴 𝑒𝑛𝑡𝑎𝑖𝑙𝑠 𝐵
𝐴 𝑐𝑜𝑢𝑝𝑙𝑒𝑠 𝑐𝑜𝑢𝑝𝑙𝑒𝑑 𝑤𝑖𝑡ℎ 𝐵 can be regarded as a relation.

And this is represented by 𝑅 on a space 𝑋 × 𝑌 where this 𝑋 is the universe of discourse

for the generic variable 𝑥. And 𝑌 is the universe of discourse for generic variable small y

778
if 𝐵 is the fuzzy set. So, this needs to be noted. So, when we have a relation when we have
a fuzzy relation in between 𝐴 and 𝐵 we can always write this as 𝐴 × 𝐵 we have already
done this. So, 𝑅 = 𝐴 𝑒𝑛𝑡𝑎𝑖𝑙𝑠 𝐵 = 𝐴 × 𝐵 you can see here.

So, this way we can write that for continuous fuzzy set this is for the continuous fuzzy set
that we have the

𝑅 = 𝐴 → 𝐵 = 𝐴 × 𝐵 = ∫ 𝜇𝑅 (𝑥, 𝑦)/(𝑥, 𝑦), ∀𝑥, 𝑦 ∈ 𝑋 × 𝑌

𝑋×𝑌

And similarly for discrete for discrete here we have the representation

𝑅 = 𝐴 → 𝐵 = 𝐴 × 𝐵 = ∑ 𝜇𝑅 (𝑥, 𝑦)/(𝑥, 𝑦), ∀𝑥, 𝑦 ∈ 𝑋 × 𝑌

𝑋×𝑌

So, what we have seen here is that if we have in fuzzy if and then rule both 𝐴 and 𝐵. 𝐴 is
coming from the antecedent part or premise part and 𝑌 and 𝐵 is coming from the
consequence or conclusion part. So, if we have 𝐴 and 𝐵 both fuzzy sets that are coming
from fuzzy rule if and then fuzzy rule then we can write the 𝑅 relation fuzzy set in terms
of 𝐴 and 𝐵; that means, 𝐴 × 𝐵.

And 𝐴 cross 𝐵 for continuous we have seen here and for discrete we have seen here as to
how we can write.

(Refer Slide Time: 12:22)

779
We have already done we have already discussed in detail a fuzzy relation. So, we can
accordingly manage to write the relation fuzzy relation matrix. So, if 𝐴 𝑒𝑛𝑡𝑎𝑖𝑙𝑠 𝐵 is
interpreted as or 𝐴 𝑐𝑜𝑢𝑝𝑙𝑒𝑑 𝑤𝑖𝑡ℎ 𝐵. So, both are same so, 𝐴 𝑒𝑛𝑡𝑎𝑖𝑙𝑠 𝐵, 𝐴 → 𝐵 is
interpreted as 𝐴 𝑐𝑜𝑢𝑝𝑙𝑒𝑑 𝑤𝑖𝑡ℎ 𝐵 then it can be interpreted by a fuzzy relation 𝑅 as we
have just discussed.

So, for continuous so, for continuous the relation can be written as 𝑅 can be written as here
𝐴 𝑐𝑜𝑢𝑝𝑙𝑒𝑑 𝑤𝑖𝑡ℎ 𝐵 is equal to 𝐴 × 𝐵 means the Cartesian product, simple Cartesian
product as we have already discussed in previous lectures. So, here the 𝜇(𝑥, 𝑦) how will
we find this 𝜇(𝑥, 𝑦) is here. So, this 𝜇(𝑥, 𝑦) you see here 𝜇(𝑥, 𝑦) this 𝜇(𝑥, 𝑦) we can get
by suitably taking the Cartesian product means we can we can get by taking by using the
T-norm.

So, here this is nothing, but this gives us 𝜇𝑅 (𝑥, 𝑦). So, this we can get by taking the T-
norm of 𝐴 and 𝐵 means we take the T-norm of 𝜇𝐴 (𝑥) and 𝜇𝐵 (𝑦). So, as we have already
done this T-norm in previous lectures. So, the fuzzy relation 𝑅 and 𝑇 let us say we have
two fuzzy relations sets right.

And then accordingly we can have the T-norms, T-norm operators we can apply. And then
we can say that there are four different fuzzy relations which are defined using four
commonly used T-norm operators what does this mean this means that we have we have
in T-norm we have multiple we have multiple T-norm operators. So, we have four
commonly used T-norm operators first is 𝐴 𝑐𝑜𝑢𝑝𝑙𝑒𝑑 𝐵 using minimum T-norm operator
and then the second is the 𝐴 𝑐𝑜𝑢𝑝𝑙𝑒𝑑 𝐵 is using algebraic product T-norm operator then
we have the third one is 𝐴 𝑐𝑜𝑢𝑝𝑙𝑒𝑑 𝐵 using bounded product T-norm operator.

And then we have the 𝐴 𝑐𝑜𝑢𝑝𝑙𝑒𝑑 𝐵 using drastic some product operator. So, this way we
can use any of these T-norm operators suitably to get the 𝜇𝑅 (𝑥, 𝑦) and this was for the
continuous on the same lines we can get the 𝑅 for discrete.

780
(Refer Slide Time: 16:36)

Relation matrix, the relation in between 𝐴 and 𝐵 for discrete fuzzy sets.

So, let us now go one by one. So, fuzzy rule interpretation as 𝐴 𝑐𝑜𝑢𝑝𝑙𝑒𝑑 𝐵 here and this
𝐴 and 𝐵 are coming from the if and then fuzzy rule. So, let us take the first case where we
have the minimum T-norm operator. So, we have let us first type of T-norm operator. So,
first type of T-norm operator is min.

So, for continuous fuzzy set fuzzy sets 𝐴 and 𝐵 where this 𝑅 is the relation fuzzy set which
is coming out by coming out from 𝐴 × 𝐵. So, the Cartesian product of 𝐴 and 𝐵. So, we
write here the relation fuzzy relation set as 𝑅𝑚𝑖𝑛 because min is here denoting that the
minimum T-norm operator that is being used for getting the fuzzy relation in between 𝐴
and 𝐵.

So, this can be represented by this and similarly when 𝐴 and 𝐵 are is discrete fuzzy sets
then there 𝑅𝑚𝑖𝑛 can be written as this. So, here we have the fuzzy relation set when we use
min. So, we can write min T-norm and then for discrete here we have when we have 𝐴 and
𝐵 both are discrete. So, this way we can represent the fuzzy relation matrix sorry fuzzy
relation set and when we have instead of min as the T-norm the algebraic product T-norm.

So, when the when we have algebraic product T-norm operator then we simply instead of
taking min we write we take the product in between 𝜇𝐴 and 𝜇𝐵 you can see here.

781
(Refer Slide Time: 19:18)

So, we take the product here we take the product here. So, 𝜇𝐴 (𝑥) × 𝜇𝐵 (𝑦). So, we take the
product in case of algebraic product T-norm operator. So, and we write this as 𝑅𝑎𝑝 you
can see here and similarly for discrete 𝑅𝑎𝑝 . So, for discrete 𝐴 and 𝐵 for a discrete fuzzy
sets 𝐴 and 𝐵 we write the relation fuzzy set by 𝑅𝑎𝑝 which is nothing, but 𝐴 × 𝐵 summation
𝑋 × 𝑌 as the universe of discourse

And then here again we have the product here again we have the product in between 𝜇𝐴 (𝑥)
and 𝜇𝐵 (𝑦) and then rest of the things remain the same.

So, only difference here is that the, this operator here this operator gets change. So, when
we have min we use we simply take the min when we use min T-norm we take the min
sign here and inverted when we take min it means we take the minimum of 𝜇𝐴 (𝑥) and
𝜇𝐵 (𝑦) and when we use product then simply we multiply.

782
(Refer Slide Time: 20:48)

Similarly, when we take the bounded product T-norm so, we use this formula we have
already done this when we have discussion discussed the all T-norm operators. So, here
we have bounded product. So, for bounded product in between bounded product of 𝜇𝐴 (𝑥)
and 𝜇𝐵 (𝑦) we write 0. And then the max sign of means the 0 ∨ (𝜇𝐴 (𝑥) + 𝜇𝐵 (𝑦) − 1),
similarly when we have discrete fuzzy set we write it this way.

So, let us understand now that as to how the various T-norm operators change the
computations. So, you here you see here the fourth one is 𝐴 𝑐𝑜𝑢𝑝𝑙𝑒𝑑 𝐵 using drastic T-
norm operators. So, here when we have drastic operators we use drastic computation for
drastic product computation this function. So, when we use this for 𝜇𝐴 (𝑥) and 𝜇𝐵 (𝑦) we
can write it like this 𝜇𝑅𝑑𝑝 (𝑥, 𝑦) and this 𝜇𝑅𝑑𝑝 (𝑥, 𝑦) can be computed by the drastic product
T-norm.

So, this way we have seen that as to how we can manage to get the fuzzy relations set by
using all the four types of T-norm operators.

783
(Refer Slide Time: 22:56)

And if we have here an we take an example here where we have a high speed S which is
characterized by a fuzzy set. So, 𝑆𝐻𝑖𝑔ℎ is the set with the universe of discourse 𝑆 =
(20, 25, 30, 45, 50). And we have another fuzzy set here 𝑃𝐻𝑖𝑔ℎ and with the universe of
discourse 1, 2, 3, 4. So, we have two we have two discrete fuzzy sets. So, so both I can
write here both the fuzzy sets are the discrete fuzzy sets.

So, 𝑆𝐻𝑖𝑔ℎ and 𝑃𝐻𝑖𝑔ℎ both are discrete fuzzy sets where 𝑆, 𝑃 both represent the speed and
brake pressure respectively. So, determined the implication relation, that means that
relation fuzzy set for this. So, 𝑆𝐻𝑖𝑔ℎ and then here 𝑆𝐻𝑖𝑔ℎ 𝑐𝑜𝑢𝑝𝑙𝑒𝑑 𝑤𝑖𝑡ℎ 𝑃𝐻𝑖𝑔ℎ using the
interpretation the same interpretation that we have just discussed.

784
(Refer Slide Time: 24:25)

So, let us quickly use all four operators four T-norm operators.

And let us see how we are getting various relation fuzzy sets. So, we have four T-norm
operators. So, for the first T-norm operator that is the min T-norm operator we can quickly
find the 𝑅𝑚𝑖𝑛 the relation fuzzy set the relation fuzzy set 𝑅 and since we are using min T-
norm operators. So, we use the min here see the here we use min operator as the T-norm.
So, we have already done this exercise in previous lecture. So, when we do that we are
going to get this as the fuzzy relation matrix. So, this is fuzzy relation matrix 𝑅𝑚𝑖𝑛 .

(Refer Slide Time: 25:46)

785
Similarly, when we use the algebraic T-norm operator here, so, then now the computation
becomes little bit different, so instead of taking min of the membership values we multiply
the respective membership value you can see here, so like 0.2 multiplied by 0.4. So, this
is the case when we have the algebraic product. So, we simply take the product of the
membership values.

So, this fuzzy relation matrix 𝑅𝑎𝑝 . So, please understand that this 𝑅𝑎𝑝 is a fuzzy relation
set which is represented in the matrix form. So, 𝑅𝑎𝑝 is a fuzzy relation set. So, fuzzy
relation set 𝑅𝑎𝑝 which is represented in the form of a matrix. So, when we take the
multiplication when we do the multiplication in between the respective membership
values. So, now, what we are getting is this. So, this is nothing but the fuzzy relation
matrix, fuzzy relation matrix 𝑅𝑎𝑝 when we have used the algebraic product T-norm.

(Refer Slide Time: 27:40)

Similarly, let us now take the third type of T-norm operator. So, when we take the bounded
product T-norm let us see what happens. So, when we take bonded product T-norm so, we
use this formula this relation this expression for finding the bonded product T-norm and
this is nothing, but we take the max in between 0 ∨ (𝜇𝑆𝐻𝑖𝑔ℎ (𝑠) + 𝜇𝑃𝐻𝑖𝑔ℎ (𝑝) − 1). So, this

way when we apply this to all the pair of membership values here you can see all these
values.

786
And please note that when we are taking the when we are finding the relation in between
𝐴 and 𝐵 discrete fuzzy set here and since we are taking the 𝐴 × 𝐵 for getting the relation
fuzzy set 𝑅𝑏𝑝 maybe whatever type of T-norm we apply, but 𝐴 and 𝐵 must be multipliable.
So, this means the order of 𝐴 and 𝐵 should be in such a way that they can be multiplied.
So, these two matrices can be multiplied.

So, whatever we are getting after this is here. So, when we use the bounded product T-
norm the relation the fuzzy relation fuzzy relation matrix is if it if I write it by 𝑅𝑏𝑝 . So,
this is fuzzy relation matrix and this comes out when we you in between when we use 𝐴
and 𝐵 discrete fuzzy sets and we use the bounded product T-norm operator.

(Refer Slide Time: 29:59)

So, now let us take the fourth kind of T-norm operator that is drastic product T-norm
operator and we all know that when we use drastic product in our operator this expression
applies this means that if 𝜇𝑃𝐻𝑖𝑔ℎ (𝑝) is 1 then 𝑅𝑑𝑝 (𝑠, 𝑝) = 𝜇𝑆𝐻𝑖𝑔ℎ (𝑠). And then if we have

𝜇𝑆𝐻𝑖𝑔ℎ (𝑠) is equal to 1 then 𝑅𝑑𝑝 (𝑠, 𝑝) = 𝜇𝑃𝐻𝑖𝑔ℎ (𝑝), otherwise the 𝑅𝑑𝑝 (𝑠, 𝑝) = 0. So, when

we do that when we apply this condition then the 𝑅𝑑𝑝 which is here which is the fuzzy
relation matrix based on the drastic product T-norm operator they get here 𝑅𝑑𝑝 .

787
(Refer Slide Time: 31:26)

So, this way we have seen that as to how we can get the fuzzy relation matrix by using
various kinds of T-norms all four kinds of T-norms that we have used in previous lectures.
So, the first was the minimum T-norm operator and then the second one was the algebraic
product T-norm operator the third one was the bounded product T-norm operator and the
fourth one was the drastic product T-norm operator.

So, all these four types of T-norm operators we have already studied and we are applying
this here to get the relation built by using all these four kinds of operators T-norm
operators.

788
(Refer Slide Time: 32:19)

So, with this I would like to stop here in the next lecture we will continue with the fuzzy
rules and fuzzy reasoning.

Thank you.

789
 Fuzzy Sets, Logic and Systems and Applications
Prof. Nishchal K. Verma
Department of Electrical Engineering
Indian Institute of Technology, Kanpur

Lecture - 49
Fuzzy Rules and Fuzzy Reasoning

Welcome to the lecture number 49 of Fuzzy Sets, Logic and Systems and Applications. In
this lecture, we will discuss Fuzzy Rules and Fuzzy Reasoning.

(Refer Slide Time: 00:17)

So, here when we talk of fuzzy rules, fuzzy rules are always in the form of if the linguistic
variable is some linguistic value, then again the linguistic variable as the output is some
linguistic value. So, the fuzzy rule has a particular syntax, the syntax is like this. Like if
we have any fuzzy rule, fuzzy rule will look like this fuzzy rule will have 𝐼𝐹 part and
𝑇𝐻𝐸𝑁 part. So, any fuzzy rule will have 𝐼𝐹 and 𝑇𝐻𝐸𝑁 both the words and please
understand that any fuzzy system will have a set of fuzzy rules. Without a set of fuzzy
rules, there can be no fuzzy system or in other words there can never exist any fuzzy system
without a set of fuzzy rules.

So, a fuzzy if-then rule like 𝐼𝐹 𝑥 𝑖𝑠 𝐴 𝑡ℎ𝑒𝑛 𝑦 𝑖𝑠 𝐵 and we already know that
𝐼𝐹 𝑥 𝑖𝑠 𝐴 is called as antecedent or premise and in this rule the 𝑇𝐻𝐸𝑁 𝑦 𝑖𝑠 𝐵 is
called as consequence or conclusion part. So, any fuzzy rule has two parts, the first part is

790
the premise part or the antecedent part and the second part is the conclusion part or the
consequence part.

So, this fuzzy rule in this form, the fuzzy rule 𝐼𝐹 𝑥 𝑖𝑠 𝐴 𝑇𝐻𝐸𝑁 𝑦 𝑖𝑠 𝐵 can be written
as 𝐴 → 𝐵. So, this means we have 𝐴 and then 𝐵 this means that if a particular fuzzy
variable or the linguistic variable belongs into a particular fuzzy reason that is the linguistic
value that is represented by a fuzzy set, then what is 𝐵? What is the output? what is the
linguistic value out of it? So, this how it is written. So, 𝐴 → 𝐵 is also written for a
particular fuzzy rule.

So, there are two ways to interpret a fuzzy 𝐼𝐹 − 𝑇𝐻𝐸𝑁 rule. First is
𝐴 𝑐𝑜𝑢𝑝𝑙𝑒𝑑 𝑤𝑖𝑡ℎ 𝐵 and then the second one is 𝐴 𝑒𝑛𝑡𝑎𝑖𝑙𝑠 𝐵. So, here are the two ways
as I mentioned, 𝐴 𝑐𝑜𝑢𝑝𝑙𝑒𝑑 𝑤𝑖𝑡ℎ 𝐵; 𝐴 𝑒𝑛𝑡𝑎𝑖𝑙𝑠 𝐵. So, this suggests that a fuzzy 𝐼𝐹 −
𝑇𝐻𝐸𝑁 rule can be defined as a fuzzy relation here. A fuzzy relation and we already know
that how we get a fuzzy relation from fuzzy sets.

So, if we have a; if we have a set of fuzzy sets like if we have two fuzzy sets 𝐴 and 𝐵
and if we are interested in finding the relation fuzzy set. So, relation fuzzy set will be
nothing, but the Cartesian product of these two fuzzy sets. So, and then of course, the
universe of discourse of this fuzzy relation set will be the cross product of the universe of
discourse of the first fuzzy set and the second fuzzy set.

So, for any continuous fuzzy sets 𝐴 and 𝐵, if we are interested in the relation fuzzy set
say 𝑅 so 𝐴 so, this 𝑅 can be written as the fuzzy relation set can be written as a forward
arrow, that means the 𝐴 𝑐𝑜𝑢𝑝𝑙𝑒𝑑 𝑤𝑖𝑡ℎ 𝐵 = 𝐴 × 𝐵 and we can always write the same
with this expression. We have already done in the previous lectures.

So, integral sign and then the universe of discourse of it will be X cross Y and then we
have mu r of x, y oblique x comma y and then for every x comma y will be belonging into
the universe of discourse of X capital X cross capital Y. So, this is for the continuous one
for the continuous fuzzy relation set.

Similarly, if we have A and B both are discrete fuzzy sets then of course, the fuzzy relation
set will be the discrete set, discrete fuzzy sets. So, here we have already done this in the
previous lectures, so I am not going into this again. So, here this is the fuzzy relation set
fuzzy, in other words I can say the discrete fuzzy relation set discrete fuzzy relation set R.

791
So, that is how we get fuzzy relation set out of a fuzzy set A and fuzzy set B by taking the
cross product of it or I would say the Cartesian product of it.

(Refer Slide Time: 07:20)

So, here let us now go one by one. So, fuzzy rule interpretation when we do with by using
𝐴 𝑐𝑜𝑢𝑝𝑙𝑒𝑑 𝑤𝑖𝑡ℎ 𝐵. So, when we interpret A, 𝐴 → 𝐵 arrow forward, 𝐴 → 𝐵 the
interpret by 𝐴 𝑐𝑜𝑢𝑝𝑙𝑒𝑑 𝑤𝑖𝑡ℎ 𝐵 then it can be interpreted by a fuzzy relation R as I have
mentioned here. R we know how we how did we get, see here. So, the same will involve
here when we say A coupled with B will involved the T-norm, T-norm or S- co norm

So, here also we have T-norm in the discrete version of it. So, we can simply write the
fuzzy relation set by R and then we use T-norm when we interpret the fuzzy rule by
𝐴 𝑐𝑜𝑢𝑝𝑙𝑒𝑑 𝑤𝑖𝑡ℎ 𝐵. And since we are using T-norm here and we know that we have four
types of T-norms normally available. So, we can use here the same as
𝐴 𝑐𝑜𝑢𝑝𝑙𝑒𝑑 𝑤𝑖𝑡ℎ 𝐵 for the relation fuzzy set using minimum T-norm operator. So, we
know that the first and the most basic T-norm operator is the minimum. So, we take the
minimum. The first kind of T-norm is minimum, then the second kind of T-norm here is
the algebraic product.

So, we can use the algebraic product here as one of the T-norms or S-co norms. We already
know that T-norm is also called as S-co norm and similarly S-norm is also called as T co
norms. So, the third one is when the relationship, the relation fuzzy set can be obtained by
using the bounded product as the T-norm operator.

792
Similarly the forth one here is the drastic product as the T-norm operator. So, here we can
use any of the four T-norm operators to get the fuzzy relation set for either discrete fuzzy
sets or the continuous fuzzy sets and both the versions are here as mentioned.

(Refer Slide Time: 10:40)

Now, fuzzy rule interpretation as 𝐴 𝑒𝑛𝑡𝑎𝑖𝑙𝑠 𝐵 here. So, we have if 𝐴 → 𝐵 is

interpreted as 𝐴 𝑒𝑛𝑡𝑎𝑖𝑙𝑠 𝐵, then it can be defined by the following four different forms.
So, first form here is the material implication and the second form is propositional calculus.
The third form is extended propositional calculus, the fourth form is the generalization of
modus ponens. So, material implication is represented by 𝑅𝑚𝑖 = 𝐴 → 𝐵 = ¬𝐴 ∪
𝐵 ,propositional calculus is represented by 𝑅𝑝𝑐 = 𝐴 → 𝐵 = ¬𝐴 ∪ (𝐴 ∩ 𝐵).

Similarly, we have extended propositional calculus and here this is represented by 𝑅𝑒𝑝𝑐 =
𝐴 → 𝐵 = (¬𝐴 ∩ ¬𝐵) ∪ 𝐵 . Then fourth one is the generalization of modus ponens
̃ 𝐵. So, 𝐴 𝑒𝑛𝑡𝑎𝑖𝑙𝑠 𝐵 here represents the fuzzy
represented by 𝑅𝑔𝑚𝑝 = 𝐴 → 𝐵 = 𝐴 <
relation 𝑅.

793
(Refer Slide Time: 12:48)

So, let us go one by one and see what is the material implication. So, material implication
is defined as I have already mentioned by 𝑅𝑚𝑖 = 𝐴 → 𝐵 = ¬𝐴 ∪ 𝐵.

So, if we have a continuous fuzzy set let say the 𝑅𝑚𝑖 will be like this and if we have a
discrete fuzzy set the 𝑅𝑚𝑖 be for discrete fuzzy set will be expressed by this expression.
Similarly the continuous one will be this. So, this is also called the Zadeh’s arithmetic rule.
This is very important to note that the Zadeh’s in arithmetic rule which follows the ¬𝐴 ∪
𝐵 by using the bounded sum operator for union.

(Refer Slide Time: 13:55)

794
Now, the propositional calculus where the 𝑅 , the fuzzy relation set 𝑅𝑝𝑐 = 𝐴 → 𝐵 =
¬𝐴 ∪ (𝐴 ∩ 𝐵). So, when we apply this to continuous fuzzy sets 𝐴 and 𝐵, we are going
to get this expression.

Similarly, for discrete fuzzy sets when we apply the propositional calculus, we get a
relation on propositional calculus 𝑅𝑝𝑐 by this expression and this is called the Zadeh’s
max-min rule. This is very important to note which follows the ¬𝐴 ∪ (𝐴 ∩ 𝐵) by using
min for intersection and max for union. So, wherever we have the min we use the wherever
we have the min we use intersection and for max we use union.

(Refer Slide Time: 15:23)

Now, the third one that is extended propositional calculus. So, here we have 𝑅𝑝𝑐 and
𝑅𝑝𝑐 = 𝐴 → 𝐵 = (¬𝐴 ∩ ¬𝐵) ∪ 𝐵. So, when we apply this for continuous fuzzy sets 𝐴
and 𝐵 , the relation fuzzy set which is 𝑅𝑒𝑝𝑐 , the extended the relation fuzzy set for
extended propositional calculus will become 𝑅𝑒𝑝𝑐 and this is represented by this is found
by this expression for continuous fuzzy sets.

Similarly, for discrete fuzzy sets for discrete fuzzy sets, we get like this. The 𝑅𝑒𝑝𝑐 we get
like this and this is called the Boolean fuzzy implication; this is called the Boolean fuzzy
implication using max for ∪ means wherever we have max wherever we have a union,
we use max.

795
(Refer Slide Time: 17:09)

So, then comes the forth one here, the generalization of modus ponens and the fuzzy
̃ 𝐵 and this is
relation set that comes out of this is represented by 𝑅𝑔𝑚𝑝 = 𝐴 → 𝐵 = 𝐴 <
represented by 𝑅𝑔𝑚𝑝 . And here this is for the continuous fuzzy set where we have the
𝑅𝑔𝑚𝑝 (𝑥, 𝑦)/(𝑥, 𝑦) here. So, 𝑅𝑔𝑚𝑝 is computed simply by this expression here.

And if we have a discrete one discrete fuzzy sets 𝐴 and 𝐵, then the fuzzy relation set that
is

𝑅𝑔𝑚𝑝 = ∑ 𝜇𝑅𝑔𝑚𝑝 (𝑥, 𝑦)/(𝑥, 𝑦)

𝑋×𝑌

̃ 𝐵 by using the
This is called as the Goguen’s fuzzy implication which follows 𝐴 <
algebraic product for T norm operator.

So, this way we can very easily compute the fuzzy rule interpretation as 𝐴 𝑒𝑛𝑡𝑎𝑖𝑙𝑠 𝐵
for fuzzy sets, 𝐴 and 𝐵 to get the fuzzy relation set.

796
(Refer Slide Time: 18:49)

Now, let us take an example here to understand this fuzzy rule interpretation as
𝐴 𝑒𝑛𝑡𝑎𝑖𝑙𝑠 𝐵. So, for this example the 𝑆ℎ𝑖𝑔ℎ and 𝑃ℎ𝑖𝑔ℎ where the universe of discourse
for 𝑆ℎ𝑖𝑔ℎ is this; for 𝑆 is this and for universe of discourse for 𝑃 here is 1, 2, 3.

So, we have two discrete fuzzy sets here 𝑆ℎ𝑖𝑔ℎ and 𝑃ℎ𝑖𝑔ℎ is represented by the two fuzzy
expressions, two fuzzy sets.

(Refer Slide Time: 19:37)

797
Now, since we have 𝑆ℎ𝑖𝑔ℎ and 𝑃ℎ𝑖𝑔ℎ so, we can now further go ahead and get the 𝑅𝑚𝑖 .
So, this is 𝑅𝑚𝑖 , the material implication. So, this is very simple here when we use 𝑅𝑚𝑖
so, we use for 𝑅𝑚𝑖 , we use this negation of 𝑆ℎ𝑖𝑔ℎ ∪ 𝑃ℎ𝑖𝑔ℎ here and this we already know
as to how we get here.

So, if we want to compute this, the 𝑅𝑚𝑖 simply for computing the membership values of
𝑅𝑚𝑖 ; we use the 𝜇𝑆ℎ𝑖𝑔ℎ here; 𝜇𝑆ℎ𝑖𝑔ℎ this one and 𝜇𝑃ℎ𝑖𝑔ℎ . So, both the membership values,

we take and then we compute the membership value of 𝑅𝑚𝑖 . So, 𝑅 is the relation fuzzy
set for material implication.

So, this way we get this relation matrix this relation; this fuzzy relation matrix and finally
when we compute, we are going to get the 𝑅𝑚𝑖 here like this. So, we can write here if
fuzzy relation matrix 𝑅𝑚𝑖 and mi is nothing but the material implication with we can
write here with material implication.

(Refer Slide Time: 21:28)

Similarly, we use the same two matrices, we use same fuzzy sets 𝑆ℎ𝑖𝑔ℎ and 𝑃ℎ𝑖𝑔ℎ , both
are discrete and when we use propositional calculus, so for propositional calculus we use
the ¬𝑆ℎ𝑖𝑔ℎ ∪ (𝑆ℎ𝑖𝑔ℎ ∩ 𝑃ℎ𝑖𝑔ℎ ). So, when we do that here, we get the a we get the fuzzy
relation matrix fuzzy relation matrix 𝑅𝑝𝑐 . So, this is what we are going to get after using
the propositional calculus, the expression is here. So, we know has to how we are going to

798
compute. And then finally, we are going to get 𝑅𝑝𝑐 which is the fuzzy relation matrix
based on the propositional calculus.

(Refer Slide Time: 22:47)

Now, the third one is based on extended composition. So, for computing the extended
propositional calculus, we use the, we take basically the (¬𝑆ℎ𝑖𝑔ℎ ∩ ¬𝑃ℎ𝑖𝑔ℎ ) ∪ 𝑃ℎ𝑖𝑔ℎ here.
So, when we do that, we use this expression and finally, we are getting 𝑅𝑒𝑝𝑐 here. Again
this is nothing, but the fuzzy relation matrix 𝑅𝑒𝑝𝑐 is nothing but the extended
propositional calculus.

(Refer Slide Time: 23:41)

799
Now, the forth one is generalize generalization of modus ponens. When we use this, so
̃ 𝑃𝐻𝑖𝑔ℎ , you can see here. So, since we have 𝑆𝐻𝑖𝑔ℎ and 𝑃𝐻𝑖𝑔ℎ . So, when
𝑅𝑔𝑚𝑝 = 𝑆𝐻𝑖𝑔ℎ <
we find this 𝜇𝑅𝑔𝑚𝑝 (𝑠, 𝑝), please understand this 𝑠, 𝑠 here is nothing, but it is the generic

variable and p also is the generic variable here. So, this way 𝜇𝑅𝑔𝑚𝑝 , we can compute here
with this expression with this condition here. And when we use this, we are going to get
the fuzzy relation matrix, fuzzy relation matrix 𝑅𝑔𝑚𝑝 . So, this way we get the fuzzy rule
interpretation as 𝐴 𝑒𝑛𝑡𝑎𝑖𝑙𝑠 𝐵.

(Refer Slide Time: 25:02)

So, that is how we have been able to manage to get various kinds of relation fuzzy relation
set in the form of fuzzy relation matrix and with this I would like to stop here. And in the
next lecture, we will study the fuzzy reasoning

Thank you.

800
 Fuzzy Sets, Logic and Systems and Applications
Prof. Nishchal K. Verma
Department of Electrical Engineering
Indian Institute of Technology, Kanpur

Lecture – 50
Fuzzy Inference System

(Refer Slide Time: 00:19)

Welcome, to lecture number 50 of Fuzzy Sets Logic and Systems and Applications. In this
lecture we will discuss the heart of any fuzzy system that is the Fuzzy Inference System.
So, fuzzy inference system basically is very important system, that means, the fuzzy
inference system is having certain processes that is being carried so in nutshell I will say
that the process of mapping from a given input to an output using fuzzy logic.

So, fuzzy inference system basically does what? Fuzzy inference system basically maps
the given input into the output using fuzzy logic. So, if I have any fuzzy system let us say
fuzzy inference engine, in short we call this fuzzy system as FIS. So, I can just write here
the fuzzy systems say fuzzy inference system as FIS here.

Of any fuzzy system here if I say FIS so it takes some input and it produces some output.
So, fuzzy inference system basically maps a given input to an output using fuzzy logic.
So, this FIS will take the help of what, the fuzzy logic, this is based on fuzzy logic. There
are so many such systems available which helps us in managing to map the input into the

801
output, but here the fuzzy inference system does what, it also does the same thing means
it maps the input into the output, but the logic that it uses here is the fuzzy logic.

So, basically we can say that it involves all the pieces that we have discussed in our
previous lectures. So, we have covered so many topics and the fuzzy inference engine
almost uses everything so that we have covered so far. So, some of them some of these
can be, like membership functions, fuzzy logic operators, if then rules, composition rules,
etcetera etcetera so many almost all of the topics that we have covered so far in this course
so far will be used in the fuzzy inference system.

So, here a non-linear mapping by the FIS basically it helps us in deriving its output based
on fuzzy reasoning and a set of fuzzy if-then rules. So, the domain and range of the
mapping could be the fuzzy sets are the points in a multidimensional spaces. The fuzzy
inference systems have been successfully applied in multiple domains so many domains
and basically it helps us in managing to give us a suitable model.

So, modeling, control, etcetera etcetera and then it has been applied into various domains
for example, computer vision, control, automation, data mining, machine learning etcetera
etcetera. So, here because FIS is a multidisciplinary entity, the FIS is known by its various
names so the name could be fuzzy rule based system, fuzzy expert system, fuzzy model,
fuzzy associative memory, fuzzy logic controller, fuzzy system. So, FIS is known by its
multiple names. So, because fuzzy logic can be applied to many interdisciplinary areas and
the areas using the FIS in with multiple names.

802
(Refer Slide Time: 05:08)

So, let us now understand first the building blocks of any fuzzy inference system. So, any
fuzzy inference system will have 4 building blocks. First block here is this is the first block
which is fuzzifier, the second block is the inference engine, the third block is the fuzzy
rule base, fourth block is the defuzzifier.

So, as I have already mentioned that a fuzzy inference engine a typical fuzzy inference
system basically takes the input and it maps suitably to a particular output, and this in
between the input and output we have a block which is called the fuzzy inference system,
and this fuzzy inference system has 4 blocks. First block is the fuzzifier, second block is
the inference engine, third block is the fuzzy rule base. So, fuzzy rule base or fuzzy
knowledge base this has a set of fuzzy rules and as I have already mentioned when we
discussed fuzzy rules, fuzzy rules are always in the form of if and then rules.

So, we have set of fuzzy rules available to help the inference engine and finally, fourth
block is the defuzzifier and then defuzzifier produces the suitable output or output
comparison, comparative to the input that is given to the fuzzifier. So, this is a typical
architecture, this is a typical architecture of a fuzzy inference system.

803
(Refer Slide Time: 07:27)

So, let us go one by one to all the blocks and then see what these blocks are doing for us.
So, the fuzzifier basically, in the fuzzy inference systems takes the input from the outside
world. This input can be either, this input can be either a crisp input or the fuzzy input. So,
if the input is a crisp input, if the input is a crisp input let us say, if the input is a crisp input
then the fuzzifiers job is to converts this crisp input into a suitable fuzzy value.

So, here we get some fuzzy value, some fuzzy value, means the fuzzy quantity comparable
to the crisp input that is fed to the fuzzifier or fed to the fuzzy inference system. If the input
is fuzzy input then fuzzifier normally does not do any job. So, this fuzzy quantity
automatically gets transferred here as the fuzzy value.

So, finally, the output of fuzzifier is a fuzzy value which is the input to the inference
engine. So, first block here as I already mentioned. So, first block is a fuzzifier and
fuzzifier if the input to the fuzzifier are FIS, the fuzzy inference system is crisp it produces
the fuzzy value comparable to the crisp value which is fed.

If the input is already fuzzy then fuzzifier normally doesn’t do any job if this fuzzy value
is directly transferred to the fuzzy inference engine or inference engine as its input. Now
in short we can say the fuzzifier converts the crisp input to a linguistic variable or linguistic
value here, using the membership function.

804
So, basically the fuzzifier takes the help of membership functions and these membership
functions are helping in order to convert the crisp value into the fuzzy value. So, here
basically comparable to the crisp value that is fed it produces it gives it assigns some
membership value which with the help of the membership functions.

(Refer Slide Time: 10:51)

Now, next is the, next block here is the inference engine. So, now since the fuzzy, now
since here the fuzzy quantity is already available this is a fuzzy quantity or fuzzy value is
available as input to the fuzzy inference engine or inference engine, and this inference
engine basically computes, basically inputs basically infers, based on the fuzzy value. With
the help of fuzzy knowledge base fuzzy rule base and it produces the fuzzy output,
normally the fuzzy value.

So, here also we have the fuzzy value available. So, the inference engine takes the fuzzy
value and comparable to this fuzzy value it produces the output in terms of the fuzzy value.

805
(Refer Slide Time: 12:04)

So, this FIS, this this fuzzy inference basically does certain reasoning. So, this inference
engine basically contains the fuzzy reasoning. So, fuzzy reasoning here basically as we
have already discussed in the previous lecture that the this here is the approximate
reasoning and it is an inference procedure that derives the conclusions from the set of
fuzzy. If then rules and these if then rules that are available already in another block
another basket, the fourth one the third one and then based on certain compositional rule
of inference.

So, here we have the compositional rules we have already done this in previous lecture so
I am not going into the detail we have certain compositional rules like max-min
composition here and then based on that we have you know certain output based on the
processing, inside the inference engine. And as I have already mentioned that the fuzzy set
the fuzzy rule set are the rule base helps finally in managing this composition.

806
(Refer Slide Time: 13:40)

So, this way we get the fuzzy output here out of the inference engine, and here you can see
that as to how these rules are helping the. So, if let us say we have a set of r rules the 1st
rule, 2nd rule and then the 𝑟 𝑡ℎ rule. So, we have a set of r rules which are all producing us
the fuzzy outputs here the, so fuzzy values here based on the input which is whether the
crisp input or the fuzzy input and these fuzzy values these fuzzy outputs out of all the rules
fuzzy rules these are finally aggregated together.

So, here you see the these values are aggregated suitably, and when these are aggregated
means normally the aggregation is here as the maximum the union. So, here we use union
of all these fuzzy values and then the outcome of this fuzzy, fuzzy this aggregator is the
fuzzy value again you can see here. So, here we have the fuzzy value as the outcome the
output so this fuzzy value is since it has to since the fuzzy system the fuzzy inference
engine, inference system that we are using here it has to interact with the outer world which
is working in the, working with the crisp value crisp logic.

So, the output should be the crisp. So, here the fuzzy value is further defuzzified to convert
this into the crisp value that is 𝑦 here you can see. So, for this we use the defuzzifier block,
I will be discussing in the next slide. So, basically what is this? This is nothing, but the
fuzzy inference or I can say the fuzzy, just the inference engine, inference engine this is
inference engine.

807
And inside inference engine we use composition, compositions with the fuzzy rules or the
fuzzy rule compositions and then we aggregate and finally, we get the output based on the
input which is fed into it whether it is normally it is the fuzzy value, and which is which
we get with the help of the fuzzifier. And finally, the output of it goes into the defuzzifier
and this defuzzifier converts this fuzzy value into the crisp value.

(Refer Slide Time: 16:47)

So, now let us move to the third block the third block is the fuzzy knowledge based
knowledge base. So, fuzzy knowledge base here is nothing but a set of rules, for fuzzy if-
then rules a set of fuzzy if then rules. So, a rule base referred to as the knowledge base,
normally this is called the knowledge base also. So, either we say the fuzzy set of fuzzy
then rules are fuzzy rule base or knowledge base here in fuzzy inference system all are
conveying the same meaning.

So, a rule base basically contains a number of fuzzy if-then rules. So, this must be
understood that, without the fuzzy rule base nothing can be done. So, all these similarly
all other blocks are also very important as we say the fuzzy knowledge base is important
because without the rule the inference engine cannot function.

So, inference engine takes the help of the fuzzy rule base, fuzzy if-then rules and then
produces the output suitable output based on the input that it takes.

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(Refer Slide Time: 18:17)

So, the output of it as I have already mentioned that the output is fuzzy value and then we
come to the fourth block here which is defuzzifier. So, this fuzzy value which is the
outcome of which is the output of inference engine which is a fuzzy value. So, this
defuzzifier helps us in converting the fuzzy value into a crisp value into a suitable crisp
value.

So, there here we have so many strategies available for converting fuzzy value into the
crisp value, but we use some commonly used methodologies for defuzzifing the fuzzy
value into crisp.

So, some of them are listed here so some commonly used defuzzification methods you can
see the one is the centre of area, the other one is the bisector of area, and then mean of
maximum, smallest of maximum, largest of maximum. So, these are some of the
commonly defuzzification strategies that we can use to convert a fuzzy value into crisp
value.

809
(Refer Slide Time: 19:36)

Now, let us go a little more into the detail of managing the inferencing. So, when we go
into the inferencing as I have already mentioned that we have the reasoning and this
reasoning actually is managed with the help of compositions and the fuzzy rule base. So,
here we can have the fuzzy reasoning like this so let us start with very basic.

So, the first case could be first kind of first kind of scenario could be like we have a single
rule in and in FIS. So, single rule with single antecedent. So, please understand that we
have a rule fuzzy rule, a fuzzy rule has two parts first part is the IF part see here and then
second part is THEN part this is called antecedent or premise, here this is called
consequence or conclusion.

So, here this IF part normally has, I mean it can have single antecedent like if I take a fuzzy
general generic variable 𝑥 so this can be like this 𝑥 is 𝐴. So, if we have a single antecedent
and of course, the output can be like this something like this 𝑦 𝐵. So, I can have either a
single antecedent or I can have multiple in antecedent by using some connective like AND
or OR whatever.

So, and then I can have here 𝑥1 is 𝐴 and 𝐴1 and let us say 𝑥2 is 𝐴2 . So, we have two
antecedent so like likewise we can have multiple kinds of scenarios. So, first scenario here
is that I can have single rule with single antecedent, means the FIS has only one rule which
is here and this fi this, fuzzy rule has only the single antecedent means 𝑥 is equal to is 𝑥 is

810
𝐴 and then 𝑦 is 𝐵. Then the second scenario could be a single rule with multiple antecedent
as I just mentioned.

So, multiple antecedent means we can have 𝑥1 𝐴1 and 𝑥2 is 𝐴2 or something like this.
Similarly, while I can have 𝑥3 , 𝑥4 as the generic variable, generic variables. So, here in
these two scenarios we have only a single rule, then we can have multiple rules in the third
scenario. So, multiple rules with multiple antecedent, means we can have multiple rules
like this. I mean if 𝑥1 is 𝐴1 and 𝑥2 is 𝐴2 then 𝑦 is 𝐵.

Similarly, we can have another rule if 𝑥1 maybe because we have to change the value of
the generic variable. So, we can say we can use another symbol here let us say 1 so 𝑥11 or
maybe we can use 𝑥1 is let us say 𝐴11 or something like that to differentiate the generic
variable values lying in two particular fuzzy reason.

(Refer Slide Time: 24:40)

So, similarly we can have multiple rules with multiple antecedents so let us understand
this by taking some examples here. So, we have the first scenario which is we see here the
first scenario is the single rule with single antecedent. So, what is happening here is the
we have a fuzzy rule you can see as I already mentioned, we have IF let us say 𝑥 is 𝐴 this
is already given THEN 𝑦 is 𝐵.

811
So, this is the fuzzy rule, single fuzzy rule that is available that is given, this is a first
scenario single fuzzy rules so we say this as the single fuzzy rule. So, here we have the
single fuzzy rule this is single fuzzy rule.

Now, when we have this as the single fuzzy rule this is already given. Now if we are
interested in using this fuzzy rule in our fuzzy inference system and if an unknown input
which is coming to this fuzzy inference system what will be the output comparable to the
input that is coming to the fuzzy inference system.

So, here the input that is coming to the fuzzy inference system. So, let us first understand
what I am saying, I am saying that if I have a fuzzy inference system and my input here is
an input which is coming to this fuzzy inference system and this fuzzy inference system
involves only the single rule with single antecedent. So, this is the first scenario and this
input can be either fuzzy this input can be here can be fuzzy or crisp, fuzzy or crisp.

So, let us first take the fuzzy case when the input is fuzzy. So, we have the IF part mean
the antecedent part the single antecedent so we have the generic variable 𝑥 here and the
output here is 𝑦. Single and antecedent means we have only 𝑥 is equal to 𝐴, I mean which
is given to us. Now a new input which is not known to us is coming to the FIS here, so f
the new input which is fuzzy is coming as 𝐴′ here so the 𝐴′ you can see here 𝐴′ is coming
and as I have already mentioned that this is a fuzzy input.

So, fuzzy input, fuzzy I will write here the fuzzy quantity so fuzzy quantity is already
always in the form of a fuzzy set. So, fuzzy quantity let us say this fuzzy quantity is in the
form of a fuzzy set and this fuzzy set is characterized by a Gaussian function which is this.
So, here this is a fuzzy quantity, fuzzy set 𝐴′. Now when 𝐴′ comes to this FIS which has
the fuzzy rule and the rule has a single rule is, in the first scenario the rule is the single rule
with single antecedent.

(Refer Slide Time: 29:23)

812
So, what is happening? So, now we will first try to superimpose 𝐴 and 𝐴′. So, when we do
that see here 𝐴 and 𝐴′ both are superimposed. So, when we superimpose since both are,
both are the 𝐴 and 𝐴′ both are fuzzy quantities. So, when these are superimpose you see
here is that cross section.

So, you see here this is the cross section. I am just encircling here. So, this point is found
this cross section is found and we normally say this is 𝑤. So, this w is taken and. So, this
w is nothing but the cross section point and then, then corresponding to this cross section
point we have the mu, mu is the membership value so this also called the degree. So, let
us say this has certain mu and then for, then the output here you will see the 𝑦, 𝑦 is already
a fuzzy quantity.

So, what we do here is we with the same degree we truncate the fuzzy set. So, you see
when we truncate the fuzzy set here, we are truncating the fuzzy set given fuzzy set 𝐵 so
what we are getting is the shaded portion the blackened area you see that some mu we are
getting see here. So, we get some 𝑤, I will write the w and this w is along this line we are
truncating here so, you see here we are truncating like this and this is what is the outcome
this is what is the outcome.

813
(Refer Slide Time: 31:11)

(Refer Slide Time: 31:23)

So, this blackened area or the truncated area, I will write truncated area, truncated I would
say blackened area or the shaded area the shaded area is the output corresponding to the
input, that is 𝐴′.

814
(Refer Slide Time: 31:34)

So, what is that which the FIS is producing, corresponding to the input the fuzzy input 𝐴′.
So, 𝐴′ let us say this the truncated fuzzy set that we have is 𝐵′. So, here we are getting 𝐵′
produced. So, what I have said here what I have mentioned here is that the input to FIS is
fuzzy. Now on the same lines we can have the output when we take the crisp input. So, if
we take the crisp input for example, if I have a crisp input instead of 𝐴′ we have some crisp
input like this here.

So, here also let’s say 𝑥 is some value 𝑥 is 𝑥′, 𝑥1 . So, this is the crisp value and if we use
this crisp value and then again when we superimpose this, 𝑥 or on 𝐴 so then if we get any
intersection point and along this intersection point we truncate the output fuzzy set.

(Refer Slide Time: 33:36)

815
So, here also on the similar lines we can get the fuzzy output. So, what is interesting here
is that the output in both the cases we are getting as the fuzzy. So, here we are getting a
fuzzy output, here we are getting as a fuzzy output.

So, what we have seen here is that if we have any fuzzy inference system, which is having
a single rule with single antecedent we are going to get the output which is the fuzzy output
whether the input is crisp or fuzzy and we have seen as to how we are getting the output.
Now when we, as I have, as I have already mentioned that when we take the crisp we can
say it like this, like we have crisp output.

(Refer Slide Time: 34:52)

816
I can just draw it here let us say this is 𝑥1 𝑥 is equal to 𝑥1 so here we have the intersection
point and then we get the degree which is basically some membership grade. So, wherever
it cuts wherever it intersects and then when it intersects and the same value is used for
truncating the output fuzzy set as we have seen in the case of the fuzzy input.

So, here what we are getting is again the fuzzy value, fuzzy value. So, this FIS is giving
us the input in this case the input is crisp and the output that we are getting is what? The
𝐵′ which is fuzzy value. So, this is how we can manage to get the output through the FIS
having single rule with single antecedent.

(Refer Slide Time: 36:16)

Now, let us go to the second scenario, second scenario that we have here is, the second
scenario is single rule with multiple antecedent. So, when we say multiple antecedent it
means that we have IF and then we have say, THEN part like this. So, here let us say we
have THEN part is 𝑧 is 𝐶.

Then, IF we have 𝑥 is 𝐴 and this can be this is connective and it can be either AND or OR
whatever. So, this can have any connective so I am just writing with the capital letter here
AND or it can be OR O R and then the second antecedent here is 𝑦 is 𝐵.

So, here we have only two antecedents, but we can have similarly multiple antecedents
like multiple linguistic variables 𝑥, 𝑦 and so on and so forth. So, here we have this kind of
situation. So, let us understand that, if we have FIS here we have the FIS and in the FIS

817
we send some input, which can be either crisp or fuzzy. So, I can write here the input either
the crisp, I can write can fuzzy first and then fuzzy or crisp.

So, comparable to this output or with respect to this input we get certain output which is
here the fuzzy output. And let’s see how we get it. So, if the input is fuzzy and this fuzzy
value is let us say, since there are two antecedents. So, we have I am writing here as inputs
so here we have multiple inputs multiple antecedent means multiple inputs.

So, we have here 𝐴′ and 𝐵′. So, since 𝐴 is already there in the rule 𝐵 is already there in the
rule these are known and here we are applying 2 fuzzy inputs this is fuzzy input, 2 fuzzy
input, I can write here the first fuzzy input the first input which is 𝐴′. Then the we have
this second input so and here the second input is 𝐵′ first input is 𝐴′ and second input is 𝐵′.

So, let’s see how we can manage to get the output, how we can, how the map how we
compute the output when we have this kind of situation, what does the inference engine
do? So, we see here that we have only a single rule, single rule means we have only one
rule which has multiple antecedents and this in our case we have 2 antecedents 𝑥 is 𝐴, 𝑦
is 𝐵 and these are known. So, if 𝑥 is 𝐴 and 𝑦 is 𝐵 then 𝑧 is 𝐶 this is known this is the rule,
this is already existing in our rule base in the FIS.

So, then we make use of this rule and we apply this rule to the input, inputs that are fetched
into the FIS. So, the inputs that are fetched here in the FIS 𝐴′, 𝐵′. So, here also we
superimpose 𝐴′ with 𝐴 and 𝐵′ with 𝐵 and we see that 𝐴 is 𝐴 and 𝐴′ both are intersecting
here and 𝐵′ and 𝐵 are intersecting here. So, since both of these are intersecting at certain
points let us say this is 𝑤1 and this is 𝑤2 .

So, these two points are taken and then we take min of this. So, we take the minimum of
these two points here in this case we see here min is applied. So, the minimum of the 𝑤1
and 𝑤2 , and then since the minimum of 𝑤1 𝑤2 is 𝑤2 and with this 𝑤2 we truncate the
output value the output 𝐶 the fuzzy set. So, when we truncate this 𝐶 we are getting 𝐶′ as
the output this is what is the fuzzy output.

So, this is very interesting to note that as to how we are getting the output mapped
corresponding to the input that we are, inputs that we are feeding to our fuzzy inference
system so this is the fuzzy output. Now since this is a fuzzy output that we are getting now
we can use further the defuzzifier to get the crisp output and this in the previous case also

818
on the similar lines we can manage to get the crisp output comparable to the fuzzy output,
that we are getting here.

Now, the question comes what would have happened when we have used the crisp inputs?
So, the crisp inputs here, if we would have, we would have used crisp input. So, on the
same lines as we have discussed in the first scenario we here also would have gotten first
the intersection points and then we would have taken the minimum of again the 𝑤1 𝑤2 and
on the same way, same lines we got this truncated the output fuzzy set truncated and
whatever comes as the shaded area here, shaded truncated fuzzy set the shaded area shaded
fuzzy set this is the output.

So, fuzzy output the shaded area. So, there is the fuzzy output, now this fuzzy value can
be converted into the suitable crisp value if needed we use as I have already mentioned in
the in this lecture previous slide you can refer as to how we can move forward here from
here to defuzzify this 𝐶′ by using either center of area or there are so many other ways.

(Refer Slide Time: 44:23)

Now, the third scenario here is that we have the FIS, we have the FIS and this FIS takes
the multiple inputs, the inputs again it can be fuzzy or crisp whatever. And then it produces
the fuzzy output, and remember this fuzzy inference system has, fuzzy inference system
has, this fuzzy inference system has multiple rules like the one which you have seen in the
just the previous slides the in second scenario that likewise we have multiple rules
available.

819
So, first rule is like this, this is the first rule I am just writing this as 𝑅1 and then the second
rule here so we are here we have 2 rules and the multiple in antecedent here would mean
that we have multiple linguistic variables. So, here we have 2 antecedents like in the second
scenario we have.

So, we have this same scenario as we have in the second scenario here but apart from that
we have the multiple rules. So, instead of multiple rules we have taken 2 rules or instead
of more rules like 3, 4, 5 or 𝑛 number of rules, we have only 2 rules just to make you
understand very simple.

So, we have 2 rules 1st rule 𝑅1 and 2nd rule is 𝑅2 and we have multiple antecedents here
we have 2 antecedents. So, we already have discussed as to how we proceed to get the
output here when we apply 𝐴′ and 𝐵′ as the input to the FIS. So, 𝐴′ 𝐵′ we already know
and we have already seen that we are getting 𝐶′ so, let us say the 1st rule produces the 1st
rule or I would write here as the 𝑅1 , 𝑅1 produces or generates produces 𝐶1′ .

Similarly, 𝑅2 produces 𝐶2′ . So, this is produced by the rule number 2. So, here we have 2
outputs first output is from the rule 1 and second output is from rule 2. Similarly, if we
would have more rules more number of rules we would have more number of outputs. So,
every rule here is producing the output and these outputs are fuzzy outputs and the input
also here is fuzzy, but again if we let us say take the crisp input then accordingly as we
have discussed in the previous slides in this lecture that the crisp or input also produces
here the fuzzy output.

So, now every rule is contributing to the output. So, please understand that, every rule is
not always contributing to the output, means every rule corresponding to the input is not
producing the output. Why? Because there may be cases where some of the rules may not
be applicable may not be fired so those rules normally not produces, those rules normally
do not produce any output. So, these rules are simply we say that these rules are not fired
so we simply exclude these rules.

So, we take only those rules which are applicable. So, here let us say these 2 rules are
applicable for 𝑥 is equal to 𝐴′ and 𝐵 and y is equal to 𝐵′ and these 2 antecedents are here
being used as the inputs to the FIS and these 2 are producing the corresponding output 𝐶1′

820
and 𝐶2′ . So, we see that we have 2 outputs the first output is the 𝐶1′ and the second output
is 𝐶2′ .

So, now we need to aggregate these outputs 2 fuzzy output, because these both the rules
are producing the separate fuzzy outputs, 𝐶1′ 𝐶2′ . Now we have to convert this into the crisp
value with respect to the inputs that are fed to the system FIS the fuzzy inference system.
So, first of all what we do here in the FIS the fuzzy inference what we do we aggregate so
here we aggregate.

So, how do we aggregate is, we take the union of. So, what we do here is we take the union
of the 𝐶1′ which is the outcome of the first rule and then we take the union of 𝐶1′ with 𝐶2′ .
So, union of 𝐶1′ and 𝐶2′ is going to give this as the output. So, on the same scale on the
same axis if we draw this so graphically we can get this as the output the aggregated.

So, when we aggregate we get this aggregated area and this is what is the fuzzy output.
So, since this is a fuzzy output, we may be interested in the crisp output compare
comparable to this. So, when we are interested in that then as I already mentioned that we
use defuzzifier to get the suitable crisp value.

So, we can write here as the fuzzy output. So, here it is already written so I can simply this
is the fuzzy output. So, when we use multiple rules with multiple antecedents this is the
way as to how we get the output comparable to the input. And here we have multiple
antecedent that means we have multiple inputs.

821
(Refer Slide Time: 52:28)

So, this is how the FIS work. So, we have the FIS that is the fuzzy rule base fuzzy inference
system. So, various kinds of fuzzy inference systems are available and if we classify these.
So, the first class is the non-additive here and then the second class is additive and then
under non-additive we have Mamdani and Larsen additive we have the Tsukamoto and
TSK. So, this is how the crude classification of fuzzy inference system available.

(Refer Slide Time: 53:21)

And we will discuss the FIS, these kind of FIS in more detail and we’ll stop here and the
in the next lecture we will go further and discuss the non additive fuzzy inference system

822
and the first model that will come in that discussion will be Mamdani model Mamdani
fuzzy model. So, with this I would like to stop here.

Thank you.

823
 Fuzzy Sets, Logic and Systems and Applications
Prof. Nishchal K. Verma
Department of Electrical Engineering
Indian Institute of Technology, Kanpur

Lecture – 51
Mamdani Fuzzy Model

Hi, welcome to lecture number 51 of Fuzzy Sets, Logic and Systems and Applications. In
this lecture I will discuss Mamdani Fuzzy Model.

(Refer Slide Time: 00:13)

So, the Mamdani Fuzzy Model basically is the first fuzzy model which was developed in
1975, by Professor E. H. Mamdani. At this model was used to control a steam engine and
boiler combination by a set of linguistic control rules obtained from human operators, that
means, the experience, human experience.

So, Mamdani model is a very interesting model, in the sense that we use the set of fuzzy
rules of a specific type. So, Mamdani model we use when we have fuzzy rules, fuzzy if-
then rules of the type here, where in the fuzzy if-then rules we have see here the premised
part and here we have the consequent part. So, when both the parts of the rule, the premise
as well as the consequent both the parts are fuzzy, then we use Mamdani model.

So, in other words we can say that, Mamdani model is used when we have a set of rules,
set of fuzzy rules available is like this where the premise part of the rule is fuzzy and the

824
consequent part of the rule is also fuzzy. So, this means that both the parts the premise and
consequent are fuzzy. Please also note that the Larsen also, Larsen model which is also
one of the fuzzy models is also using the same kind of if-then rules.

So, since we are discussing here, the Mamdani fuzzy model. So, we will be dealing all the
rules of this type, all the fuzzy rules of this type.

(Refer Slide Time: 02:42)

So, we already know that we have a fuzzy inference system and fuzzy inference system
has multiple components. How many components? 4 components. The first component is
the fuzzifier which I am denoting by 𝐹. And the second component here is the inference
engine, I am denoting this by 𝐼𝐸. The third component here is the fuzzy rule base, I am
writing here as FRB and the fourth component here is the defuzzifier. I am denoting this
by DF.

So, whenever any input comes to the fuzzy inference system here, I will just mark the
fuzzy inference system like this here by here. So, this is what is the fuzzy inference system,
a typical fuzzy inference system. And inside it we have 4 blocks, 4 components which are
acting and to give the suitable output corresponding to the input data fed, input data fed.
So, here we have the output.

So, exactly the same are mentioned here, you see here the fuzzification the first the
fuzzification and then the rule evaluation means we should have a set of fuzzy rules, fuzzy

825
if-then rules which basically helps in helps the inference engine to generate the suitable
output, suitable fuzzy output. So, it’s written over here that, if a given fuzzy rule has
multiple antecedent. So, a fuzzy rule can have can be a various kind which I will be
discussing. And the fuzzy operators 𝐴𝑁𝐷 or 𝑂𝑅 is used to obtained a single number that
represents the result of the antecedent evaluation this number is then applied to the
consequent membership function.

So, in nutshell what is happening here is that, we have a fuzzy rule base which helps the
inference engine to give the output to generate the output corresponding to the input the
fuzzy input that is fed to 𝐼𝐸.

(Refer Slide Time: 06:25)

And when it is done, the output is basically the every rule is generating certain output. So,
we aggregate all the outputs corresponding to the fuzzy rules and then the aggregated
output is in Mamdani fuzzy model we have the aggregated output which is fuzzy output,
which is a fuzzy quantity. So, then we need to defuzzify this output to generate a crisp
output, crisp value.

826
(Refer Slide Time: 07:05)

So, let us understand how a Mamdani fuzzy model works for the following cases. So, I am
going to talk about 3 different cases, where we have the Mamdani model and Mamdani
model can use the max-min composition or max-product composition which is mentioned
here. And then again the input that is fed to the fuzzy model can be either the fuzzy or
crisp.

So, then we have 3 cases, first case here is that the single rule when we have a model with
single rule, with single antecedent. So, this means that if I have let us say a fuzzy model,
which is having only a single rule and that rule is with the single antecedent. And then we
have another case where the model can be with single rule with multiple antecedents.

Then the third case could be the multiple rules with multiple antecedents. So, all these 3
scenarios will be discussed in coming slides, in this lecture. And again these 3 scenarios
we will be discussed when we will be using max-min composition, max-product
compositions and again for the fuzzy inputs and crisp inputs. So, let us go one by one.

827
(Refer Slide Time: 09:30)

So, now let us start with the first scenario, where we have a Mamdani fuzzy model. And
this Mamdani fuzzy model has only single rule with single antecedent.

(Refer Slide Time: 09:48)

You can see the rule here in the Mamdani model, Mamdani fuzzy model. So, the rule
basically says 𝐼𝐹 𝑥 𝑖𝑠 𝐴 𝑇𝐻𝐸𝑁 𝑦 𝑖𝑠 𝐵, so we already know what is 𝑥. So, 𝑥 is nothing but
the generic variable, the input and this input is nothing, but the generic variable.

828
Similarly, 𝐴 is a fuzzy region, 𝐴 is some fuzzy, some fuzzy value; that means, the fuzzy
set. And what is 𝑦 here? 𝑦 is the output. And 𝐵 here is basically, B here is again the fuzzy
value. So, when we have a single rule and the input is coming to the model, let us see what
happens. So, as I have already mentioned in FIS, when we discussed FIS we saw that we
have 5 block, 4 blocks the first block takes the input, and it converts the input into a fuzzy
value.

So, this means that this the input is first fuzzified, then it passed on to the inference engine
where various compositions and aggregations are done with the help of fuzzy rule set that
is available. And then this output that is generated out of the inference engine is normally
a fuzzy output and which is converted into crisp value by the defuzzification, defuzzifier
or defuzzification.

So, let us now give an input here to this fuzzy model and let us see what happens. So, we
have a single rule only, only one rule with one antecedent, means with a single antecedent.
So, when we say single antecedent means we have the only one input, this means 𝑥 is the
single antecedent, 𝑥 𝑖𝑠 𝐴 is the single antecedent.

(Refer Slide Time: 13:11)

So, here since the rule is already with us, here we have the rule and this rule is in this form,
𝑖𝑓 𝑥 𝑖𝑠 𝐴 𝑡ℎ𝑒𝑛 𝑦 𝑖𝑠 𝐵. And rule is single as we can already see and also here the antecedent
also is single. So, I can write here a single antecedent, antecedent. So, single antecedent

829
and single rule, all right. So, now, when we have this kind of scenario in Mamdani model
or this kind of scenario of Mamdani model.

If we substitute or if we feed an unknown input, what is this fuzzy model is going to give
us? So, this means what? This means that if we have let us say model here, a fuzzy model
and this form fuzzy model is the Mamdani model, Mamdani type of model Mamdani
model. And if I am giving an input here some input 𝑥 is equal to something and this model
is going to give us some output here which normally is the crisp output. So, let us see what
we are going to get.

And as I have already mentioned that this model here in this case has this Mamdani model
has a single fuzzy rule. So, this model has a single fuzzy rule, and also this rule has only a
single antecedent, which I have already mentioned. Now, if a new input comes, if a new
unknown input is comes unknown means this that input which this model has never seen.
So, if some input is coming to this model as input and then this model is going to generate
some output corresponding to that input.

Now, what is this output is going to be if my input is a fuzzy input or maybe a crisp input.
So, when we say fuzzy input it means we provide, we feed a fuzzy value as the input. So,
there can be 2 kinds of input; one is here is the fuzzy input, directly fed to the model or
maybe the crisp input fed to the model. So, when we say fuzzy input. So, fuzzy input
means a fuzzy set or fuzzy value is directly given to the model. And when we say crisp
input then we feed some value of 𝑥. So, let us take the first case where we are giving the
fuzzy input. So, when we say fuzzy input it means at 𝑥 is a fuzzy value.

So, here we see that we have the fuzzy input. So, fuzzy input means 𝑥 is, 𝑥 as the input is
fuzzy. So, the model is ready, model is known when we say model is known means this
rule is known here, they and this model has a single rule and the premise part of this rule
is 𝑥 𝑖𝑠 𝐴, which you can see here, 𝑥 𝑖𝑠 𝐴 𝑎𝑛𝑑 𝑦 𝑖𝑠 𝐵. So, this means that for any 𝑥 which
is falling in 𝐴 fuzzy region, corresponding to that the output is going to fall in 𝑦 in 𝐵 fuzzy
region.

830
(Refer Slide Time: 18:25)

So, here let us now, let us now apply the input to this fuzzy model, where we have a single
rule with single antecedent.

So, let us do this quickly and see what we are going to get. So, our input is 𝐴′ , so our input
is x which is 𝐴′ . 𝐴′ is here this is a fuzzy quantity, this is a fuzzy quantity. So, fuzzy
quantity is always in the form of a fuzzy set. So, this is a fuzzy quantity or fuzzy value or
it’s a fuzzy set basically. So, we see that here we have a fuzzy set, which is fed to the
model as input. So, now, let us apply this and see what is the corresponding output that we
are going to get.

831
(Refer Slide Time: 19:33)

So, what we have done here is that we have the rule that is with us single rule and this is
the rule 𝑥 𝑖𝑠 𝐴 𝑡ℎ𝑒𝑛 𝑦 𝑖𝑠 𝐵.

Now, the 𝐴′ , the fuzzy see here the fuzzy value as the input that is fed to the model, fuzzy
value here. So, when this is fed to the model, now let us superimpose this 𝐴′ with 𝐴 and
when we superimpose these 2 fuzzy sets. So, 𝐴′ is the fuzzy value which is given as the
input to the model and 𝐴 was already known, 𝐴 is already existing here in the fuzzy rule.

Now, when we superimpose these 2 fuzzy sets, 𝐴′ and 𝐴, we see here that these 2 are
intersecting to some point and this is the point of intersection. So, here when we see we
find that corresponding to this intersection point we its membership value is 0.58. So, we
note this point 0.58, the intersection point and please note that there may be multiple points
which point of intersection you might get.

So, when we get multiple points, then we take the maximum of these two and the
maximum will apply. So, we take the maximum of the intersection points and then the
with the maximum we will proceed.

So, the next step here is two, in this case we have 0.58, 0.58 as the value of the point of
intersection, this we call as the weight. So, we call this as the w, we call this as the w. So,
this is the point of intersection. Now with this value, with this weight we truncate the

832
output. So, output you see here and with this value of 𝑤, see here this is my 𝑤 and with
this value if we truncate the output fuzzy set 𝐵.

(Refer Slide Time: 23:15)

So, what is that we are getting here is the this solid area the truncated area which is
blackened. So, this area is the output corresponding to the fuzzy input 𝐴′ , we may call this
fuzzy area which is the blackened one, the shaded one is 𝐵 ′ . So, here the corresponding to
the fuzzy value, is fuzzy value is fuzzy input we are getting the output and again this output
is the fuzzy output.

So, this means that when we have a fuzzy when we have a Mamdani fuzzy model with
single rule with single antecedent we get the output like this corresponding to fuzzy input.
So, now here in this case the output has been fuzzy has been corresponding to the fuzzy
input. Now, what if we have the crisp input for the same model?

833
(Refer Slide Time: 24:51)

So, let us and before that here I would like to mention one more thing that since we are
using the max-min composition. So, here we have the 𝑤 which is here the 𝑤, if we would
have multiple antecedents this min would have been applicable. Because here we have a
single point of intersection, so we are getting only one 𝑤.

So, single antecedent I would say single antecedent. So, for single antecedent we have
single 𝑤, even if we would have multiple point of intersections in this case we would have
avoided the conflict by taking the max and we for single antecedent we will have single
w.

But if we would have multiple antecedents, so for multiple antecedents we would have
either used max-min composition or max-product composition. So, even if we take the
min of 𝑤, we are going to have the same value that is point 0.58. So, with this value we
truncate. Now, we can have the max-product composition. So, since we have here the w
which is again this single value.

So, for the single value whether you take the min or product both will remain the same.
So, here also will have the same value of 0.58. So, here in this case the both the max-min
or max-product both are going to give the same truncated fuzzy value. Now, comes here
after this the output of this single rule, now we take the max of it. So, since we have only
one rule even if we take max the same value is going to come.

834
So, that is why whatever is here is coming here directly. So, max of this same value is
going to give us the same fuzzy value.

(Refer Slide Time: 27:24)

So, in today’s lecture we have discuss the Mamdani fuzzy model using max-min and max
composition, max-product compositions for single rule with single antecedent for fuzzy
input. In the next lecture we will continue our discussion with Mamdani fuzzy model.

Thank you.

835
 Fuzzy Sets, Logic and Systems and Applications
Prof. Nishchal K. Verma
Department of Electrical Engineering
Indian Institute of Technology, Kanpur

Lecture – 52
Mamdani Fuzzy Model

Hi welcome to the lecture number 52 of Fuzzy Sets, Logic and Systems and Applications.
In this lecture, we will continue our discussion with Mamdani Fuzzy Model.

(Refer Slide Time: 00:19)

So, in Mamdani fuzzy model, let us understand the fuzzy reasoning of Mamdani fuzzy
model for the following. Mamdani fuzzy model using Max-Min and Max-Product
composition for fuzzy and crisp inputs. We have already discussed the fuzzy input for
Mamdani fuzzy model using Max-Min and Max-Product compositions in our previous
lecture.

So, we have already discussed the Mamdani fuzzy model using Max-Min and Max-
Product compositions for fuzzy input with single rule and single antecedent.

836
(Refer Slide Time: 01:07)

When we feed the input as the crisp value so, this means that when we apply 𝑥 as crisp
means the input means the crisp input we apply.

(Refer Slide Time: 01:27)

So, we see here that the rule we have a single rule in Mamdani fuzzy model. So, and in
this rule we have a single antecedent here, only one antecedent. When we say only one
antecedent, it means we have 𝑥 is 𝐴, but when we say multiple antecedent or two
antecedent and then here we have we would have 𝑥1 is 𝐴1 or and 𝑥2 is may be 𝐴2 or
something like that.

837
So, multiple antecedent scenario would have been like that, but since here we have 𝑥 𝑖𝑠 𝐴,
it means single input, we are we have in this fuzzy model. So, let us now apply the crisp
input here like this.

(Refer Slide Time: 02:22)

So, the value of 𝑥 here is the value of 𝑥 input the value of input 𝑥 is a crisp value which is
6.8. So, let us now apply this. So, what we do here is we simply take this fuzzy set which
is here as the input.

So, we have 𝐴, let us have 𝐴 fuzzy set like this, then we find the intersection point. We
draw a line at you see here in between 6 and 8, we have point 6.8. So, this line is this line
gives us 𝑥 is equal to 6.8. So, this value of the generic variable corresponding to this, we
have the membership value 0.34. So, 0.34 we may get here.

Please note that when we have a crisp input so, the point of intersection is going to be only
the single value. So, here there will not be any conflict as we have seen in the case of the
fuzzy input. So, in fuzzy input, we can have multiple pointer points of intersection and
then if you would have gotten the multiple points of intersections, you would have taken
the maximum of these two. But here we have a single point of intersection.

So, this is our w which is the weight. So, we have a single 𝑤. So, even if we take min that
is going to remain the same or even if we have we are going to take product, the 𝑤 is going
to remain the same because we have a single value. So, with this we will move forward

838
and we use this value to truncate the output fuzzy set; output fuzzy set here is 𝐵. This is
the 𝐵 fuzzy set.

(Refer Slide Time: 04:53)

So, when we do that we see that point corresponding to the input 𝑥 is equal to 6.8, we have
this as the 𝑤 and we use this 𝑤 after either min product. This remains the same 0.34
because we have the single 𝑤.

So, we proceed further and we use this value to truncate the output fuzzy set 𝐵 and again
as I mentioned since we are using Max-Min composition so whatever we are taking first
min and then we take the max here we, whatever we get we take max. So, max would have
been applicable when we would have multiple rules applicable. And so, the outputs of all
these rules would have been aggregated or output of all these rules would have been taken
as a union of all these.

So, the max is not max is going to give the same value as we have here. So, this is the 𝐵 ′
and this 𝐵 ′ will remain the same even if we take max, same here.

839
(Refer Slide Time: 06:22)

So, that is how the corresponding to 𝑥 is equal to 6.8 is giving us the output as 𝐵 ′ and this
is nothing, but what? This is the fuzzy value fuzzy value, but this fuzzy value has no
meaning unless we understand it like what is the corresponding crisp value out of it. We
have fed the crisp value to the model, but we are getting the fuzzy value.

So, this fuzzy value has to be defuzzified to be further used. So, will discuss in the last as
to how we convert this fuzzy value into the crisp value by various kinds of defuzzification
models. So, here this fuzzy model which is basically having the single rule with single
antecedent based on the crisp input that is 𝑥 is equal to 6.8 is giving us the fuzzy output,
the fuzzy value as the output which is nothing, but 𝐵 ′ .

840
(Refer Slide Time: 07:51)

Similarly, the max-product composition for the same input is going to give us the same
output. Because the product is also the product of the 𝑤 here, the weight we have this
single weight here.

So, the product is also going to remain the same and with this value with this weight value,
we move further and we truncate and we are going to get these same output. So, 𝑥 is equal
to 6.8 is going to give us the same output. This is the fuzzy value as the output. So, whether
we have here in this case the composition Max-Product or Max-Min, both are giving the
same output.

And we know the reason because in both the cases here in this case only single rule with
single antecedent is applied. So, single antecedent will produce only the single weight
here, single weight 𝑤 and the whether we use the max, whether we use the min or product
this is going to give us the same. So, that is why the Max-Product or Max-Min both the
composition are going to give us the same result.

841
(Refer Slide Time: 09:42)

Now, let us move to this second scenario where we have the single rule with multiple
antecedents. So, we have a Mamdani model; we have a Mamdani model here which has
only one rule only one rule, but this rule has multiple antecedent means the premise part
has multiple antecedents.

(Refer Slide Time: 10:19)

So, let us see how does it look like. So, you can see here we have a rule, this rule here and
in this rule, we have the premise part which is this and then the consequent part. So, this
is the premise part; premise part and this is the consequent part. So, in premise part, we

842
see that we have two antecedents. What are those two antecedents here? First antecedent
is 𝑥 𝑖𝑠 𝐴 and the second antecedent is so, second antecedent is 𝑦 𝑖𝑠 𝐵.

So, this is the 𝐼 𝑠𝑡 antecedent, 𝐼 𝑠𝑡 antecedent and 𝑦 𝑖𝑠 𝐵 is II, the 𝐼𝐼 𝑛𝑑 one the 𝐼𝐼 𝑛𝑑
antecedent. So, these two antecedents are connected by 𝐴𝑁𝐷, so 𝐴𝑁𝐷 is nothing but a
connective this could be by 𝑂𝑅 as well. So, we have a fuzzy model Mamdani fuzzy model
which has a single rule, but two antecedents under the multiple antecedent case. And here
also we have the inputs, but the inputs are two inputs here because we have two
antecedents.

So, 𝑥 and 𝑦 both are the inputs to the model. Now this 𝑥 and 𝑦 can be either crisp or fuzzy,
we will discuss both the cases. Now coming to the consequent part in consequent part, we
have 𝑧 as the output and 𝐶 is the fuzzy value. So, consequent part is fuzzy. So, so when
we see this rule, this rule has the premise part fuzzy because 𝐴 and 𝐵 both are fuzzy and
the consequent part is also fuzzy because 𝐶 is fuzzy. So, the rule type here is that the
premise part is fuzzy and then the consequent part is also fuzzy.

Now, coming into the premise part, let us first understand that we have two antecedents.
The first antecedent here is 𝑥 𝑖𝑠 𝐴, second antecedent is 𝑦 𝑖𝑠 𝐵 and both these antecedents
are connected by connective 𝐴𝑁𝐷, here this 𝐴𝑁𝐷 in place of 𝐴𝑁𝐷 there could be other
connectives like or, but so, but in this case we have 𝐴𝑁𝐷 as the connective. So, we have
in this premise part which is fuzzy we have two antecedents and these two antecedents
have basically 𝑥, 𝑦 as the generic variable values as the input.

So, 𝑥, 𝑦 are the inputs. So, I hope this is very clear now. So, with this the fuzzy Mamdani
fuzzy model is available. Now what happens when we apply the inputs 𝑥 and 𝑦 to this
model, what is the kind of output that it generates? So, there can be two scenarios. The
first scenario here is in this case that 𝑥, 𝑦 can be the fuzzy input as we have seen in the
previous case; 𝑥 and 𝑦 can be the fuzzy inputs, 𝑥 can be 𝑥 and 𝑦 can be the crisp inputs
and then here also we can have Max-Min composition and Max-Product composition.

843
(Refer Slide Time: 15:16)

So, let us now go ahead and see the various cases. So, here we see that we have the model
available and we have the rule. In this rule, we have the antecedent two antecedents. So,
here we have the 𝐼 𝑠𝑡 antecedent. So, this is our 𝐼 𝑠𝑡 antecedent where we have 𝑥 𝑖𝑠 𝐴. So,
here this is the 𝐼 𝑠𝑡 antecedent, the 𝐼𝐼 𝑛𝑑 antecedent is here. So, I can write here the 𝐼 𝑠𝑡
antecedent, the 𝐼𝐼 𝑛𝑑 antecedent then we have the output; this is the output.

So, this since this model is known it means 𝐴, 𝐵, 𝐶. All these three fuzzy sets are known.
Now let us apply the input 𝑥, 𝑦. So, we call this as the fact here. So, let us apply the input
𝑥 and 𝑦 and let us apply fuzzy inputs. When we say fuzzy inputs, it means we give the
fuzzy value and fuzzy value we all know that fuzzy value means a fuzzy set. So, let us give
𝑥 is equal to 𝑥 is equal to 𝐴′ , 𝑦 is equal to 𝐵 ′ , 𝑥, 𝑦, 𝑧. So, now, let us do that.

So, since here 𝑥, 𝑦 both are fuzzy values so, they are represented by the fuzzy sets, you see
here.

844
(Refer Slide Time: 17:29)

A fuzzy set 𝐴′ fuzzy set, 𝐵 ′ fuzzy set; both the fuzzy sets are here. Now let us apply this
was the rule which is already known and this model Mamdani fuzzy model. When we
apply this input means 𝑥 is equal to 𝐴′ , 𝑦 is equal to 𝐵 ′ , let us see what it is going to
produce.

So, again like in the previous scenario, we superimpose these two fuzzy sets to their
corresponding inputs input fuzzy sets. So, with 𝐴′ here is superimposed to 𝐴 because 𝑥 is
𝐴′ and then 𝑦 is 𝐵 ′ . So, this also is superimposed to the fuzzy set which is already known.

So, we see that when we superimpose these two, when we superimpose these two we see
the point of intersection here and we call this as the 𝑤1. So, let us represent this by the 𝑤1.
So, this is 𝑤1, the intersection point. So, there is the first weight. Similarly here when B
and 𝐵 ′ intersects let us say this is 𝑤2 . So, these two are two different values of weights
and these weights are nothing but the corresponding, you see here the intersection
corresponding membership values.

So, you can easily get the corresponding membership value and you call this as the 𝑤1.
Here also you get the corresponding membership value and you call this as the 𝑤2 . So,
these two weights are available. Now, if we apply Max-Min composition. So, we take min
of these two. So, min of these two values min of 0.58, min of 0.58 and 0.36,obviously, the
min of these two is going to be 0.36. So, we will take 0.36 and we proceed truncation with
this value.

845
(Refer Slide Time: 20:21)

So 0.36, we use for truncation and this is what we are getting as the min value min of the
two; 𝑤1 and 𝑤2 . So, this is the output of the first rule. I mean the we have only one rule
so, the output of this rule. If we would have multiple rules where we will see that in coming
sides, then you would have taken max of all this, we would have aggregated the outputs
of all the rules.

So, here since we have only the single rule so, this will become the output corresponding
to the inputs 𝑥 is equal to 𝐴′ , 𝑦 is equal to 𝐵 ′ connected with 𝐴 connected with 𝐴𝑁𝐷.

(Refer Slide Time: 21:16)

846
So, you see this is what is the output that we are getting. The same output here because the
max is here we come since we are using Max-Min composition, max is not really
applicable because we have the single value, single rule, the output is single. So, the max
is giving us the same thing whatever is here is coming as the output. So, this way in the
second scenario, we get the output. So, here in this case the input is fuzzy. So, you see here
the input is fuzzy values fuzzy values or the fuzzy sets.

(Refer Slide Time: 22:18)

Now, in the second scenario what happens when we use when we use the Max-Product
composition? So, say with the same input same set of fuzzy inputs. So, everything remains
the same as we have already seen that the intersection point is designated as the 𝑤1 and
here 𝑤2 , the weights and weights we have already seen that 0.58 and 0.36.

So, here since we are using Max-Product composition instead of Max-Min where we have
taken a min of 𝑤1, 𝑤2 . Here we are taking the product of 𝑤1 and 𝑤2 because we are using
Max-Product composition. So, we will take the product of 𝑤1 and 𝑤2 and the product of
0.58 and 0.36, you see there is 𝑤1 and there is 𝑤2 . When we take the product, we are going
to get the final value that is the product of this 𝑤1 and 𝑤2 , we are getting 0.21.

So, since we are getting 0.21, we truncate with this membership value with this weight
value which is 0.21. So, we see as compared to the previous case, the fuzzy value is lesser
than that of the Max-Min case. So since we have now, we cannot take the max because

847
max is really not applied, we have only single rule. So, the same output is transferred over
here.

So, we see that we have the 𝐶1 because this as the 𝐶1 as the output. So, max Max-Product
is giving us again the fuzzy output. Here is a again the fuzzy output and this we can
defuzzify to get the crisp output; this fuzzy output. Fuzzy output is there in the form of a
fuzzy set. So, in the second scenario, we have seen that how Max-Product, Max-Min
produces the output for fuzzy inputs to two antecedents.

(Refer Slide Time: 25:15)

Now next is in the second scenario itself instead of fuzzy inputs, let us take the crisp inputs.

848
(Refer Slide Time: 25:32)

So, again on the same lines, we see that we have the rule here which is the single rule, but
multiple antecedents. So, here we have two antecedents instead of 𝐴, under multiple
antecedents we have two antecedents. This 𝐼 𝑠𝑡 antecedent 𝐼 𝑠𝑡 and the 𝐼𝐼 𝑛𝑑 ; 𝐼 𝑠𝑡 antecedent,
𝐼𝐼 𝑛𝑑 antecedent and here we have the output. So, when we apply the inputs as crisp values,
it is very simple instead of fuzzy where we applied the fuzzy value in form of fuzzy set.

(Refer Slide Time: 26:26)

So, there we could have multiple points of intersection and we could have avoided this by
taking the max within that antecedent. Here that case will never exist means the conflict

849
will never exist rather than we are going to get always the single value single intersection
value, you see here for 𝑥 is equal to point for 𝑥 is equal to 6.5 for here 𝑥, 𝑦. There is a
small 𝑥 𝑦; here also 𝑥 ,𝑦, 𝑧.

So, when we take 𝑥 is equal to 6.5, 𝑦 is equal to 8.5; we get the point of intersection as 𝑤1,
point of intersection as 𝑤2 and this way we have two points of intersection. Now for Max-
Min composition, we have to take min of these two, min of 𝑤1, 𝑤2 .

So, when we take the min of 𝑤1 and 𝑤2 , we get again the we get 0.36 here and since we
have only one rule so, maximum of the Max-Min composition does not apply here and the
truncated value here, the fuzzy value will become the final output. So, this is the final
output of the model corresponding to the fuzzy input 𝑥 is equal to 6.5, 𝑦 is equal to 8.5.

(Refer Slide Time: 29:11)

Now, what happens when we take the Max-Product composition? Since we already have
this 𝑤1, the weight 𝑤2 the other weight corresponding to the 𝑥 is equal to 6.5 and 𝑦 is
equal to 8.5. So, when we multiply this when we take the product of these two weights,
we see here we get 𝑤 is equal to 0.29. So, with this value that is 𝑤 is equal to 0.29. So,
with this value, we truncate the fuzzy set 𝐶.

So, with this value we get the output this is the output out of the first rule. This is the fuzzy
output the truncated shaded area is the output that is fuzzy output basically fuzzy output
and then again the max is not applied here because we have only one rule, single rule. So,

850
we do not have to do anything further and the same output is the becomes the final output.
So, the same is the final output that is the fuzzy output which is in the form of fuzzy set
the truncated fuzzy set, I can write here if fuzzy output.

Now, we can use suitable defuzzifier to convert this fuzzy output into the crisp value.

(Refer Slide Time: 31:13)

So, in today’s lecture we have discussed the following. Mamdani fuzzy model using Max-
Min and Max-Product compositions for single rule with single antecedent for crisp input.
And also we have discussed the Mamdani fuzzy model using Max-Min and Max-Product
compositions for single rule with multiple antecedents for fuzzy and crisp inputs. And in
the next lecture, we will continue our discussion with remaining part of the Mamdani fuzzy
model.

Thank you.

851
 Fuzzy Sets, Logic and Systems and Applications
Prof. Nishchal K. Verma
Department of Electrical Engineering
Indian Institute of Technology, Kanpur

Lecture - 53
Mamdani Fuzzy Model

Hi, welcome to the lecture number 53 of Fuzzy Sets, Logic and Systems and Applications.
In this lecture we will continue our discussion with the remaining part of the Mamdani
Fuzzy Model.

(Refer Slide Time: 00:18)

Here, let us understand the fuzzy reasoning of Mamdani fuzzy model for the following.
Mamdani fuzzy model using max-min and max-product compositions for fuzzy and crisp
inputs for multiple rules with multiple antecedents. In the previous lecture what we have
done was, the Mamdani fuzzy model using max-min here max-min and max-product
compositions for fuzzy as well as the crisp inputs for single rule with single antecedents
and single Rule with multiple antecedents.

852
(Refer Slide Time: 01:04)

Now, let us discuss the third scenario where we have a Mamdani fuzzy model with
multiple rules more than one rule. And, even in the rules itself we have multiple
antecedents. So, let us now discuss this scenario.

(Refer Slide Time: 01:29)

So, here we are taking for simply simplicity we are taking 2 rules only under this multiple
rule case with multiple antecedents here also we are taking only 2 antecedents. So, let me
just make it very clear that we have we have this as the 𝐼 𝑠𝑡 antecedent, 𝐼 𝑠𝑡 antecedent.

853
And then we have the 𝐼𝐼 𝑛𝑑 antecedent and we have 2 rules. So, 2 rules 2 antecedents and
we all already know that this is the premise part and this is the consequent part this premise
part, this is consequent part. And here also we see that premise part is a fuzzy and the
consequent part is also fuzzy.

So, both premise and consequent both parts are fuzzy, when we say fuzzy it means we
have this premise and consequent both part are designated by the fuzzy values like here in
this case 𝐴1 , 𝐵1, 𝐶1 , 𝐴2 , 𝐵2, 𝐶2 are fuzzy values.

(Refer Slide Time: 03:16)

Now let us on the same lines as we have discussed the two previous scenarios, let us
discuss this also. So, here we have rule number 1 where we have 𝑥 is 𝐴1 , 𝑦 is 𝐵1 this
𝑥 this 𝑦 this is 𝑧, 𝑥, 𝑦 this is rule number 2. So, we have two rules and then two
antecedents. So first antecedent is this, second antecedent is this. And these two
antecedents are connected again by AND, you can see here and so both the rules are
containing AND as the connectors.

Now, these two rules are given in the fuzzy model Mamdani fuzzy models. Or in the other
words we can say this this Mamdani fuzzy model is described by or characterized by the
2 fuzzy rules and these fuzzy rules are with 2 antecedents.

854
(Refer Slide Time: 04:52)

And which is clear case of the multiple rules with multiple antecedents. So, let us first take
the fuzzy input. So, Rule 1 and this is Rule 2. So, here let us take the same input here 𝑥, 𝑦
and 𝑥 is fuzzy 𝑦 is also fuzzy, when we say fuzzy again fuzzy means of fuzzy value and
fuzzy values always characterized by or represented by a fuzzy set. So, we are taking a
fuzzy set alright. So, now, let us apply this input to the fuzzy model and let us see what we
are getting as the output to the corresponding input.

So, this is fuzzy input, first fuzzy input. So, here 𝑥 is the fuzzy input and what is this
fuzzy set here? Fuzzy set 𝐴′ , here 𝑦 is again is a is also a fuzzy set which is 𝐵 ′ this is
𝑦 alright. So, now, let us apply this. So, this is very simple and as we have done in second
scenario, so let us take the Rule number 1 first and take these two 𝑥, 𝑦 fuzzy values
superimpose.

And then finds the points of intersection and these points of corresponding to the points of
intersections, the membership values will become the 𝑤’s like 𝑤1 and 𝑤2 like this here.

855
(Refer Slide Time: 07:18)

So, when we superimpose this is our Rule number 1 and this is Rule 2. So, when we
superimpose 𝐴′ , so you see here 𝐴 is superimposed on 𝐴1 , 𝐵 ′ is superimposed on 𝐵1.

So, 𝐴′ is superimposed on 𝐴1 and we are getting here, we are getting here as the 𝑤1,
here also we are getting the point of intersection as 𝑤1. Now, here in this case since we
are using the max-min composition. So, now, let us take 𝑤1 this 𝑤2 . So, let us take the
minimum of the 𝑤1 and 𝑤2 and when we take minimum of the 𝑤1 and 𝑤2 we are
getting 𝑤 as 0.36.

(Refer Slide Time: 08:29)

856
So, this is 𝑤1 this is 𝑤2 , so 𝑤 is. So, let us take this as the w dash, this is also w dash
and when we take this as the 𝑤1 and 𝑤2 , 𝑤1′, 𝑤2′ let us call this as 𝑤1 ; 1 is for the
Rule here this for the first rule, first rule it is for the first rule.

(Refer Slide Time: 09:18)

So, when we get 𝑤1 here as the minimum of the 𝑤1′ this was the 𝑤1′ this is 𝑤2′ alright.

So, 𝑤1 is coming out to be 0.36 and again we use this 𝑤1 to truncate 𝐶1 and this is the
output is the fuzzy output corresponding to the fuzzy inputs if fuzzy output from Rule 1,
from Rule 1. Similarly, we do the same exercise using min composition here also we are
getting 𝑤1′ , 𝑤2′ . When we take 𝑤2′ here in this case.

So, when we take min because we are using we are taking max min composition, we take
min. So, 𝑤2 which is the outcome which is corresponding to the second rule. So, 𝑤2
with this 𝑤2 we truncate 𝐶2 and this becomes 𝐶2′ . So, this is this we can write as the
fuzzy output from Rule 2.

Now, here we are using max-min composition. So, min is utilized here now whatever
outcomes that are coming from each Rule we are taking the maximum of it, maximum
means we are taking the union of it. So, let us take the let us aggregate it with union.

857
(Refer Slide Time: 11:56)

So, when we take the union, so we are getting this kind of thing you see here when we plot
it we are going to get the union see here. So, we aggregate the output from Rule 1 and then
we get the output from Rule 2 both outputs are aggregated, we are taking the union and
the union is giving us see here the max is giving us the fuzzy output corresponding to the
fuzzy input. Now, this fuzzy output can suitably be defuzzified by using suitable
defuzzifiers. Now, in this scenario the third scenario let us see when we use the max
product composition.

(Refer Slide Time: 12:57)

858
So, this again very simple here when we superimpose the fuzzy inputs here. So, 𝐴1 is
superimposed to 𝐴′ is super imposed 𝐴1 this is 𝑤1′, this is 𝑤2′ similarly this is 𝑤1, 𝑤2′ .
So, since here we are using the max product composition, so we take max we take the
product of these two.

So, product is going to give us 0.31 and this is the output of the Rule output from the Rule
fuzzy output that we are getting, the fuzzy output from Rule 1. Similarly, here also we are
getting some output on the same lines here also we have the 𝑤1′, 𝑤2′ when we use max-
product max-product composition.

So, we take product in this case product of 𝑤1′, 𝑤2′ . So, when we do this we are getting
here 𝑤1′, 𝑤2′ and yes these are 𝑥, 𝑦, 𝑧; 𝑥, 𝑦, 𝑧; 𝑥, 𝑦, z everywhere.

So, this way we are getting the again now coming back to the third scenario, when we use
max-product composition. So, in max product composition we takes here again the product
of the 𝑤1′, 𝑤2′ and this truncated portion becomes the and we truncate with the value 𝑤2 .
So, this is 𝑤2 yes 𝑤2 which is nothing but 0.20 and here 𝑤1 is 0.31.

So, this way we are getting two outputs, the first output here is corresponding to Rule 1
and the second fuzzy output is corresponding to Rule 2. So, here I can write fuzzy output
from Rule 2. So, every rule is going to give us some output, but these rules should be
applicable.

So, this way when we take the max, so here we take the union of this means we take the
maximum which union right, we aggregate. So, when we do this we are getting this
structure, this is fuzzy set fuzzy value. So, this as we can see there is a fuzzy value as the
output corresponding to the fuzzy inputs 𝑥 is 𝐴′, 𝑦 is 𝐵′. So, we can call this as the
fuzzy output.

And this can be suitably converted into crisp value by using some defuzzifiers. Now, here
we have discussed when 𝑥, 𝑦 both were fuzzy values. Now, let us see what happens when
we use 𝑥, 𝑦 both are crisp values means the inputs are inputs that are fed or crisp values.

859
(Refer Slide Time: 18:12)

(Refer Slide Time: 18:13)

860
(Refer Slide Time: 18:16)

So, let us go ahead and see. So, here we have Rule number 1 here we have Rule number
2.

And we see that when we take 𝑥 is equal to 7 means 𝑥 the value of 𝑥 is 7. The value
of 𝑦 is 6.5 and here please note that, all these 𝑥 that we have written here is 𝑥, 𝑦, 𝑧,
𝑦, 𝑧. So, this is Rule number 1, Rule number 2 and we see that we have to antecedents
here both the antecedents are connected by AND. Here also we have 𝑥, 𝑦, 𝑧, 𝑥, 𝑦, 𝑧.
Now, here we see that we have these inputs the values of the inputs 𝑥, 𝑦 are crisp values.

So, I can write here the values of 𝑥 and 𝑦 here are crisp values ok. So, when we have
crisp values now we apply these crisp values and when we apply this we see if there is any
intersection here on 𝐴1 for 𝑥 is equal to 7. So, we see that we have one intersection. So,
we call this as 𝑤1′ similarly for 𝑦 value we see whether there is any intersection here yes
for 𝑦 is equal to 6.5, we have on 𝐵1 we have 𝑤2′ and similarly for Rule 2 also we see.
So, here also we see that we have the intersection w 1 dash and then we have the 𝑤2′ . Now
for the first Rule first, so we are taking the max min composition.

So, we will take the min of 𝑤1′ and 𝑤2′ , 𝑚𝑖𝑛(𝑤1′ , 𝑤2′ ) and this is going to give us 𝑤1.
So, let us see what we are getting here when we take the min we are going to get obviously
the min 𝑤1 is going to give us 𝑤2 𝑤1′ which is nothing, but zero 0.28 𝑤1′ which is
nothing but 0.28 here.

861
(Refer Slide Time: 21:47)

So, here we see that 𝑤1 which is the strength of this Rule is the 0.28 and with this value
0.28 we truncate 𝐶1 , 𝐶1 is the consequent part. So, we truncate here also we write small
alright. So, this is the output of the first Rule corresponding to the crisp inputs 𝑥 and 𝑦.

So, the output from Rule 1. Similarly, when we for the first for the second Rule we are
getting the this as the output the fuzzy output after truncation here. So, this is the output
from Rule 2. Now, since here we are using max min composition. So, we have to take the
maximum of all the outputs.

(Refer Slide Time: 23:50)

862
So, we take the aggregation we take the union of these two outputs coming from the Rule
number 1 and Rule number 2 and this way we are getting this structure this fuzzy value as
the output corresponding to the inputs as crisp values 𝑥 is equal to 7 and 𝑦 is equal to
6.5. So, this is what is the fuzzy value as the output ok.

So, we see that the output in all the cases here we are getting fuzzy. So, Mamdani model
Mamdani fuzzy model always produces the fuzzy output and we use the defuzzifier
suitable defuzzifier to convert this fuzzy output into the crisp value. Now, what happens
when for the same input 𝑥 is equal to 7, 𝑦 is equal to 6.5 as a crisp values when we use
max-product composition.

(Refer Slide Time: 25:01)

So, when we use max product composition we see that the 𝑤1 is 0.27 we multiply 𝑤1′,
𝑤2′ here also 𝑤1′, 𝑤2′ and this is our 𝑤1′, this is our 𝑤2′ , this is 𝑤1′, this is 𝑤2′ . So, these
two outputs when we aggregate we are getting this as the output.

So, in this case here this is the fuzzy outcome this is the output this is the fuzzy output and
this we can get we can convert as the convert into the crisp value again by the suitable
defuzzifier.

863
(Refer Slide Time: 26:07)

So, this way we have seen that we have discussed all the three scenarios where we have
seen multiple cases with the fuzzy model Mamdani fuzzy model with single Rule single
antecedent.

And again we saw all these cases with max-product, max-min compositions. Similarly,
multiple antecedents with single Rule; then we discussed the multiple rules with multiple
antecedents with both the compositions the max-min composition, max-product
compositions and all of these cases with the fuzzy inputs and crisp inputs.

And we saw that we all the cases in all the cases we found always found the output in the
fuzzy value, as the fuzzy value. And this fuzzy value can further be converted into the
crisp value by using suitable defuzzification methods. Alright so, this way we have
understood as to how with certain inputs to the Mamdani fuzzy model we can get the
suitable output or the corresponding output.

864
(Refer Slide Time: 27:49)

Now, when it comes to the defuzzification. So, there are several defuzzification methods,
but here we will be discussing very commonly used defuzzification methods. So, the first
one is the centroid of area method through which we can get the crisp value of the fuzzy
value. And then we have the bisector of area, mean of maximum, smallest of maximum
and largest of maximum.

So, these are some of the common commonly used defuzzification methods that we use
for converting the fuzzy value into the crisp value.

(Refer Slide Time: 28:48)

865
Let us discuss this one by one quickly. So, center of area centroid of area is COA and here
is the formula through which we get the crisp value when we have the fuzzy value and we
use this formula the center of area is very simple.

So,

∫𝑧 𝜇𝐴 (𝑧)𝑧 𝑑𝑧
𝑧𝐶𝑂𝐴 =
∫𝑧 𝜇𝐴 (𝑧) 𝑑𝑧

Where, 𝜇𝐴 (𝑧) is the aggregated output membership function means this is the output that
the fuzzy output that fuzzy model generates.

So, by applying this, we can get crisp value corresponding to the fuzzy value. So, this is
the this is the center of area which is used for converting the fuzzy value into the crisp
value.

(Refer Slide Time: 30:27)

Now, the second one is bisector of area. So, bisector of area expression is here this
basically does nothing, but when we have any fuzzy set as the output fuzzy value as the
output of the model. So, what does what it does is basically it bisects it divides into 2.

And each of these sectors each of the parts of this equal area. So, this area is same as this
area here in this case.

866
(Refer Slide Time: 31:20)

Similarly, we have the mean of maximum this is very simple, there is no calculation
normally needed. So, mean of maximum here is that we have the maximum. So, we see
that the maximum is here in this case.

So, we first take the maximum here all the maximum all the here the maximum power
portion and then we take the mean of that. So, we basically divide this into 2 here in this
case because we are taking the maximum in the mean of maximum. So, max we take the
maximum and then we take the mid value.

And this value the corresponding to this value we take as the crisp equivalent to the fuzzy
value that we are taking for the conversion. So, there is the crisp value as the output.

867
(Refer Slide Time: 32:30)

Then comes the fourth one here. The fourth here is the fourth one here is the smallest of
maximum. So, in this case if we have the this as the fuzzy value mean fuzzy value.

So, we first take the maximum in this fuzzy value. So, maximum is here you see the
maximum is here this portion is maximum. So, here we choose the smallest of this
maximum section. So, smallest of the maximum section is this and then corresponding to
this we take the output. So, this output is nothing, but the crisp value.

So, this is the crisp value this is the crisp value corresponding to three fuzzy value fuzzy
value and this is fuzzy value, ok.

868
(Refer Slide Time: 33:58)

So, then comes fifth one and fifth one is the largest of maximum. So, if this is the fuzzy
value here. So, this section again is the maximum. So, in this section we see which one is
the largest. So, largest of the maximum will lie here this this end point corresponding to
this output here the output value that is 𝑧 is the crisp value.

So, this is the crisp value if we follow this criteria. So, here when we apply largest of
maximum, if we have been given this as the fuzzy output and corresponding to this we
will get the crisp output converted.

(Refer Slide Time: 35:00)

869
So, this way we have seen as to how the Mamdani fuzzy model operates for different
scenarios, like we have the single Rule single antecedent, single Rule multiple antecedents,
single multiple rules with multiple antecedents. And then with different compositions like
max-min, max-product and again all of these with fuzzy inputs and the crisp inputs.

And that after that we discussed the some commonly used defuzzification methods. And
with this I stop here and in the next lecture we will continue with some examples of
Mamdani fuzzy model.

Thank you.

870
 Fuzzy Sets, Logic and Systems and Applications
Prof. Nishchal K. Verma
Department of Electrical Engineering
Indian Institute of Technology, Kanpur

Lecture – 54
Mamdani Fuzzy Model

Hi, welcome to the lecture number 54 of Fuzzy Sets, Logic and Systems and Applications.
In this lecture we will cover the remaining part of the Mamdani Fuzzy Model.

(Refer Slide Time: 00:16)

So, in this lecture, I will discuss an interesting example on Mamdani fuzzy model having
single antecedent with three rules.

871
(Refer Slide Time: 00:27)

So, let us have a look here at the example here. So, this is the example and this example
basically has. So, this is the example and this example basically has single antecedent with
three rules. So, what is this example let me just read. An example of single input you can
see here single input single output that means, this is a SISO Mamdani fuzzy model.

And which is shown below for the antecedent and consequent membership functions with
universe of discourse 𝑋 here, which is from −10 to 10. And the universe of discourse for
output that is 𝑌, which is from 0 to 10 respectively. And again for every 𝑥 for every 𝑥 here
and for every 𝑦 belonging into the 𝑋 and 𝑌 respectively.

So, here what we have is that we have the fuzzy reason for the input 𝑥. So, this is the fuzzy
reasons. And this fuzzy reasons have basically three these fuzzy reasons are characterized
by three fuzzy sets, first fuzzy set is the fuzzy set for small, the second fuzzy set here is
the fuzzy set for medium, the third fuzzy set here is the fuzzy set for large. So, this means
that the total input region. So, total input 𝑥 that is the generic variable.

So, this x has been divided into three fuzzy reasons; small, medium, large. And this small
medium large is within the universe of discourse −10 to 10 you can see here is −10; that
means, 𝑥 is equal to −10 to 𝑥 is equal to plus 10. So, this means this 𝑥 is within the
universe of discourse −10 to 10. Similarly the output 𝑦 so, 𝑥 is our input and 𝑦 is the
output. So, 𝑦 is again having the universe of discourse right from 0 to 10.

872
So, you can see here the 𝑦 starts from 0 here and this ends at 10. So, the 𝑦 of the output
again is divided into three fuzzy reasons, the first fuzzy reason is characterized by the
fuzzy set small. And then the second fuzzy reason is characterized by a fuzzy set for
medium. And the third one is the fuzzy set for the fuzzy region is characterized by the
fuzzy set for large. So, we can see here. So, we have the left side, left hand side here we
have the fuzzy regions and the corresponding fuzzy sets for the input. And here the right
side we have the fuzzy sets for the outputs for the fuzzy regions.

So, as it is mentioned that this fuzzy model has single input and single output. So, we can
say this SISO fuzzy model. And this fuzzy model is a Mamdani type fuzzy model because
you see all the rules are with a special type of rules where the premised part is fuzzy and
the consequent part is also fuzzy. So, you can see that the premised part of each rule is
fuzzy and the consequent part is also fuzzy.

So, and we know that Mamdani fuzzy model, Mamdani fuzzy model deals with such kind
of rules. So, we have a set of three rules; the rule number 1, rule number 2, rule number 3.
First rule says that, if 𝑥 is small then 𝑦 is small. Then the second rule says if 𝑥 is medium
then 𝑦 is medium; third rule says if 𝑥 is large then 𝑦 is large. So, if a model is known if
Mamdani model is known, this means that all these three rules are known the parameters
of all these three rules are also known.

So, when we say any model is already known, it means all the parameters here of the rules.
Like small for input small for output fuzzy reason medium for input medium for output
large for input large for output all these are known; means all these fuzzy reasons are
known. Means the fuzzy sets for characterizing these reasons are also known. Now, this
model this Mamdani fuzzy model is known.

Now, the question is that since this model is known if we have an unknown input and we
are giving this unknown input to the model what will be the output corresponding to this
input? So, we all know that the Mamdani fuzzy model can take the input in the form of a
fuzzy set are in the form of a crisp value. So, here in this problem in this example basically,
we have the input which is given as the 𝑥 is equal to −3.9. So, when we supply this input
−3.9 what will be the output? So, precisely the question is that what will be the output for
and input 𝑥 −3.9?

873
So, this is what is the question. So, let us now go ahead and see how we can manage to
solve this, how we can manage to get the answer for this. So, the question is that, we have
an input and the input the value of the input is −3.9 and comparable to that are
corresponding to this input what will be the 𝑦. So, I can write y here. So, let us now use
the Mamdani fuzzy model and find the output corresponding to 𝑥 is equal to −3.9. Here
in this example we have three rules and each rule has single antecedent.

So, we can see premise part has single antecedent, means single inputs. And we also see
that all these premise part of the rules are fuzzy like small, medium, large. And the type of
input that we can supply here in fuzzy can be can be fuzzy can be crisp, but here in this
case the type of input that we are supplying is crisp.

(Refer Slide Time: 09:33)

So, let us now go ahead and see what is going to be the output corresponding 𝑥 is equal to
−3.9 here. So, what we first do is, we first try to get the points try to get the input.

So, our 𝑥 is equal to −3.9 is here see here. And here we try to see if we draw a line a
straight line at parallel to the 𝜇(𝑥) axis this is the membership grade axis this is 𝜇(𝑥). So,
parallel to the 𝑦 axis are 𝜇(𝑥) axis I would say. So, when we draw a line we see that the 𝑥
is equal to −3.9 is giving us the two intersection points, two points of intersection, one
point of intersection is at 0.3 another point of intersection is at 0.7. And then 𝑥 is equal to
−3.9 is cutting or intersecting the fuzzy set is small at 0.3 and it is intersecting medium
fuzzy set at 0.7.

874
So, we have two intersect points of intersection. So, two fuzzy sets are being intersected
by the line which is parallel to 𝜇(𝑥) and this is at 𝑥 is equal to −3.9 here. So, once we
have now come to know that, we have the point of intersection and we also see that when
we draw this line parallel to 𝜇(𝑥) axis. Then we see what are the fuzzy reasons, which are
applicable which are being which are belonging to this particular input. Or in other words
we can say that what are the fuzzy reasons which are relevant to this input 𝑥 is equal to
−3.9.

So, for −3.9 we have two fuzzy reasons; one is a small and the another one is medium.
Large is not relevant because the 𝑥 is equal to −3.9 is not intersecting the fuzzy set large.
Or in other words we can say 𝑥 is equal to −3.9 is not intersecting fuzzy set or fuzzy region
large. So, we have two points of intersection. So, once we finish this then we now come
to the rules. So, the given rules are here for the fuzzy set. So, rule number 1 so, we have
two fuzzy sets which are relevant.

So, we first seen what are the rules which what are the fuzzy reasons which are relevant
for this input. So, for this input we have a small and medium and these two are relevant.
So, applicable rule is I will just tick. So, we since we have the input which is falling in the
fuzzy reason is small.

So, this rule will be applicable, means rule 1 will be applicable. And then the medium rule
number 2, since our input is also falling under medium fuzzy reason. So, this second rule
the rule number 2 will also be applicable. So, these two rules will be applicable. Third rule
will not be applicable because our input 𝑥 is equal to −3.9 is not falling under the large
fuzzy set.

So, this way we see that we have two rules that are applicable. Now, when we know now
that there are two rules which are applicable. Now, let us find the output value
corresponding to each rule as we have done, when we have discussed the Mamdani fuzzy
model for single antecedent multiple rules.

875
(Refer Slide Time: 14:34)

So, we can see that this two rules which are applicable here, both the rules are highlighted
here only two rules are applicable. And now let us find the outputs of the corresponding
rules for the input value 𝑥 is equal to −3.9.

(Refer Slide Time: 15:12)

So, here we have the first rule where this is rule number 1. So, first rule says that, my input
is falling in the small, then the output is also falling in small. Second rule says the input is
falling in medium, then the output is also falling in medium. So now, let us find the point
of intersection here for these two rules. So, the rule number 1 and rule number 2. So, since

876
we have the x that is the input that is equal to −3.9. So, corresponding to this value x is
equal to −3.9, we see here that we are getting two points of intersection.

So, let us call this as the 𝑤1 and this as 𝑤2 point of intersection. So, we have single
antecedent, but multiple rules, so multiple rules means here we have two rules. So, now,
as we have done while we discussed the Mamdani fuzzy model having single antecedent
multiple rules. So, in that case, what we did was that for max-min comp max-min
composition we could get the minimum of this like.

(Refer Slide Time: 16:49)

So, since we have single antecedents. So, minimum will be 0.3 only here, so 𝑤1 which is
0.3. So, since we have only one value that is 𝑤1 and the minimum will also be 𝑤1. So, here
this is going to be the 𝑤1, which is 0.3. Now, we have to truncate the small fuzzy set, the
fuzzy set for a small like this here and we have to only keep this value that the truncate
here.

And similarly this was for the rule number 1, this was for rule number 2. So, corresponding
to rule number 1, means when we apply the input to rule 1, the output of the rule is this,
you can see here the output of the rule here the fuzzy output I can write here fuzzy output
corresponding to rule number 1.

Similarly, now here we have the point of intersection which is nothing but the weight. So,
𝑤2 let’s say and this is 0.7. Now, corresponding to 𝑤2 in other rule the second rule, when

877
we truncate the fuzzy set for medium and we choke chop it off and then the remaining part
of medium fuzzy set will be the output of.

So, output fuzzy output corresponding to corresponding to here the it is for input 𝑥 is equal
to −3.9. Similarly, here also the fuzzy output corresponding to rule number 2 for input 𝑥
is equal to −3.9. So, now, since we have two fuzzy outputs, we had two rules that are fired
that are applicable in this for this input 𝑥 is equal to −3.9 and these two inputs now have
to be aggregated.

So, these two input because we are applying they are taking the max-min composition,
max min composition. So now we have to take the maximum of the two inputs means we
have to aggregate.

(Refer Slide Time: 20:41)

And then, when we aggregate when we take max you see here and basically max means
we are taking the union of these two. And we are getting here the after taking the union
we are getting and in an irregular shape basically and which is nothing, but a fuzzy value.

So, this is a fuzzy value. Now, our next job is to get the crisp value, we have to get the
crisp value after converting this fuzzy value. So, this is fuzzy value now we need to convert
it.

878
(Refer Slide Time: 21:24)

So, let us defuzzify here using the center of area. Let us now go ahead and use the center
of area formula, which I discussed in the previous lecture. And here the center of area is
very simple, here we all know that the center of area we can find by integrating here like
this and then we divide it by the as you mentioned here.

∫𝑦∈𝑌 𝜇(𝑦)𝑦 𝑑𝑦
So, 𝑦𝐶𝑂𝐴 = . So, since this is an irregular shape, so what we do is we first find
∫𝑦∈𝑌 𝜇(𝑦) 𝑑𝑦

all these vertex vertices all these points here, the first point here is the start point that is
basically the 0 and then 0, 0.3. And then similarly we have this vertex here as 1.7515, 0.3
similarly here, so I just use different colors so, that. So, here we have this I am using green
color for all showing all the vertex here and then this vertex and then this vertex.

So, all these vertex are known because we know the shape. And if we draw this on the
graph paper or even otherwise also we can know the vertex. And then when we know this
then we can use the formula for centroid of area or center of area here and we apply this
formula. So, we can get 𝑦𝐶𝑂𝐴 you can see here 𝑦 coordinate of the center of area. So, what
we do here is that, we integrate the 𝜇(𝑦), the 𝜇(𝑦) from here to here you see for the first
segment 𝜇(𝑦)𝑦. So, 𝜇(𝑦) is all throughout 0.3 and then we multiply this by 𝑦 and then we
take 𝑑𝑦.

So, we have one integral and this limit is from 0 to 1.7518. So, the limit is here this limits
0 to 1.7518. So, when this is done then so, this part is over. Now, now let us go to this part

879
this part. So, I will show it like this. So, here we have first part then second part area and
then third part and then fourth part here. So, we have basically four parts. So, first part is
here this is first part. Now, this is the second part that we have here since its second part.

So, second part is basically here, we have the equation of line here which is you can see
here the 𝜇(𝑦). So, 𝜇(𝑦) is point 0.9747 𝑦 −1.4074. So, this is the expression for the 𝜇(𝑦)
here for this segment the second segment. So, this we write as it is here you can see. And,
when we write this then we multiply this the whole thing by 𝑦 and we put the limit here as
you can see here the limit. The limit of the second segment starts from 1.7518 to 2.1622.

So, this is the limit this one and this one, so from this to this alright. So, now, once this is
done now we move to the third segment, third segment is this is the third segment. Third
segment 𝜇(𝑦) we see that here 𝜇(𝑦) this is the this is for the second segment. So, let me
first name it. So, we have the first segment this is the integral for first segment. And then
we have the second segment and then here we have let me write this also and then is and
then here we have the third segment.

So, third segment here 𝜇(𝑦) is constant and this value is nothing but here 0.7. So, 𝜇(𝑦) is
0.7 for this structure and we multiply this with 𝑦. So, 0.7 into 𝑦 and then we take 𝑑𝑦, we
write 𝑑𝑦 because we had to integrate it with respect to 𝑦. And the limit will be right from
the start of the third segment to end of the third segment. So, this is going to be 2.1622 to
5.5956 you can see here. So, this way we write the expression for the third segment.

Now, fourth segment is here this is fourth segment. So, for fourth segment now we get the
equation of 𝜇(𝑦) here 𝜇(𝑦), 𝜇(𝑦) which is here you see 𝜇(𝑦). And this 𝜇(𝑦) is nothing
but −0.4984 𝑦 + 3.489. So, we use this value here as it is and then we multiply this by 𝑦.
So, we multiply with 𝑦 and then we write 𝑑𝑦 because we have with respect to 𝑑𝑦 we have
to integrate.

So, here we see that, this is the expression for the fourth segment. And once this is over
then comes the denominator part where we have the expression for the integrate
integrations here. And then when we solve this with all these limits what we get here is
this.

So, 𝑦𝐶𝑂𝐴 means the 𝑦 coordinate of the center of area or centroid of area here for 𝑥 is equal
to minus 3.9, we are going to get 3.6327. Output is by this method 𝑦 is equal to 3.6327.

880
Now, as I have already mentioned that we have other methods also to defuzzify the fuzzy
output.

(Refer Slide Time: 30:05)

So, let us now see what we are getting when we use the bisector of area. So, bisector of
area method is basically the defuzzified value here is the crisp value and we defuzzify the
fuzzy value. So, here this bisector of area method, it actually divides the whole fuzzy value
into two parts and the line which divides which bisects basically is the representation of
the 𝑦 value the output.

So, the formula is here the expression here is here for finding the bisector area. So, let us
Β
assume that we have the 𝑦𝐵𝑂𝐴 you see here ∫𝑦 𝜇(𝑦)𝑑𝑦. So, where and then we have
𝐵𝑂𝐴
𝑦
here alpha and then the ∫𝛼 𝐵𝑂𝐴 𝜇(𝑦)𝑑𝑦. And you see that this left hand side is equal to the
right hand side, where 𝛼 = 𝑚𝑖𝑛(𝑦|𝑦 ∈ 𝑌) and 𝛽 = 𝑚𝑎𝑥(𝑦|𝑦 ∈ 𝑌).

So, when we write the expression here. So, our 𝛼 is here you see the 𝛼 is 0 and then because
this is the minimum value minimum 𝑦; means the lower limit and the 𝛽 here is 7 this value.
So, when we apply this formula and we write the expression here corresponding to our
problem, so with all the limits and all. So, when we solve this, we are getting the output
through the bisector of area method. So, 𝑦 we call this as 𝑦𝐵𝑂𝐴 , for 𝑥 is equal to the input
−3.7, so then we get this as the crisp value. So, this is the output that we get output.

881
(Refer Slide Time: 32:52)

Similarly, when we use the defuzzification method, mean of maximum we have already
discussed this in previous lecture. So, we have this shape here see. And we find the
maximum first, so maximum is this. And then we take mean of this we know these limits
here these two limits, then we take the mean of this. So, we see that you see here 2.16 here
and then 5.5956 and then when we take mean we divide we add these two together and
then we take the average of it.

So, when we do this 𝑦𝑀𝑂𝑀 is giving us the output corresponding to 𝑥 is equal to minus 3.9
as 3.8789. So, this is the output.

882
(Refer Slide Time: 34:03)

Similarly, now when we use another defuzzification is key, let us say which is smallest of
maximum. So, smallest of maximum is going to be this here the maximum we know and
then the smallest is the 2.1622. So, just by looking at the fuzzy value as the output we can
get the crisp value through this defuzzification method.

So, here we have got the corresponding to the input x is equal to −3.9, we get 2.1622.
Similarly now, we when we use the largest of maximum we get this value as the defuzzy
defuzzified value that is the crisp value corresponding to 𝑥 is equal to −3.9. So, that is
how we are getting these values as the crisp value corresponding to the fuzzy value through
the various defuzzification schemes.

883
(Refer Slide Time: 35:10)

So, here basically in this example that we just discussed, we have input fuzzy regions; you
see here we have three input fuzzy regions; the first one was the small and then second is
the medium the third is the large. And similarly, the output also is divided into three fuzzy
regions. So, when we here in this problem, we took only one value of the input that it 𝑥 is
equal to −3.9. So, let us say when we take various values of 𝑥 as input and then what
would be the corresponding output?

So, here when we take the 𝑥 value right from −10 to 10 what we see here is the output
will be like this. So, we can say that for other inputs input values of 𝑥 we can obtain the
crisp outputs, the corresponding plot between the different values of input and output can
be shown as below. So, this exercise I am leaving for you and you can try for various
values of 𝑥 and see that your output will be falling on this input output characteristic curve.

884
(Refer Slide Time: 36:40)

So, at this point I would like to stop. And in the next lecture, I will discuss an example on
Mamdani fuzzy model having two antecedents with four rules.

Thank you.

885
 Fuzzy Sets, Logic and Systems and Applications
Prof. Nishchal K. Verma
Department of Electrical Engineering
Indian Institute of Technology, Kanpur

Lecture – 55
Mamdani Fuzzy Model

Hi, Welcome to the lecture number 55 of Fuzzy Sets, Logic and Systems and Applications.
In this lecture, we will discuss another example which is based on the Mamdani Fuzzy
Model.

(Refer Slide Time: 00:18)

So, here in this Mamdani fuzzy model, we will have two antecedents with four fuzzy rules.

886
(Refer Slide Time: 00:34)

So, let us quickly go through this example and in this example we have a fuzzy model
which is also called a controller here. So, in this controller, the input is 𝑥 and another input
is 𝑦. So, we have two inputs input 𝑥 and then another input is 𝑦. And corresponding to
these two inputs we have the output which is 𝑧 the 𝑧 is here.

So, this input 𝑥 input 𝑦 input 𝑧 are partitioned as shown in the figure, figures here and the
applicable rules are given. So, I will not say applicable rules rather I will say the rules that
are there for this model, the available rules are available here. So, we have four rules that
are available. So, we have rule number 1 and then we have rule number 2, rule number 3,
rule number 4.

So, rule number 1 says that if input 𝑥 is very high or input 𝑦 is medium, then output 𝑧 is
high. Similarly, rule 2 says that if input 𝑥 is medium or input 𝑦 is low, then output 𝑧 is
low. Similarly, rule 3 says if input 𝑥 is medium or input 𝑦 is high, then output is medium.
Rule 4 says, if output 𝑥 is low or input 𝑦 is medium, then output is medium. So, let us first
understand the rule here, the rule we have four rules, four fuzzy rules in this Mamdani type
of fuzzy model.

So, the in this fuzzy rule basically we have two antecedent. So, if we look at the premise
part here of each rule we have two inputs 𝑥 and 𝑦. So, that is why we can say that we have
two antecedents see here. So, we have two antecedent 𝑥 and 𝑦. 𝑥 is low and 𝑦 is medium
for example in rule number 4.

887
So, we have two antecedents and now how many rules do we have here is? We have four
rules four fuzzy rules. So, we can say this is a class of multiple antecedents and multiple
rules. And we see that the inputs are divided input fuzzy regions are basically input is
divided into multiple fuzzy regions. So, input 𝑥 you see that here please look at this input
𝑥 is divided into multiple fuzzy regions in each region is represented by a fuzzy set low,
medium, high, very high.

Similarly, the input 𝑦 also is divided into multiple fuzzy regions. So, that is here in this
case we this is divided into 3 fuzzy regions, the low, medium, high and the output here is
divided into again 3 regions which is low, medium and high. One more thing I would like
to mention here that in this fuzzy rule in this fuzzy Mamdani fuzzy model where these four
rules multiple fuzzy rules are there and these rules have the connectives and the connective
here is OR instead of AND.

So, when we have OR connective it means we take the for all for the both the inputs when
we find the when we compute we use for OR we take the union. So, we will see when we
will be discussing this ahead. So, let us understand here that the membership functions for
input and output for inputs and output are defined as follows. So, 𝑥 is this 𝑥(𝐿𝑂𝑊) we
have gaussian here gaussian function which is x and then we have the mean and variance.
So, the mean here is 0 and variance here is the standard deviation here is 10.

Similarly, 𝑥(𝑀𝐸𝐷𝐼𝑈𝑀) is triangular membership function and you can see here triangular
membership function, then 𝑥(𝐻𝐼𝐺𝐻) is trapezoidal membership function and then
𝑥(𝑉𝐸𝑅𝑌 𝐻𝐼𝐺𝐻) is gaussian membership function.

Similarly, we have 𝑦 fuzzy set, the membership function for 𝑦(𝐿𝑂𝑊) is triangle triangular
membership function and the vertices are 0, 0, 40, Similarly, for 𝑦 we have 𝑦(𝑀𝐸𝐷𝐼𝑈𝑀),
we have gaussian membership function with 50 mean and 10 as the standard deviation.
Similarly, 𝑦(𝐻𝐼𝐺𝐻) is trapezoidal membership function. And then we have output
𝑧(𝐿𝑂𝑊), 𝑧(𝑀𝐸𝐷𝐼𝑈𝑀) and THEN 𝑧(𝐻𝐼𝐺𝐻), the trapezoidal membership functions
respectively.

So, what is the problem here? What is the, what we need to do here is we need to find here
the OUTPUT 𝑧 for INPUT 𝑥 and 𝑦. So, when the INPUT is 55, INPUT 𝑥 is equal to 55,
𝑥 is equal to 55 and OUTPUT and INPUT 𝑦 is equal to also 45 here. So, we have two

888
inputs, one input is 55 another input 𝑦 is the 45. So, when these two inputs are there,
corresponding to these two inputs 𝑥 and 𝑦 we need to find the 𝑧.

So, let us now first find that as to how many rules are applicable, how many fuzzy regions
are relevant for these two inputs as we have seen in the previous example which was
discussed in the last lecture, the previous lecture.

So, let us find first the output and then of course, yes because this is a Mamdani fuzzy
model the output is going to be the fuzzy value finally, after getting the aggregated value.
So, and then this aggregated value will be converted into the crisp through it is a
defuzzification method. So, these defuzzification methods are mentioned here that the you
have to find the crisp value by using centroid of area, bisector of area, smallest of
maximum method, largest of maximum method, mean of maximum method. So, these we
have to use.

(Refer Slide Time: 08:18)

So, let us first find the for 𝑥 is equal to 55 and for 𝑦 is equal to 45. So, 𝑥 is equal to I am
writing here for 𝑥 is equal to 55 and let me write here no the for INPUT 𝑥 is equal to 55
𝑦 is equal to 45. So, let us find the OUTPUT. So, this is what is the question in this
example.

889
(Refer Slide Time: 08:56)

So, the target here is that you have to find the OUTPUT. So, let us first find that as to how
many fuzzy rules are applicable. So, for finding the applicable fuzzy rules, we have to first
look at the relevant fuzzy regions which are relevant for the particular INPUT. So, when
we see that for 𝑥 is equal to 55, you can see here for 𝑥 is equal to 55, this 𝑥 is equal to 55
is falling under the region of high only. So, this 𝑥 is equal to 55 is falling in the high region,
no other region is visible to be relevant.

So, maybe because we have very high which is a gaussian. So, maybe it may be relevant
here, because this is not very visible, but very negligible value we will be getting, but let
us see. So, since very high is a gaussian so, of course, this is not going to be 0 for 𝑥 is
equal to 55 similarly, for 𝑦 is equal to 45. So, 𝑦 is equal to 45, we see that only medium is
applicable because low is having 0 value for corresponding to 𝑦 is equal to 45 and similarly
high is also having 0 value. So, I can write here fuzzy regions are applicable are relevant
for 𝑥 is equal to 55.

So, I should write it like this 𝐼 𝑠𝑡 fuzzy region is high region and the 𝐼𝐼 𝑛𝑑 region is the very
high region. Here only one fuzzy region is relevant, only one fuzzy region that is medium
is relevant for y is equal to 45. So, here we have three fuzzy regions. So, 𝐼𝐼 𝑛𝑑 fuzzy region
here, 𝐼𝐼𝐼 𝑟𝑑 fuzzy region here is that the low is also there.

890
(Refer Slide Time: 12:20)

And before this let us first find the values also like the points of intersection, the
intersection values. So, here the values are there you can just look at these values. So, we
can come to this later also and these are the membership functions which are characterizing
the corresponding fuzzy sets.

So, the combination of rules that are obtained here is. So, all these combinations are there.
These are the input combinations. So, since we have these input combinations here
𝐿𝑂𝑊, 𝑀𝐸𝐷𝐼𝑈𝑀, 𝐻𝐼𝐺𝐻, 𝑀𝐸𝐷𝐼𝑈𝑀, here 𝑉𝐸𝑅𝑌 𝐻𝐼𝐺𝐻, 𝑀𝐸𝐷𝐼𝑈𝑀.

(Refer Slide Time: 13:34)

891
Now, let us check whether these combinations are existing in the rules that are provided
or not. So, we see that we have been given only four rules and in this four rules we have
two rules which are taking our combinations. 𝐿𝑂𝑊 − 𝑀𝐸𝐷𝐼𝑈𝑀, that means, the
highlighted ones and then the 𝐿𝑂𝑊 − 𝑀𝐸𝐷𝐼𝑈𝑀 is here and then we have 𝑉𝐸𝑅𝑌 −
𝐻𝐼𝐺𝐻 𝑀𝐸𝐷𝐼𝑈𝑀. So, we see that these two are there which are existing in the set of rules
that are given to us, when we say the set of rules that are given to us means the model is
having these rules.

So, then when we look at the second combination that 𝐻𝐼𝐺𝐻 − 𝑀𝐸𝐷𝐼𝑈𝑀 which is not
there in the set of rules. So, we just we will just discard it. So, we’ll only take this rule
number 1 and rule number 4 and we will proceed further.

(Refer Slide Time: 14:46)

So, when we do that. So, for the 1𝑠𝑡 rule, 1st rule where we have if 𝑥 is 𝐿𝑂𝑊, 𝑦 is
𝑀𝐸𝐷𝐼𝑈𝑀 and then what we have here is the 𝑧 is 𝐻𝐼𝐺𝐻, 𝑥 is 𝐿𝑂𝑊 and 𝑦 is 𝑀𝐸𝐷𝐼𝑈𝑀 then
𝑧 is 𝑀𝐸𝐷𝐼𝑈𝑀. So, the 𝑀𝐸𝐷𝐼𝑈𝑀 is here. Similarly the rule number similarly the rule
number 2 is here. The rule number 2 is that we have 𝑉𝐸𝑅𝑌 𝐻𝑖𝑔ℎ if 𝑥 is 𝑉𝐸𝑅𝑌 𝐻𝑖𝑔ℎ, 𝑦 is
𝑀𝐸𝐷𝐼𝑈𝑀 then 𝑧 is 𝐻𝐼𝐺𝐻. So, this way when we as we have already done in the previous
example.

So, when we see the intersection points. So, the intersection points as these are computed
here from the membership functions 𝜇(𝑥) these are the 𝜇(𝑥) and this is 𝜇(𝑦). So, we see
that we have two membership functions membership grades. So, here the first intersection

892
point is 2.69 × 10−7 which is very low, but we have we need to write this here because
we are computing and then we have another value which is 0.9 and since we are taking
here the max-min composition.

So, what we are doing here is that since we have the connective OR please understand we
have OR combination we have the OR connective. So, for OR connective we will use max
instead of the min. So, that is why we are taking the max of the 2.69 × 10−7 and 0.9. So,
this is going to be the, if we use 0.9 for truncation. So, we use this value as the fuzzy
output.

So, this is the fuzzy output from rule number, from rule 1 for the input 𝑥 is equal to 55 and
𝑦 is equal to 45. Now on the same lines if we move for rule number 2, we get point of
intersection here for very high 4.0 × 10−5 and 𝑦 is equal to for 45 for 𝑦 is equal to 45, we
get 0.9. Here also we get 0.9 as the max of these two values and this when we use this
value for truncation we are getting this is this as the fuzzy output here. And this fuzzy
output is coming out of coming out of the fuzzy rule, fuzzy rule 2. So, now since we have
the output from both the rules since only two rules are applicable here.

(Refer Slide Time: 18:26)

So, these two rules basically, now further when we aggregate these rules will be after get
up after aggregating this I am writing here the aggregation, aggregation of the outputs and
what we do here is we take the MAX of the two. So, we take the MAX, I am just writing
the MAX ok, means we take the maximum of this. We take the union of this. So, I am

893
writing the MAX. So, when we take MAX so, what we are getting here is, this structure
and this is nothing but, a fuzzy value the output here is the fuzzy value.

Now, it is quite interesting here, this structure this fuzzy value is the irregular structure
irregular shape. So now we have to convert this into crisp and this irregular structure,
irregular shape can be converted into the crisp, that means, that means, it is a fuzzy value
which needs to be converted into crisp. So, we have the methods for defuzzification that
are available.

(Refer Slide Time: 19:54)

The first of the methods is here, first method is the centroid of area. So, when we use
centroid of area. I am not going to discuss the again because we have discussed in detail
in the previous example in detail. So, we get corresponding to 𝑥 is equal to 55, 𝑦 is equal
to 45, we are getting the output as 𝑧 ∗ which is 69.84 all right. And then when we use
bisector area we are getting 𝑧 the output here as 𝑧 ∗ is equal to 72.

894
(Refer Slide Time: 20:38)

Similarly, when we use the smallest of maximum method here, we get the 𝑧 ∗ here, 𝑧 ∗ is
our 𝑧 ∗ which is the output corresponding to 𝑥 is equal to 55 and 𝑦 is equal to 45 that is
49.65. Similarly, when we use largest of maximum our 𝑧 ∗ is here 100. So, this you can see
from the figure here the structure irregular shape you can very easily get these values.

(Refer Slide Time: 21:15)

And similarly, when we take the mean of maximum the 𝑧 ∗ that is the output corresponding
to 𝑥 is equal to 55 and 𝑦 is equal to 45, we are going to get 69.41.

895
(Refer Slide Time: 21:35)

So, this way we see that the output that we are getting are the values which we have
discussed from various methods of defuzzification and that is how we get we obtain these
values, the test values. So, with this I would like to stop here and in the next lecture we
will discuss the Larsen fuzzy model.

Thank you.

896
 Fuzzy Sets, Logic and Systems and Applications
Prof. Nishchal K. Verma
Department of Electrical Engineering
Indian Institute of Technology, Kanpur

Lecture - 56
Larsen Fuzzy Model

(Refer Slide Time: 00:14)

Hi, Welcome to lecture number 56 of Fuzzy Sets Logic and Systems and Applications.
Today in this lecture I will discuss Larsen Fuzzy Model. We have already covered the
Mamdani fuzzy model and now in today’s lecture, I will discuss with you the Larsen fuzzy
model which is very similar to the Mamdani fuzzy model, in the sense that the output here
is the fuzzy.

So, if we have fuzzy rule base with n number of rules and our input membership functions
could be 𝐴𝑖 , 𝐵𝑖 and so on with the universe of discourses 𝑋 and 𝑌 and so on respectively.
So, the output membership function of the resultant fuzzy set let us say here in this case
𝐶𝑖 , if we are taking only two antecedents where the first antecedent is 𝑥 is 𝐴𝑖 and the
second antecedent here is 𝑦 is 𝐵𝑖 and both are connected by the connective either A-N-D
AND, or, O R. So, both the antecedents are connected here by the connective A-N-D or
O-R.

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So, here we have taken a typical fuzzy rule the typical fuzzy rule here is that as you see if
𝑥 is 𝐴𝑖 and then we have the connective AND oblique or so, either of these could be and
then the other antecedent here is 𝑦 is 𝐵𝑖 . So, these two are the antecedents and then we
have then part; that means, the consequent part starts now. So, then 𝑧 is 𝐶𝑖 , so here we
have the consequent part and this is the premise part or antecedent part.

So, we have a rule a fuzzy rule where we have two antecedents, here this can have multiple
antecedents or even the single antecedents also. So, we can have multiple cases which we
have discussed in the coming slides and here in this particular rule we have two
antecedents, first antecedent and the second antecedents and then we have the connective
here. So, these two antecedents here will be connected with any suitable connective either
AND or by OR.

So, as I already mentioned that this premise part can have either only one antecedent or it
can have multiple antecedents. So here for simplicity, we are taking only a typical rule
with two antecedents and then we have in the consequent part a fuzzy output. So, in
Mamdani fuzzy model also we saw the rule set and these rule sets were basically having
the antecedent part fuzzy and the consequent part fuzzy.

So, here also we have for the Larsen fuzzy model the set of fuzzy rules that will be
applicable will be the antecedent part fuzzy and the consequent part fuzzy. So, both
Mamdani and Larsen will have the same kind of fuzzy rules applicable. So, here 𝐴𝑖 in 𝐴𝑖
and 𝐵𝑖 we have written, see here and this 𝐴 and 𝐵 are nothing but the fuzzy reasons or
fuzzy value corresponding to the input 𝑥 and 𝐵𝑖 is the fuzzy value which is nothing but
fuzzy set corresponding to 𝑦. 𝑖 here signifies the this fuzzy set for 𝑖 𝑡ℎ rule.

So, if this is the 𝑖 𝑡ℎ rule I can write here 𝑅𝑖 , or in other words I can write here 𝑅𝑖 . So, I
represents the 𝑖 𝑡ℎ rule I can write here the 𝑖 𝑡ℎ rule. Now if we have let us say n number of
rules, so if my 𝐴𝑖 is like this, 𝐵𝑖 is like this, we know this just the 𝐴𝑖 is a fuzzy set for 𝑖 𝑡ℎ
rule, 𝐵𝑖 is the fuzzy set for antecedent of the 𝑖 𝑡ℎ rule, similarly 𝐶𝑖 is the fuzzy set for the
output.

Now, the firing strengths of the 𝑖 𝑡ℎ rule here is defined as we have already done in
Mamdani. So, firing strength is also known as the weight. So, I have use weight word in
the Mamdani fuzzy model. So, weight and firing strength both are the same. So, here

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weight we write by 𝑤𝑖 , 𝑤 subscript 𝑖. So, if I have let us say first rule so, for the first rule
I will have the 𝑤1 as we have written here written there in case of Mamdani.

So, 𝑤𝑖 we can find by simply taking the min of these two antecedents corresponding
membership values and these we could find by the points of intersection. So, we can have
max min and then we can have max products. So, here if we have let us say two
antecedents. So, if we have two antecedents are more in antecedents right. So, for two or
more antecedents in max min, we will have the we will take here for max min we will take
min and in case of max product, we will multiply simply. We will take the product of the
antecedent values the weight values to get the final weight.

So and when we have done this then comes the membership function for the fuzzy
output 𝐶′. So, let us say we have here in the as in this case we have the output fuzzy set 𝐶.
So, then if we have 𝐶 it and accordingly after the truncation with the help of the weight
values are firing strength of the rule, we will be getting the truncated fuzzy value which if
it is denote by 𝐶′.

So, then membership value of the membership function of the this 𝐶′ fuzzy set can be like
this here. And if we have let us say n number of rules so, we take the maximum of all the
fuzzy outputs. So, for that we have used the maximum. In other words the final output we
can say that we add them we take the maximum or we say that we take the union of these.

So, the fuzzy output 𝐶′ can be further whatever for the output that we will be getting will
be fuzzy. So, 𝐶′ basically is fuzzy and then this 𝐶′ can be further defuzzified suitably as
discussed before to obtain a crisp value.

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(Refer Slide Time: 08:53)

So, let us now take some of the cases which we have already discussed for the Mamdani
fuzzy model as well. So, here in Larsen fuzzy model, we have the cases the first case is
single rule with single antecedent. You can see here single rule with single antecedent then
we can have the single rule with multiple antecedents and then we have the third case
multiple rules with multiple antecedents.

So, all these three cases for Larsen fuzzy model I will discuss in this lecture and then again
all these three cases we will discuss for the composition that is max-min and the max-
product composition. So, both the composition will be discussed here in this lecture.

So, we have two compositions; first composition is max-min composition so, both the
compositions, the first is the max-min composition and the second one is the max-product
composition. So, both the compositions will be discussed here for all the cases and then
again we will discuss the all these cases will be discussed with respect to the fuzzy input
and the crisp input both. So, let us now first take up this case, there is a first case where
we have single rule with single antecedents.

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(Refer Slide Time: 10:46)

So, single rule with single antecedent, we have the rule of this type. So, here we have the
rule please see here. So, here we have the single antecedent and the single rule. So, we
have a Larsen fuzzy model which has only one rule here as mentioned here and then in the
rule we have the antecedent only one so, means the single antecedents. So, these we can
see here in the premise part, we have only one antecedent that is 𝑥 is 𝐴.

So, since we have only one antecedent, we do not need any connective here. So, we have
only one antecedent that is 𝑥 is 𝐴; 𝑥 is 𝐴. 𝑥 here is nothing but, the input variable input
generic variable and 𝐴 here is nothing but the fuzzy region in which this 𝑥 can fall. Now,
if we have only one rule here as the Larsen fuzzy model or in the Larsen fuzzy model so,
then if we have any unknown input comes as 𝑥 is 𝐴′.

So, if any unknown input that comes as the input to this model and this input please
understand is the fuzzy input means a fuzzy input is fed to the fuzzy model and for this
unknown input what will be the corresponding output. So, let us understand how we can
get it. So, if we have the input here.

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(Refer Slide Time: 12:56)

We have, they are same as the premise part remains the same as the Mamdani fuzzy model.
So, we as we have already seen in Mamdani model we have the premise part antecedent.

So, the antecedent have we have as 𝑥 is 𝐴 so, this is already known. Now this input is
coming, this is the unknown input, this is unknown input that is coming and this unknown
input is a fuzzy input. So, I can write here the unknown fuzzy input. So, when this input is
coming this input is superimposed here. So, we can clearly see that 𝐴′ is superimposed
here over 𝐴. So, when both are shown together so, 𝐴′ and 𝐴 both are superimposed we see
a point of intersection here as 𝑤1, I can write here 𝑤1. So, this we call as the firing strengths
for single antecedent type of rule fuzzy rule.

So, 𝑤1, we get the point of intersection here we have already discuss this in the Mamdani
fuzzy rule. So, there should not be any confusion over here and this 𝑤1 comes out to be
so, what is 𝑤1 here is the point of intersection. We might get two points of intersections
since we are taking fuzzy value fuzzy set so fuzzy set is characterized by a fuzzy
membership function. So, we might get multiple points of intersections normally two at
the max.

So, in that case what we will do? We will take only the max of all the intersection points.
So, here we are getting only one point of intersection that is 𝑤1. Here this is 0.58 and we
have designated this as 𝑤1, this is called the weight of the rule number 1. So, here since
we have only one rule so, we do not worry about the number.

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So, we have only one rule. So, we can call either this by 𝑤 or 𝑤1 so, we have this as the
𝑤1. This is as I mentioned this is the weight of the rule this is this is also called as the firing
a strength of the rule. So, boths of these terms the weights and the firing strengths remain
the same they are used interchangeably. So, now, what next? So, next is that we take this
point five eight value here and with this value we find the output.

So, how do we do that? So, here since 𝐵 is given this is given membership function, this
is the given output fuzzy set and here we have the corresponding fuzzy set that is 𝜇𝐵 . So,
how do we get the corresponding output fuzzy set corresponding to the 𝐴′? So, here this
value the highest value is scaled down to 0.58, here we have 1, the highest value here is
the highest the height of the fuzzy set 𝐵 is 1.

Now, the height of the resulting fuzzy set has to be the maximum height of this resulting
membership function or the fuzzy set has to be 0.58. So, this has to be 0.58. How do we
get that? And correspondingly what we need to do here is all the other points, all the other
edges have to be is scaled down. So, how do we get this? So, for this as I have already
mentioned that we take this weight here we have w 1 and then we multiply this with the
𝜇𝐵 (𝑦) and when we multiply this with mu y, we are going to get the membership function
of the resulting output fuzzy set that it 𝐵′. So, I can write here 𝜇𝐵′ (𝑦).

So, this 𝜇𝐵′ (𝑦), I can write it like this 𝜇𝐵′ (𝑦) in this case, this 𝜇𝐵′ (𝑦). So here, so with this
we have gotten the output fuzzy set here which this output fuzzy set is this fuzzy set which
we have got just multiplying 𝑤1 the weight of the fuzzy rule with the 𝜇𝐵 (𝑦).

So, since we have only one antecedent here, since we have only one antecedent and only
one rule. So, since we have only one rule which means there is no other rule so, we cannot
take the max. So, here the max of this will remain the same. So, we are just taking this
here and we say that corresponding to 𝐴′ as the 𝐴′ as the unknown fuzzy input to Larsen
fuzzy model the output is 𝐵′ the output is 𝐵′.

And this 𝐵′ can be denoted by this 𝐵′ can be denoted by simply I can write here this is the
output and this 𝐵′ and this is nothing but the summation 𝜇𝐵 or 𝑦 𝜇𝐵 . So, 𝜇𝐵′ (𝑦) and then
𝑦 here we have the universe of discourse 𝑌. So, this is how we can get the output.

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(Refer Slide Time: 20:37)

Now if we compare this output here with Mamdani fuzzy model, we see that this is the
output which is which we get by using the Larsen fuzzy model, we can compare this fuzzy
output this output with the Mamdani fuzzy model. So, we see that in Mamdani fuzzy model
this 𝑤 was just used for truncating the fuzzy set 𝐵, but here in Larsen fuzzy model we get
the trunk we get the scaled down output. So, this is how we can compare the Mamdani
fuzzy model and Larsen fuzzy model for single rule with single antecedent fuzzy input 𝐴′.

(Refer Slide Time: 21:38)

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Now let us take the fuzzy the max product composition here. So, max-product
composition is going to give us the same result because we have only one rule we have the
single rule. So, max so, whether we take min product that there is we have here this single
antecedent. So, this product are min are not going to play any role in single antecedent
case. So, since we have here the single antecedent product and min both are going to give
us the same output. So, we can see here both in both the cases we are getting the same
output here.

(Refer Slide Time: 22:23)

(Refer Slide Time: 22:31)

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If we compare the max-min composition and max product composition here, we see that
we are getting both same. So, I can write here the same output.

(Refer Slide Time: 22:59)

Now, let us move ahead and take the second case in this the so, the this this in this case
what we do here is instead of the fuzzy input what happens when we use the crisp input,
so the rule here in this is actually in the first case. So, first case where we have the single
rule single antecedent, but here we have the instead of fuzzy input. We are taking crisp
input. So, here if we are taking the crisp input; that means, 𝑥 is equal to 𝑥1 , let us see what
is the output that we are going to get.

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(Refer Slide Time: 23:51)

So, we have here the 𝐴 fuzzy region which is represented by a fuzzy set 𝐴 and similarly
here we have the input that is 𝑥1 , we have just taken 𝑥1 is equal to 6.8, we can take any
input here in the fuzzy region. So, if 𝑥1 is equal to let us say point let us say 6.8, then here
we get the point of intersection here as so we try to find the point of intersection with 𝑥 is
equal to 1. So, what we get here is the point of intersection for 𝑥 is 𝑥1 is equal to 6.8 on
this membership function for the fuzzy set 𝐴.

So I can write here I can represent this by the 𝑤1 . So, this is also called as the weight or
the firing strength of the rule. So, with this firing strength, with this weight again we scale
down the 𝐵 members 𝐵 fuzzy set, the membership function of this fuzzy set we scale down
here to this height 0.34, 0.34 height. So, how do we do that? Here again we use the same
we have the 𝑤.

So, here in this case we have 𝑤1 or 𝑤 whatever since we are here we have only single rule
we can write only 𝑤 or 𝑤1. So, what is the outcome here? What is the membership let us
say if this fuzzy set is 𝐵′, the outcome fuzzy set is the output fuzzy set is 𝐵′ so what is the
corresponding membership function? What is the membership function for this 𝐵 fuzzy
set 𝐵′ fuzzy set is basically, I can write it like this 𝜇𝐵 let us say this is my 𝐵 fuzzy set.

So, the membership function of 𝐵 fuzzy set, let us say this is 𝑦. So, this is going to be 𝑤1
into 𝜇𝐵 (𝑦). So, this is how we can get the resulting output the scale down membership

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function. And this is 𝐵′ so, we can. So, this will be nothing but 𝐵 ′ = ∫𝑌 𝜇𝐵′ (𝑦)/𝑦 and then.
So, this is how the output will be represented. And since we have only one rule that means,
the single rule so the max is not applicable here. So, whatever is here the output here will
be transferred here. So, this is the output 𝐵′.

So, corresponding to, corresponding to the crisp input here crisp input 𝑥1 is equal to 𝑥, 𝑥1
is equal to 6.8, the fuzzy output the fuzzy output is 𝐵′. And as I have already mentioned
that the output here is fuzzy so we can use any suitable defuzzification technique,
defuzzification methodology to convert this fuzzy value into the crisp value as we have
discussed this in Mamdani fuzzy model in our last lecture.

(Refer Slide Time: 28:22)

Now, how will it look like? So, here we have the corresponding to the crisp input 𝑥1 is
equal to 6.8 here, we are getting this output so, we have presented here and then the fuzzy
Mamdani fuzzy model for the same input how will it look like. So, we see that if we would
have used Mamdani fuzzy model the output will be like this. So, here we have compared
the output for 𝑥1 is equal to 6.8 as the input, I can write here as the input.

So, we can see we can compare the input so, in the Mamdani model we see we use the
weights are the firing strength of the rule to truncate the fuzzy set 𝐵. But here in Larsen
fuzzy model the we use the weight or the firing strength of the rule to scale down the fuzzy
set 𝐵, the output fuzzy set 𝐵. So, we have scaled down the fuzzy set and the corresponding

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membership function, you can see here and similarly the and similarly it can be shown by
𝐵′. So, we can clearly see the difference here.

(Refer Slide Time: 30:05)

Now, with the same crisp input, now if we use max-product composition here so, since we
have only one antecedent, so this product or min is not going to make any difference. They
are only going to make difference when we have more than 1 antecedents. So, the output
is going to remain the same here whether we use the max-product composition or max-
min composition.

(Refer Slide Time: 30:41)

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So, here we when we compare this again with the Mamdani and Larsen we see the
difference. And now when we see the difference in all the four cases like we have the all
the two cases like for max-product composition and max-min composition, we see the
output in case of Mamdani fuzzy model and the Larsen fuzzy model with the respect to
input the crisp input that we have fed to the system fed to the model fuzzy model, Larsen
fuzzy model so, here we have fed 𝑥1 is equal to 6.8.

So, what we are interested to note is that that for single antecedent whether the we use
max-product composition or we use the max-min composition, the result is going to remain
the same the output is going to remain the same. So, as I have already mentioned that as
output we are getting a fuzzy value and this fuzzy value can be defuzzified by using
suitable techniques of defuzzification and we can get as a result the crisp value.

(Refer Slide Time: 32:12)

So, in today’s lecture, we have discussed the Larsen fuzzy model using max-min
composition and max-product composition for single rule with single antecedent and with
this, I would like to stop here. And in the next lecture, we will discuss the Larsen fuzzy
model for single rule with multiple antecedents.

Thank you very much.

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 pFuzzy Sets, Logic and Systems and Applications
Prof. Nishchal K. Verma
Department of Electrical Engineering
Indian Institute of Technology, Kanpur

Lecture - 57
Larsen Fuzzy Model
(For Single Rule with Multiple Antecedents)

Hi, welcome to the lecture number 57 of Fuzzy Sets, Logic and Systems and Applications,
in this lecture I will continue our discussion on Larsen Fuzzy Model. Here in this lecture
I will discuss the single rule with multiple antecedents for Larsen fuzzy model.

(Refer Slide Time: 00:29)

Now, coming to the second case here of the Larsen fuzzy model and in this second fuzzy
second case, second case of Larsen fuzzy model. We have the model which is
characterized which is defined by single rule only one rule with multiple antecedents. So
this is very important to note. So, we already know that how will the rule of this kind of
model will look like.

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(Refer Slide Time: 01:04)

So, we see that we have the rule here and this rule, since the model is defined by only one
rule. So, you see only one rule we have a single rule, but this model is defined by single
rule with multiple antecedents. So, we can have multiple antecedents for simplicity we
have here only two antecedents but it can be even more antecedents.

So, we have here two antecedents and both the antecedents here 𝑥 is 𝐴 and 𝑦 is 𝐵 are
connected with any with a connective that is AND here, but it can be OR also, O-R. So,
either we use AND connective or we use OR connective or we use or we can use any other
connective also in between the two antecedents, any two antecedents.

So, we see that this kind of rule will look like this, this kind of fuzzy rule will look like
this. So, if 𝑥 is 𝐴 and 𝑦 is 𝐵 then 𝑧 is 𝐶. So, this is our premise part and this is our
consequent part. So, we have in premise part to antecedents as I have mentioned. Now, if
we have a Larsen fuzzy model which is characterized by this rule this kind of rule; now if
any unknown input comes to this model. So, this input can be either fuzzy or crisp.

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(Refer Slide Time: 02:50)

So, let’s now discuss this one by one. So, here in this case we have the fuzzy rule the single
rule is our single rule and we have multiple antecedents. So, we have first antecedent and
then we have the second antecedent, this must be noted here that we have in this kind of
multiple antecedent we have only two antecedent. We can have so many other antecedents
also.

So, I mean we can have few more antecedents are more antecedents. So, here for simplicity
I have taken only two antecedents. So, first antecedent here is 𝑥 is 𝐴 and the second
antecedent is 𝑦 is 𝐵 and both are connected with the connective AND. So, we can we could
have as I mentioned we could have any other connective like OR, OR.

So, let us now apply the fuzzy input first. So, here we have the this as the fuzzy inputs. So,
we are now supplying Larsen fuzzy model this fuzzy input. So, I can write this fuzzy
inputs. So, if we have supply in this fuzzy input as 𝐴′, 𝑥 is 𝐴′ which is fuzzy and 𝑦 is 𝐵′
which is also fuzzy.

So, when we supply this input and so, 𝑥 is going to the 𝑥 part means 𝑥 antecedent first
antecedent and then 𝑦 is 𝑦 the fuzzy input 𝐵′ is going to the second antecedents. So, when
we superimpose here these two fuzzy sets like 𝐵′ and 𝐴 we see that we are getting the point
of intersection. So, we are getting only one point of intersection as I mentioned earlier that
we could get many points of intersections. So, in those cases normally we get two. So, in

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all the cases if we are getting multiple points of intersection then what we do we take the
maximum of this.

So, here since we are getting only one point of intersection we call this as w 1 and here we
are getting the another intersection. So, let us call this as 𝑤2 of rule 1, of rule 1. So, I can
write here w 1 and here I can write; so, 𝑤11 and 𝑤21 . So, then what we are getting here is
that we take the min of these two weights, as I have already mentioned that the we call
either weights or we call the firing strength of the rule.

So, what is the firing strength of the rule is 𝑤 here this is the final firing strength of the
rule is called the rule weight. The rule weight or firing strength of rule all right. So, now,
since we have 2 weights here to membership values here. So, these are nothing, but the
membership values or grades. So, we take the min because we are using max-min
composition here. So, when we take max min composition we use the min of these 2. So,
we see that when we take min we are getting this value.

So, min is min of the 2 will be 0.36. Now, we will use this to a scale down the height of
the fuzzy set 𝐶. So, output fuzzy set is 𝐶. So, we will bring this down to here to this value
0.36. So, let me repeat here that the height of the fuzzy set we will bring down to the 0.36
and we know how did we get this value this value that is 0.36. So, we bring the height here
of the 𝐶 fuzzy set to 0.36 and early mentioned that we can get it very easily.

So, what we do here we just take this w as 0.36 and then we multiply it with simply 𝜇𝐵 as
the membership function of the fuzzy set 𝜇𝐵 which is characterizing the output fuzzy set
𝐶. So, this is the, this 𝜇𝐶 not 𝜇𝐵 . So, w into 𝜇𝐶 (𝑧) and this is going to give me 𝜇𝐶 ′ there is
going to give us 𝜇, I can write here 𝜇𝐶 ′ (𝑦). So, this is how we get 𝜇𝐶 ′ (𝑦) and then from
here we can write the 𝐶. So, what is 𝐶′ here? C dash will be nothing, but the fuzzy set
defined in terms of 𝜇𝐶 ′ (𝑧) now is this is not 𝑦 this is 𝑧.

So, this is z and then here we have 𝑧 a 𝑧 this is 𝑧 this is also 𝑧. So, this is how we get this
fuzzy set as the output. This is 𝑦 is equal to 𝐵′. So, this is very simple and this we have
seen that we have a Larsen fuzzy model which is defined which is having only one rule,
but with multiple antecedents. So, when we supply the fuzzy value as inputs. So, this is
how we are getting the fuzzy output and this fuzzy output is the 𝐶′ here in this case.

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(Refer Slide Time: 10:24)

So, here we have compared here means we are comparing the output values for the same
input when we use Mamdani fuzzy model and Larsen fuzzy model we get here in this case
our 𝐶′ like this and here and when we use Larsen fuzzy model we get this output.

So, we see that when we use Mamdani fuzzy model the w that we have gotten the weight
of the rule this firing strength of the rule that we have gotten is used to simply chop off the
fuzzy set output fuzzy set. Means the to truncate the fuzzy set, but here in Larsen fuzzy
model the same value is used to reduce the to scale down the output fuzzy set. So, this is
how we can see we can compare the outputs of the Mamdani fuzzy model and Larsen
fuzzy model for the same input and here the input is the fuzzy input.

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(Refer Slide Time: 11:48)

And, here in this case let us use the other composition other composition that is max-
product composition instead of max min composition. So, since we have multiple
antecedent here we have in this case in a particularly in the rule that we have here is having
two antecedent which is the case of multiple antecedent. So, here the max product
composition will give us the different result because there in the first case first composition
we have taken the max-min.

So, we took the firing strength the minimum of the firing strength minimum of the firing
minimum of the weights and which was in the previous case 0.36, but here the inputs
remain the same for the same input we are getting here these values as 𝑤11 and here we
have 𝑤21 . So, here since we are using max-product composition. So, we have to take the
product of these two. So, simply w will be nothing, but 𝑤1, I can write it 𝑤1 𝑤1 and. So,
here is also 𝑤1. So, there is 1 and then we have the 𝑤. So, w so here this is 𝑤1 there is for
the first rule a single rule that is why.

So, whether we write it or not write it does not make any difference. So, let us not use 1
ok. So, when we multiply we get multiplied the two 𝑤11 and 𝑤21 we get the rule of the firing
a strength of the rule.

916
(Refer Slide Time: 14:25)

So, this is firing a strength or the weight of rule which is 𝑤 is equal to which is 𝑤 is equal
to 0.21 here we have got 0.21 because they have multiplied the two points of intersections
these values. So, this is less than the previous case where we use max-min composition.
So, please note this and with this now when we scale down. So, the output. So, let us now
a scale down.

So, when we scale down the output here to let us a scale down this to 0.21. So, this is scale
down here. So, what we are getting here is this output because we have a single rule. So,
max does not make any difference see max of this will remain the same. So, here we are
getting this as the output the fuzzy output corresponding to the fuzzy inputs. So, this is this
is the fuzzy output and this is nothing but the 𝐶′ and what is 𝐶′, 𝐶′ is nothing but the fuzzy
set defined by here 𝜇𝐶 ′ (𝑧)/𝑧. So, this is how it is defined.

Now, let us compare this output which the Mamdani fuzzy model let us compare the
Larsen fuzzy model output with the Mamdani fuzzy model and we can clearly see the
difference here the rule strength the firing strength of the rule of the weight here is the
same in both the cases and that the same rule strength the firing rule strength we when we
truncate the output fuzzy set we get this as the output. Whereas, here in Larsen fuzzy model
we scale down the height we scale down the fuzzy set. So, we see the difference in the
output for the same fuzzy input for the same fuzzy input all right.

917
So, now let us compare the Larsen fuzzy model output again in the case of max-min
composition. So, we have two compositions that we have used here. So, here since we
have the antecedents multiple case and here in our case we have two antecedents. So, since
we have two antecedent. So, max-product and max-min will make the difference. So, here
we see the Larsen fuzzy model with max product composition the output and here we see
the max min composition but for the same fuzzy inputs.

So, we see the difference as to how we get the output for the same fuzzy inputs and we see
here again the difference in between the Mamdani fuzzy model output and the Larsen
fuzzy model output.

(Refer Slide Time: 18:44)

Now, let us go to the crisp inputs here. So, in the second case itself let us use a crisp input
instead of fuzzy inputs and see what we are getting. So, the rule remains the same because
we are dealing with the Larsen fuzzy model with single rule and multiple antecedents. So,
this remains the same here and now when we supply the input to this model here 𝑥 is equal
to 𝑥1 and 𝑦 is equal to 𝑦1 .

So, both the inputs we are supplying here and both the inputs are crisp inputs both the
inputs are the crisp inputs.

918
(Refer Slide Time: 19:51)

So, let us see what we are getting as the output when we use the Larsen fuzzy model using
first using max-min composition. So, let us take the two crisp values, that means the our
𝑥1 is let us say 6.5 here and 𝑦1 here is the 8.5 as the crisp values. So, you can choose any
value which should be applicable, which should be falling within the fuzzy region and
should be further should be in the universe of discourse within the universe of discourse.

So, here in this case we are taking for simplicity 𝑥1 is equal to 6.5 and 𝑦1 is equal to 8.5.
So, we see when we take the 𝑥1 is equal to 6.5 and when we draw a line parallel to the 𝑦
axis that means, the parallel to the ordinate that means, the 𝜇𝐴 (𝑥) here. So, we get the point
of intersection here as mu as 𝑤11 , let us say 𝑤1 and since we have only one rule. So, we did
not write this. So, let us say this is 𝑤1 is equal to this and 𝑤2 is equal to, 𝑤2 is equal to
0.36.

Now, since we are using the max min composition. So, let us take the minimum of these
two when we take minimum of the two yes. So, this was nothing, but the 𝑤1 this was
nothing, but the 𝑤2 . So, when we take the min of these two we are getting 0.36 here, now
again as we have done in the previous slides since we are dealing with Larsen fuzzy model.
So, we have to a scale down the height of the original fuzzy set of the output. So, 𝐶 that is
𝐶 here the fuzzy set. So, the height of the fuzzy set we will a scale down to we will bring
down to here to this value 𝑤.

919
So, this 𝑤 is equal to 0.36. So, we will use this value here and we will bring the height
here accordingly the fuzzy set will be scaled. So, the formula that we use here is we use
we get the new fuzzy set like this the membership function of the new fuzzy set let us say
𝜇𝐶 ′ let us say 𝜇𝐶 ′ (𝑧) and this is nothing, but the 𝑤 into 𝜇𝐶 (𝑧) ok. So, this way we will get
the new the output membership function and here this fuzzy is the output fuzzy set here
will be 𝐶′ and this we can represent by simply 𝜇𝐶 ′ .

(Refer Slide Time: 23:28)

And since we have only one rule. So, max is really not applicable here is whatever outcome
that we are getting here will be will remain the same. So, now, let us compare this Larsen
fuzzy model output for the input that we have supplied with the Mamdani fuzzy model.
So, let us now compare we see here that the for max min composition we get Mamdani
fuzzy model here truncated output truncated fuzzy set whereas, in Larsen fuzzy model we
get the scale down fuzzy set.

920
(Refer Slide Time: 24:37)

Now, let us move ahead and for the crisp input let us use the max product composition.
So, we see that here in max product composition here let us quickly use this. So, since we
have already found the points of intersection 0.81. So, we can write here as 𝑤1 and we can
write here as the 𝑤2 .

So, here in max product composition is simply instead of taking the min we will take
product of these two. So, we take the product here 𝑤1 into 𝑤2 and we get the rule strength
the firing strength of the rule or the weight final weight of the rule is equal to 𝑤 is equal
to 0.29 you can see here. Now, this value will be used to scale down the original fuzzy set
which is the output fuzzy set to this value to 𝐶′ we call this as the 𝐶′.

So here so this way we scale down the output fuzzy set and we have already seen that as
to how we write the output final output. So, here since we have the single rule. So, max is
really not applicable we have only one the whatever output comes will remain. So, fuzzy
output is 𝐶′ and 𝐶′ is what the 𝐶 ′ = ∫𝑧 𝜇𝐶 ′ (𝑧)/𝑧 and this is how this will be represent that
this output will be represent. Now, let us compare this output with the output of Mamdani
fuzzy model.

921
(Refer Slide Time: 27:13)

So, we can clearly see that when we have used the max product composition. So, how we
get the output in case of Mamdani fuzzy model and in case of Larsen fuzzy model for the
same crisp inputs. Now, let us compare all the outputs like when we have supplied the
crisp inputs and when we have used the max product composition and max min
composition, how these two models Mamdani fuzzy model and Larsen fuzzy model are
giving the fuzzy output.

And as I as I have already mentioned that this fuzzy output is not the final output basically,
the further the defuzzify this fuzzy output using the suitable defuzzyfication method to get
the crisp value. So, we can here see that the you can compare the outputs of the Mamdani
fuzzy model and Larsen fuzzy model for the crisp inputs for the same crisp inputs and in
case of the max-product composition and the max-min composition.

922
(Refer Slide Time: 28:30)

In today’s lecture, we have discussed the Larsen fuzzy model using max min composition
and max product composition for single rule with multiple antecedents. So, with this I
would like to stop here and in the next lecture we will discuss the Larsen fuzzy model for
multiple rules with multiple antecedents and we will discuss the example also on the
Larsen fuzzy model.

Thank you.

923
 Fuzzy Sets, Logic and Systems and Applications
Prof. Nishchal K. Verma
Department of Electrical Engineering
Indian Institute of Technology, Kanpur

Lecture - 58
Larsen Fuzzy Model (For Multiple Rules with Multiple Antecedents

Welcome to the lecture number 58 of Fuzzy Sets, Logic and Systems and Applications.
And here, we will continue our discussion on the Larsen Fuzzy Model for Multiple Rules
with Multiple Antecedents and this also we will discuss with max-min composition and
max-product composition for fuzzy and crisp inputs both.

(Refer Slide Time: 00:22)

(Refer Slide Time: 00:34)

924
So, since this class is having the multiple rules and multiple antecedents. So, we are taking
2 rules here for simplicity, but we can have multiple rules like n number of rules we can
have and similarly, we can have multiple antecedents. So, here also we are taking two
antecedents only.

So, 2 antecedents and 2 rules we are taking for simplicity. So, rule number 1 and rule
number 2, but if we understand this then we can apply this for multiple antecedents and
multiple rules, means we can apply to any number of rules and any number of antecedents.

(Refer Slide Time: 01:31)

925
So, here we are taking only 2 rules and let us go to the first case, where we are taking the
fuzzy input. So, we have this as the rule number 1 and this as the rule number 2. These
two are the rules already given for the fuzzy model Larsen fuzzy model and this dotted
one is the applied input.

So, we may not at the moment consider when we are discussing the rule. Now, this is the
input, this is the fuzzy input this is a fuzzy input that we supply to the Larsen fuzzy model.
And we have this fuzzy input for 𝑥 this fuzzy input for 𝑦 means the fuzzy input the fuzzy
value 𝐴′ that means the 𝐴′ is nothing, but a fuzzy set here and 𝐵′ is also a fuzzy set.

So, these two fuzzy values are applied fuzzy values are given as the input to the model.
So, when we do that, we see that for the first rule let us understand that when we apply
this so we superimpose 𝐴1 and 𝐴′. So, we superimpose 𝐴1 and 𝐴′, we get the point of
intersection as 0.86 and similarly here for 𝑦 input we superimpose 𝐵′ and 𝐵1 , 𝐵1 was
already there and 𝐵′ is the given fuzzy input. So, when we superimpose these two or we
superimpose 𝐵1 on superimpose 𝐵′ on 𝐵1 and we see that there is an intersection here
point of intersection and this point of intersection is 0.36.

So, we have two points of intersection first is for the 𝑥 first antecedent and the second one
is for the second antecedent. I can call this as 𝑤1 and I can call this as the 𝑤2 . Now, since
here we are taking max-min composition. So, we will take the min of these two weights
𝑤1 and 𝑤2 .

So, when we take the min and since this is for the first rule, where there are multiple rules
here means two rules. So, we write the symbol of weight like this. So, upper subscripts
here is for rules, so first rule similarly here the upper subscript will be the 𝑤 1 . So,
𝑤 1 basically I can ok.

So, let us have this have it like this the symbol is a bit, we can take here in this particular
case you can take this as the 𝑤11 and 𝑤12 . So, the lower subscript is for the rule and so the
𝑤1 here is the 𝑤1 is equal to the minimum of the 2.

So, 𝑤1 is equal to the minimum of 𝑤11 and 𝑤12 . So, this way we have the min of the two
as 𝑤1 𝑤1 is 0.36 and this we used to a scale down the output fuzzy set 𝐶1 . So, we can
click, we can very easily get this value the 𝐶1 . So, there is the fuzzy output here this is the

926
fuzzy output this is 𝐶1 ′ and this is nothing but 𝜇𝐶 ′ (𝑧)/𝑧 and what is 𝜇𝐶 ′ (𝑧) is 𝜇𝐶 ′ (𝑧)
is nothing but 𝜇 but 𝑤 into this is 𝜇𝐶 ′ (1).

So, 𝜇𝐶1′ (𝑧) is 𝜇𝐶1′ = 𝑤1 × 𝜇𝐶1 (𝑧). So, this is how we get this value of 𝐶1′ . Similarly, we
find the points of intersection here for the second rule, we call this as the 𝑤21 and then we
call this as the 𝑤22 . So, 𝑤2 = min(𝑤21 , 𝑤22 ).

So, this way we get 𝑤2 here and similarly here we get the 𝐶2′ . So, this is the fuzzy set and
this is nothing, but see, if this is 𝑤𝐶2′ (𝑧)/𝑧 and 𝜇𝐶2′ (𝑧) = 𝑤 × 𝜇𝐶2 (𝑧). So, this is how we
get the output fuzzy set scaled. Now, since here we have multiple rules so we have to when
we apply max-min composition.

So, now the max is relevant here max of max-min composition is irrelevant. So, we have
multiple rules here we have two rules. So, both the outputs are now included both the
outputs are now accounted and this accounting is done by taking the union of the two. So,
we take the union of the two and that is how we are getting this as the output.

So, this is the union of 𝐶1′ and 𝐶2′ . So, we can write it like this 𝐶′ and then 𝐶2′ and this is
our final output in this case. So, now, this output is the fuzzy output and as I have already
mentioned that when we are interested in crisp equivalent of this crisp value of this, then
we use suitable defuzzification methods and we get the crisp value the corresponding fuzzy
set.

(Refer Slide Time: 11:04)

927
Now, let us compare this output of Larsen fuzzy model with the Mamdani fuzzy model.

So, this means that when we have for the same inputs for the same fuzzy inputs, if you
would have used Mamdani model what will would have got against here. So, for the same
max-min composition and for the same fuzzy input, we are getting the different outputs.
So, Mamdani model is giving us here, this output this fuzzy output whereas, the Larsen
fuzzy model is giving this output

So, please look at the outputs and these two are different outputs. Now, let us go ahead
and use the other composition. Let us go ahead and use the max-product composition. So,
when we use max-product composition.

(Refer Slide Time: 12:14)

So, since we have already this as the 𝑤11 and this we have 𝑤12 . So, our 𝑤1 = min(𝑤11 , 𝑤12 ).
So, we are getting here. No this is the product this is not minimum. So, this is the product.
What is this? This is this we have 𝑤11 and this is we have 𝑤12 . So, when we multiply this
the value 𝑤1 that is the firing rule strengths is coming out to be 0.31 and we use this value
to the scale down the height of the 𝐶1 the fuzzy set 𝐶1 to 0.31.

So, this is how it is done. Now, the new membership function of 𝐶1′ the new membership
function the scale down membership function 𝜇𝐶1′ (𝑧) = 𝑤1 × 𝜇𝐶1 (𝑧) and then you can
write the 𝐶1 here we can write the 𝐶1′ the output is scaled down fuzzy set here 𝜇𝐶1′ .

928
Alright so now similarly, when we apply the second rule when we apply the input the
fuzzy inputs to the second rule. This was the rule number 1 the first rule and then we have
the second rule. Now, when we apply this 𝑤2 this is the second rule. So, we write 𝑤21 and
then we write here 𝑤22 . So, this is 𝑤21 this is 𝑤22 . Similarly, here also we have the value
that we are getting as the firing strength of the rule as 0.2.

So, this value will be used to a scale down the 𝐶2 to 𝐶2′ . So, the membership function here
of this scaled down fuzzy set will be 𝜇𝐶 ′ (𝑧) = 𝑤2 × 𝜇𝐶2 (𝑧) and the 𝐶2′ = ∫𝑧 𝜇𝐶2′ (𝑧)/𝑧. So,
this is how we get the 𝐶2′ as the output the fuzzy output. Now, since we are applying here
a max-product composition. So, we take the union of the two outputs.

So, we take the 𝐶1′ take 𝐶1′ and 𝐶2′ . So, we take the union of these 2 and this is what we
are getting here as the output. So, we see that we are finally, getting the fuzzy output and
this fuzzy output can be defuzzified further to get the crisp output and this is the output
that we obtained using Larsen fuzzy model using the third case the third case of the Larsen
fuzzy model that is multiple rules and multiple antecedents and this output is with respect
to max product composition with fuzzy inputs.

(Refer Slide Time: 17:54)

So now, let us compare this case with the this output with the Mamdani fuzzy model. So,
had it been a Mamdani fuzzy model you would have gotten this output for the same input
for the same composition, that means the max-product composition. So, see here that, we

929
are getting different fuzzy outputs for the same inputs and for the fuzzy for the same
compositions.

So, we see clearly that we have different fuzzy outputs and since we have the different
fuzzy outputs, obviously, we are going to get the different crisp values as well. Now, let
us compare this with the max-min composition and we see that all four are different when
we use max-min composition here also the Larsen and Mamdani both are producing
different fuzzy outputs.

So, we can see that as to how when we use the same even if the same max-min composition
different models are producing corresponding to the same fuzzy inputs different fuzzy
outputs and so the crisp out outputs also will be different.

(Refer Slide Time: 19:30)

Now instead of fuzzy inputs, let us use crisp inputs and see what happens. So, we have the
again for this case also we have two rules and the input here is different that means, 𝑥 is
equal to 𝑥1 instead of the fuzzy set and here we have 𝑦 is equal to 𝑦1 that means the
crisp input instead of the fuzzy input. So, let us see what we are going to get when we
apply this crisp input.

930
(Refer Slide Time: 20:08)

So, here we have let’s us assume that 𝑥1 is 7 and 𝑦1 is 6.5. So, when we take this 𝑥1 for
the first antecedent, so first antecedent is 𝑥 is 𝐴1 . So, when we take 𝑥1 is equal to 7. So,
corresponding to 𝑥1 is equal to 7 we see that this is cutting this is intersecting the 𝐴1
fuzzy set at 0.28 membership value. So this we call as 𝑤11 , for the first antecedent and first
rule this is rule number 1 this is rule number 2.

So, similarly here 𝑦1 is cutting 𝑦1 is also intersecting here at 0.97. So, we call this as 𝑤12
the second antecedent and first rule. So, if we use max-min composition, then we have to
take the min of these two. So, 𝑤1 is equal to or we can simply write here like this that we
have this as the 𝑤11 and this as the 𝑤12 . So, the value the minimum value is coming out to
be 0.28.

So, since we are taking max-min composition. So, the minimum here is 0.28. Now, we use
this value to scale down 𝐶1 to 𝐶1′ means the new fuzzy set is the scale down fuzzy set is
𝐶1′ and as I have already discussed as to how we are going to get the membership function
of the scaled down 𝐶1′ fuzzy set. So, here 𝐶1′ will be like this, 𝑤1 multiplied by 𝜇𝐶1 (𝑧)
here also we will have 𝜇𝐶1′ (𝑧) and then here we will have 𝑤1 into 𝜇𝐶1 (𝑧).

So, with this we will be getting 𝜇𝐶1′ (𝑧), the membership function of the scaled down fuzzy
set and this is scaled down fuzzy set is 𝐶1′ and 𝐶1′ is this 𝜇𝐶1′ (𝑧)/𝑧. So, similarly, when

931
we apply the same input same crisp inputs to the rule number 2. So, here we get my 𝑤21 ,
0.14 and 𝑤22 as 0.99. So, since we are taking the max-min composition.

So, minimum of the two will be 0.14 and similarly here also the membership function of
the scale down fuzzy set will be 𝜇𝐶2′ . So, I can write here 𝜇𝐶2′ is going to be 𝑤2 multiplied
by 𝜇𝐶2 (𝑧) since this is defined in 𝑧. So, we can write here 𝜇𝐶2′ (𝑧) so like that and then
we have this 𝐶2′ as 𝜇𝐶2′ (𝑧)/𝑧. So, this is how we get the expression for 𝐶2′ . Now, since we
are using here the max-min composition.

So, the outputs corresponding to rule number 1 and rule number 2 are unionized is a union
of the two outputs are taken. So, 𝐶1′ ∪ 𝐶2′ and this is what is the output that we get when
we take union we combine these two. So, either we call this as the union or we taking max.
So, this is nothing, but the union of the two outputs corresponding to the rule number 1
and rule number 2.

(Refer Slide Time: 26:37)

Now, let us. So, since this is fuzzy output. Now, let us compare this fuzzy output of the
Larsen fuzzy model with the Mamdani fuzzy model with the same max-min composition
and with the same crisp input. So, we see that the outputs again here will differ. So, Larsen
fuzzy model produces this fuzzy output whereas the Mamdani produces the different fuzzy
output compared to Larsen fuzzy model. So, similarly when we defuzzify this the crisp
outputs also will remain the different.

932
So, let us now for the same input and in the same class that means, the multiple rules with
multiple antecedents let us use the max product composition and let us see what we are
getting.

(Refer Slide Time: 27:38)

So, here we are again for the same input we are getting 𝑤11 as 0.28 as the point of
intersection here we are getting 𝑤11 . So, 𝑤12 here as the point of intersection here. So, this
part remains the same, the only thing is this multiplication here because we are using max-
product composition.

So, we multiply these two the 𝑤11 and then 𝑤12 . This is 𝑤12 . So, final value that is 𝑤1 is
coming out to be 0.27 which is the firing strength of the rule 1 and again it is needless to
mention as to how we get the 𝐶1′ here. So, what is done here is that the height of 𝐶1 is
brought down to or the is brought down to 𝐶1′ and the accordingly the whole fuzzy set is
a scaled down and this membership function of this fuzzy set 𝐶1′ that means 𝜇𝐶1′ (𝑧) =
𝑤1 × 𝜇𝐶1 (𝑧).

So, when we have this then, we can simply write the expression for fuzzy set 𝐶1′ and 𝐶1′
is 𝜇𝐶1′ (𝑧)/𝑧. So, this is how we write 𝐶1′ fuzzy set. Similarly, when we apply this crisp
input 𝑥1 is equal to 7 𝑦1 is equal to 6.5 to the second rule the output is 𝐶2′ and here this
intersection is 𝑤21 and this intersection is 𝑤22 .

933
So, this is 𝑤21 and this is our 𝑤21 𝑤22 ; 𝑤22 . Similarly, the membership function of the 𝐶2′
so the membership function of 𝜇𝐶2′ = 𝑤2 × 𝜇𝐶2 (𝑧). So, this 𝐶2′ = ∫𝑧 𝜇𝐶2′ (𝑧)/𝑧. So, this is
how we can get the membership value and the fuzzy set. So, both the outcomes are now
maximized the outcomes that union of the two membership two fuzzy sets are taken.

So, here we take the max of 𝐶1′ and 𝐶2′ , we and this is same as this is same as the 𝐶1′ =
𝐶1 ∪ 𝐶2 here and this is the outcome. So, this is the fuzzy outcome and we use the
defuzzification methods suitable defuzzification methods to get the crisp output. Now, let
us compare this output of the Larsen fuzzy model here and let us compare this with
Mamdani.

(Refer Slide Time: 32:43)

So, had it been the same input same crisp inputs same set of fuzzy rules and same
composition that that means, max-product compositions. So, 𝐶 here the Larsen fuzzy
model is giving this output and Mamdani fuzzy model is giving this output which is
different from the Larsen fuzzy model for the same crisp inputs and same max-product
composition and now let us compare the outputs of the max product composition with
max-min composition as well.

So, we see that here we see that here all the four outcomes are different means, the fuzzy
values are changing the output of the Larsen fuzzy model in all the cases in both the
compositions are different for the same input and same composition and same input and
similarly here the Mamdani fuzzy model also we have different outputs. And similarly,

934
since we have the fuzzy values are different. So, the corresponding crisp values are also
going to be different.

So, we see that, we have the outputs fuzzy outputs in as the result of Larsen fuzzy model
with crisp model crisp input and max-min composition and then the same for max-min
composition and we see that the results are the fuzzy outputs are different. The different
the outputs are different and then accordingly, we can say that the crisp values are also
going to be different.

(Refer Slide Time: 34:53)

Now, let us take a simple example a very simple example here of the case the single input
and single output Larsen fuzzy model.

So, here we have the single input it means we have the single antecedent, but here we have
multiple rules. We have single antecedent, but multiple rules and in this case we have the
single input single output that means the SISO Larsen fuzzy model and which is shown
here for antecedent and consequent membership functions with universe of discourse. 𝑥
belonging into belonging from −10 𝑡𝑜 10 and 𝑦 from 0 𝑡𝑜 10 respectively for
every 𝑥 ∈ 𝑋 for every 𝑦 ∈ 𝑌.

So, we see here that the 𝑥 is having three fuzzy regions small, medium, large and all these
fuzzy regions are defined by or represented by the corresponding fuzzy sets for small for
medium for large. Similarly in consequent part that means, the output is also divided into

935
𝑦 is also divided into three fuzzy regions and every region is represented by a fuzzy value
fuzzy set. So, small, medium, large.

So, we have the input and we have the output and here we have the three rules of the model
which is given. So, rule 1 says if 𝑥 is small then 𝑦 is going to be in the small means, if
any input 𝑥 which is falling in the small region then the output has to be fall has to fall in
the small region only. Similarly what rule 2 is saying is if 𝑥 is medium means 𝑥 is going
to be medium.

Then y is also going to be in the medium means what does this mean exactly is if any input
𝑥 is falling in the medium region then y will also be falling in the medium region.
Similarly for rule 3 if 𝑥 is large means if the 𝑥 values if the input 𝑥 is falling into large
region large fuzzy set region, then the corresponding 𝑦 will also fall in the large region.
So, these are very simple case to make you understand. So, if we have these three rules
present for this fuzzy model for this SISO fuzzy model for this SISO Larsen fuzzy model,
then let us find the output corresponding to the input 𝑥 is equal to −3.9.

So, let us apply this input. So, please understand that the output input here is the input that
we are giving to the model is a crisp input. So, we have 𝑥 is equal to −3.9 and this is a
crisp input. So, this input when we supply to the Larsen fuzzy model, let us see what is the
output that we are going to get corresponding to this input.

(Refer Slide Time: 38:42)

936
So, when we apply this input, we see that corresponding to 𝑥 is equal to −3.9, we get
here the two points of intersection. So, we see that a small region is cut at one place a small
region a small fuzzy set is cut at is intersected at one place by 𝑥 is equal to −3.9 line and
similarly the medium is also intersected at 0.7 and then when it comes to large is not at all
affected.

So, the input that we are supplying here which is 𝑥 is equal to −3.9 is applicable is falling
in to two regions a small and medium it is not falling in the large region. So, large is large
fuzzy region is irrelevant for this input. So, let us now further understand that,
corresponding to this input we are getting 0.3 as the point of intersection in a small region
and 0.7 in medium region. Now, let us check this in the given rules.

So, let us now look at the rules and see whether the rules are applicable for this input or
not. So, we have we have the input 𝑥 is equal to −3.9 which is falling in the small region.
So, if it is falling in the small region, we have the given rule we see here and in given rule
if we if our input is falling in the small region. So, we have the output also in the small
region.

So, this means that the input since our input 𝑥 is equal to −3.9 is falling in the small
region. So, this means rule 1 is applicable, no matter what is the output. So, we will first
see the input. So, input is falling in this small region, then we see the input is also falling
in the medium region. So, rule 1 and rule 2 are applicable and rule three is not applicable
because the 𝑥 is equal to −3.9 is not falling in the large region.

So, this is this rule is not applicable so only two rules are applicable. Now, let us proceed
with these two rules.

937
(Refer Slide Time: 41:34)

So, we have the small fuzzy set here the fuzzy set for a small here and then we have the
fuzzy set for medium here this is medium, this is medium fuzzy set and this is small fuzzy
set and these two are given. So, since we have two rules here, rule number 1 let us now
apply the input.

So, for rule number 1 and then we have the rule number 2. So, when we apply the input
here we find here for the first rule we are getting the point of intersection as the weight
which is 0.3; 0.3. Now, if we apply Larsen fuzzy model. So, the height of this output fuzzy
set that is small here in this case we have to bring it down to the 0.3 value here.

So, and the membership function, the new membership function let us say the membership
function of the small here the small here let us say this is y and the membership function
of they small dash let us say will be the 𝑤; that means, the 0.3 × 𝜇𝑆𝑚𝑎𝑙𝑙 (𝑦).

So, similarly here, when we apply a rule number 2 we get here the point of intersection as
𝑤2 and we scale down the medium to medium height of the medium to 𝑤2 ; that means,
the 𝑤2 is equal to 0.7 and this is the outcome that we get a scale down fuzzy set and the
corresponding membership function. So, let us call this as the dash medium dash

So, this is 𝑦 so 𝜇𝑀𝑒𝑑𝑖𝑢𝑚 . So, this will be basically, 0.7 × 𝜇𝑀𝑒𝑑𝑖𝑢𝑚′ (𝑦). And here we have
the membership function mu medium let us say 𝑦 and this is dash. So, this is how we are
getting the scale down fuzzy set and this is the membership function is the membership

938
function. We can get the fuzzy set a 𝑆𝑚𝑎𝑙𝑙 ′ = ∫𝑦 𝜇𝑆𝑚𝑎𝑙𝑙′ (𝑦)/𝑦 similarly here the fuzzy

set and so this will be the fuzzy set basically, this is the fuzzy set and here this fuzzy set
there the mu medium this is fuzzy set that you write it here. This fuzzy set is 𝜇, let us say
𝑀𝑒𝑑𝑖𝑢𝑚′ and this 𝑀𝑒𝑑𝑖𝑢𝑚′ = ∫𝑦 𝜇𝑀𝑒𝑑𝑖𝑢𝑚′ (𝑦)/𝑦 . So, that is how we get the scale

down membership function and the corresponding fuzzy set for medium dash like scale
down medium dash scale down fuzzy set.

(Refer Slide Time: 46:55)

So, the same is shown here. Now, since we are using max-min composition. So, this is for
max-min composition here, this is for max-min composition. So, we use max-min
composition. So, minimum we have already taken here the minimum was not applicable
because, we have only one antecedent. So, max-min composition or max-product
composition both will give the same result because, we have only one antecedent here.

So, that is not really going to make any difference. So, I can simply remove this because
this is applicable to both min and max-min and max-product composition. But finally, in
both the cases we have to take the max of the two output. So, both the output and what is
this output this is the small dash here this is a 𝑆𝑚𝑎𝑙𝑙 ′ and this is 𝑀𝑒𝑑𝑖𝑢𝑚′ as the fuzzy
output as the fuzzy output. So, both these outputs are now taken as the union.

So, I can write here as a small dash union medium dash. So, this is what is the output. So,
this is finally, again the fuzzy output.

939
(Refer Slide Time: 48:40)

Now, we can use any suitable methods of defuzzification to get the crisp value. So, here
this is the output and fuzzy output and when we use the centroid of area the defuzzification
gives the 𝑦 ∗ as the crisp value. So, 𝑦 ∗ is 3.5420, the formula I have already discussed all
these defuzzification I have already discussed.

So, if we use all these, we will get the center of area if you use center of area the same
fuzzy value the same fuzzy quantity is giving us different crisp values. So, when we use
center of area method of defuzzification, we get 3.5420, when we use the bisector of area
we get 3.6286 as the crisp value, when we use max of mean of maximum we get 3.7287
as the crisp value, when we use the smallest of maximum then we get 2.4625 as the crisp
value.

And then when we use the largest of maximum here we get a 4.995 as the crisp value for
the same fuzzy output. And here it is shown the fuzzy output that we have got is shown
and their vertices also all these vertices 𝑃1 , 𝑃2 , 𝑃3 , 𝑃4 , 𝑃5 , 𝑃6 are shown here, is you are
interested you can use these vertices to compute and compare the and match the results
that we have obtained.

940
(Refer Slide Time: 50:44)

So, we have discussed in today’s lecture. So, many things and I hope you enjoyed the
lecture in detail I have discussed the Larsen fuzzy model using max-min composition and
max-product composition for both the inputs fuzzy and crisp. And all these three cases we
have discussed, when we have the Larsen fuzzy model with single rule with single, single
antecedent and then this was the first case.

The second case was that we discussed was single rule with multiple antecedents and the
third case that we discussed for Larsen fuzzy model was the multiple rules with multiple
antecedents and this covered almost all the cases that we could apply. And with this I will
stop here and in the next lecture we will discuss the Tsukamoto fuzzy model.

Thank you very much.

941
 Fuzzy Sets, Logic and Systems and Applications
Prof. Nishchal K. Verma
Department of Electrical Engineering
Indian Institute of Technology, Kanpur

Lecture – 59
Tsukamoto Fuzzy Model

Hi, welcome to lecture number 59 of Fuzzy Sets Logic and Systems and Applications. In
this lecture today I will discuss Tsukamoto Fuzzy Model.

(Refer Slide Time: 00:16)

Tsukamoto fuzzy model is a very special kind of model, where we have the consequent
part of the fuzzy rule, monotonically increasing or decreasing membership function. So it
looks like that this fuzzy rule has a crisp function, but actually the consequent part of the
fuzzy rule of this fuzzy model, that means Tsukamoto fuzzy model is always represented
by a monotonically increasing or decreasing membership function. And, the highest value
of this membership function can go up to 1 not beyond that.

So, in nutshell I can say that Tsukamoto fuzzy model involves the fuzzy rule a set of fuzzy
rules of the form where its premise part is fuzzy and the consequent part is also fuzzy. But,
the consequent part here is not exactly a fuzzy set that we have seen in the case of Mamdani
model or the Larsen model. But, it is different here this fuzzy set is different here the
consequent part of the fuzzy, the consequent of a part of the fuzzy set is different. In the

942
sense that this fuzzy set of the consequent part is represented by or characterized by a
membership function, which is monotonically increasing or decreasing.

So, if this is the case then, the fuzzy set of the this type which is characterized by the
monotonically increasing and decreasing membership function will always be of this kind
like either it is left open or right open fuzzy set. So, this is right open and this is left open.

So, this kind of thing with the consequent part happens. And here as I mentioned that the
consequent part of the fuzzy rule is fuzzy, like we have in the case of Mamdani model,
Mamdani fuzzy model and the Larsen fuzzy model. The Tsukamoto fuzzy model also has
the fuzzy rule of the kind of the type where the premised part is fuzzy and the consequent
part is also fuzzy. So, this has to be noted very clearly.

Now, the inferred output of each rule please understand carefully here that the inferred
rule inferred output of each rule each fuzzy rule is a crisp value. And this is found
corresponding to the firing strength, that means the weight of that particular rule. So, the
overall output is taken as the weighted average of the output of each rule. So, Tsukamoto
fuzzy model basically avoids the time consuming process of defuzzification because here
the final output is the weighted average of output of each rule. So, that is where this makes
the difference.

(Refer Slide Time: 05:03)

943
So, let us understand the Tsukamoto fuzzy model. So, here we have a rule as we have
taken in case of Mamdani fuzzy model and the Larsen fuzzy model. So, the same rule is
being taken. So, this does not need any explanation here. And this rule has the premise
part where we have two antecedents connected by AND or OR connective and then THEN
part we have the which is known as the consequent part and this is 𝑧 is 𝐶𝑖 and 𝑖 here
signifies the 𝑖 𝑡ℎ fuzzy rule.

So, in any model any fuzzy model, we can have any number of rules say here we have n
number of rules 𝑛 number of rules. As so, that is why 𝑖 has been used here and this is our
𝑖 𝑡ℎ rule. So, I can represent this by 𝑅𝑖 . So, 𝑖 𝑡ℎ rule can be if small 𝑥 is 𝐴𝑖 AND or OR 𝑦 is
𝐵𝑖 then 𝑧 is 𝐶𝑖 , where 𝑖 can go from 1,2,3 … . . 𝑛 and fuzzy sets 𝐴𝑖 , 𝐵𝑖 and 𝐶𝑖 are expressed
as you can see here.

Similarly, 𝐵 fuzzy set and then the 𝐶 fuzzy set. The firing the strength of 𝑖 𝑡ℎ fuzzy rule is
defined by here the 𝑤𝑖 this is 𝑖 𝑡ℎ rule so, that is why 𝑤𝑖 has been written. And then here,
the 𝜇𝐴𝑖 (𝑥) and then we have a min sign, the open triangle sign and then here we have
𝜇𝐵𝑖 (𝑦). So, and this is 𝜇𝐵𝑖 (𝑦).

So, what are these values basically? These are the values corresponding to the input that
we feed that we applied to the model. And, corresponding membership values will give us
the weights and then these weights basically after taking either min or product will give us
the firing strength of the rule or weight of the rule.

So, when we have this firing a strength of the rule available, then the overall input can be
found by taking the weighted average of the output of each rule, you can see here like this.
So, 𝑧 ∗ has been used. So, please understand here that, we do not use any union of the
outputs of the rules, means we are not doing the we are not taking the union of the outputs
of the rules that are applicable for the particular input.

944
(Refer Slide Time: 08:53)

So, let us now move ahead and have the cases of single rule with single antecedent and
then single rule with multiple antecedents then multiple rules with multiple antecedents
and these cases we will be discussing with fuzzy inputs as well as the crisp inputs. Here as
I mentioned that the Tsukamoto fuzzy model does not follow is strictly the composition
compositional rule of inference what is that compositional rule of inference? This is max
min or max product.

So, since here we are not taking the union so, max is not applicable; here the output is a
weighted average of the outputs that are coming out from each rule which is which are
applicable. So, let us now, discuss all these cases one by one for the fuzzy inputs and the
crisp inputs. So, let us discuss first the single rule with single antecedent here.

945
(Refer Slide Time: 10:30)

So, as we already know that, if we have a model fuzzy model let us say called Tsukamoto
model. So, Tsukamoto model is defined by is characterized by only one rule that a single
rule and which is of this type, where we have single antecedents like if 𝑥 is 𝐴 then 𝑦 is 𝐵.
So, if this is the rule that is there for the model and this rule is known.

Now, if a new input comes to the input comes to the model as the input here 𝑥 is 𝐴′, 𝐴′ is
nothing but what is 𝐴′? A dash is fuzzy set some fuzzy value fuzzy set or fuzzy value fuzzy
set is used for describing quantifying the fuzzy value. So, I can write here the fuzzy value.
So, if this is the case, then our output corresponding to the 𝑥 input that is fuzzy here will
be this. So, we will discuss this here.

946
(Refer Slide Time: 11:58)

So, if we have a single rule with single antecedent which is here. So, 𝑥 is 𝐴 is already there,
then corresponding to this 𝑥 is 𝐴 we have the output which is 𝐵 and this is the 𝐵 fuzzy set.
And please note that here the premise part has a fuzzy set which is bell shaped fuzzy set,
whereas the consequent part you see the fuzzy set that we have here is the left open fuzzy
set. And the highest value of this fuzzy set is going up to only 1 and this is monotonically
decreasing fuzzy set.

So, let us I can write here the monotonically decreasing fuzzy set or monotonically
decreasing function the fuzzy set function. So, and this is actually the fuzzy set. So, here
this is the rule that is given to us for the Tsukamoto fuzzy model. So, here a new input this
is a new input which is coming, new input I can write here new fuzzy input that is coming
here. And if this input is coming to the model as input to the Tsukamoto model as input,
so what we do? We try to superimpose this fuzzy value this fuzzy set on the fuzzy set 𝐴
which was already given which is already given in the rule so, 𝐴 is already there.

So, 𝐴 we already have. So, we try to superimpose 𝐴′ with 𝐴 and then we find the
intersection point if any. So, here we have the intersection point, which is called weight
and I can represent this by 𝑤. Since this is the for the first rule, I mean here you can have
only you have only one rule. So, I am not writing 𝑤1 or 𝑤2 like that, but I can simply write
𝑤. So, 𝑤 represents the points point of intersection and since we have only one antecedent
here.

947
So, we are not going to use any compositional rule here, like min or product. So, this 𝑤 is
our strength of this rule. So, w is here basically 0.58 this 𝑤 is equal to 0.58. Now, when
we have this 𝑤 available, this is also known as the firing strength of the rule. So, when we
have this now, corresponding to this point here we try to find the 𝑦, we see here that this
is nothing but the 𝜇(𝑥), that means the membership function membership value
corresponding to the 𝑥 and here this is 𝜇(𝑦).

So, corresponding to 5, 0.58 we try to find the 𝑦1 which is here. So, 𝑦1 is you can see the
𝑦1 here 𝑦1 is 4.88. So, this 4.88 is the value the crisp value which we are corresponding to
the fuzzy input that we are feeding to the model to the Tsukamoto model, which is which
has a single rule which is characterized by the single rule with single antecedent and the
input is fuzzy here that is 𝐴′.

So, the interesting point here is that we always get a crisp output of the rule means even if
we provide the fuzzy input, we are getting the crisp output which is 𝑦1 here. So, 𝑦1 is for
this case 4.88 now. Since we have only one rule. So, the weighted average is going to be
the same here it is 𝑤1, which need not be there. So, I can write here 𝑤1 and this 𝑤
corresponding to this this rule the output is 0.58 we have 𝑦 ∗ is equal to 𝑤1 into 𝑦1 over 1
and so, we are getting 4.88. So, here the weighted average is going to remain the same.

(Refer Slide Time: 18:08)

948
Now, what if we have the crisp input? So, if we have crisp input 𝑥 is equal to 𝑥1 the output
here is going to be 𝑦1 and let us see how do we find the corresponding output here for the
crisp input.

(Refer Slide Time: 18:35)

So, here we have the rule and corresponding to this rule, we have a fuzzy set A that is
already given. Now, we have the input some input let us say 𝑥1 is equal to 6.8. So, we have
taken 𝑥1 is equal to 6.8 which is lying within the region of universe of discourse and also
it is lying within the region of the fuzzy set 𝐴.

If it is not lying within the region of fuzzy set 𝐴, this rule will not be applicable. So, this
input has to intersect the fuzzy set before the rule can be applicable. So, here we are taking
𝑥1 is equal to 6.8 and corresponding to this input we are getting the intersection point
which is, let us say 𝑤1 here and this 𝑤1 is equal to 0.34 and this 0.34. We have got this
called the rule strength fuzzy rule strength or strength of the fuzzy rule they also called the
weight of the rule.

So, corresponding to this now, coming to the consequent part fuzzy set which is here and
corresponding to this we are getting the corresponding to this 𝜇(𝑦) this becomes 𝜇(𝑦) is
equal to 0.34. So, corresponding to this we are getting our 𝑦1 here as 5.21. And, again
since we have only one rule so, we have only single rule. So, the weighted average is going
to remain the same.

949
So, this way even if we have the fuzzy input or the crisp input for this case, for this model
the Tsukamoto model, where we have only one rule with single antecedent we are going
to get the crisp output. And the output calculation is very simple we are not going to apply
here the compositional rule of these are not actually applicable. Because the min if we
have let us say apply the max-min composition. So, min will not be applicable or product
even will not be applicable because we have only one antecedent.

Similarly, the max is not applicable because here first of all we have only one rule, but
even if we have more number of rules then also the output is the weighted average of the
outputs of the rules.

(Refer Slide Time: 21:47)

Now, let us come to the other case the second case second case here is that we have a
Tsukamoto model and let us say it is characterized by the single rule but with multiple
antecedents. So, we have single rule let us say and then this rule has multiple antecedents.

950
(Refer Slide Time: 22:12)

So, let us now, look at this rule and see how does it look like. So, we have here this rule
where we have only one rule the model is characterized by model is represented by only
one rule, where we have here in this case we have two antecedents. So, we see that we
have first antecedent as 𝑥 is 𝐴 AND the second antecedent in this case is 𝑦 is 𝐵 and both
of these antecedents are connected by the connective AND. So, instead of AND it could
be any other connective like OR, BUT etcetera.

So, when we say single rule with multiple antecedents. So, multiple antecedents can be
any number, here we have only two we can have any number like 3, 4, 5, 2, 3, 4, 5 and so
on. So, here for simplicity we have taken only two antecedents and these are joined by
connected by the connective AND. So, this is connective as we already discuss alright. So,
next is to find the output final output corresponding to some input. So, we will go ahead
and first apply the fuzzy input.

951
(Refer Slide Time: 23:53)

So, here we have the single rule is here, but here we have two inputs the input 1 which is
𝑥 is equal to which is input 𝑥 is equal to 𝐴′, this is this means the fuzzy input is provided
and then the another input simultaneously fed is 𝑦 is equal to 𝐵′. So, these two inputs are
together supplied to the model. Now, let us see how will this rule help us in finding the
corresponding output. So, like we have done in the past, what we do here? We superimpose
the inputs the fuzzy inputs on 𝐴 and 𝐵 respectively. 𝐴 and 𝐵 are already there in the fuzzy
set in the fuzzy rule.

So, here we see that when we superimpose we find the intersection, here when we here
when we superimpose we get 𝑤1 and then we get here another intersection here which is
𝑤1. So, this is let us say 𝑤11 and this is let us say 𝑤2 . Now, since we have here two
antecedents. Now, if we use min composition from max-min. So, the 𝑤1 becomes min of
the 𝑤11 and then here 𝑤12 . So, the firing strength of the rule firing strengths of rule becomes
𝑤1 is equal to 0.36 and corresponding to this 0.36 the 𝑧1 which is the output fuzzy set we
get 𝑧1 is equal to 5.19 here and it is very clear from this picture here.

So, our 𝑧1 is 𝑧1 corresponding to 𝑥 is equal to 𝐴′ and 𝑦, 𝑦 is equal to 𝐵′ here and since we

have only one rule. So, the weighted average becomes the same. So, this is weighted
average. And since we have only one rule as I said, the 5.19 will become the final output.

952
(Refer Slide Time: 27:52)

Now, what if we supply crisp input to the Tsukamoto model characterized by single rule
with multiple antecedents. So, let us see that here we are supplying 𝑥 is equal to 𝑥1 some
crisp value and 𝑦 is equal to 𝑦1 here 𝑥1 , 𝑦1 are the crisp values that are being supplied.

(Refer Slide Time: 28:18)

So, let us take 𝑥1 is equal to 6.5, 𝑦1 is equal to 8.5 and when we apply this, when the here
we use this 𝑥1 value and 𝑦1 value we see that the rule that was given and that is
characterizing the Tsukamoto fuzzy model the rule is here is here. So, this is actually
known.

953
So, this rule is known. So, when the rule is known we can say the model is known. So, all
the parameters of this rule is known means 𝐴 is known 𝐵 is known 𝐶 is known. So, here
corresponding to this 𝑥1 we have the membership value is 0.81 which is called the weight
here. So, 𝑤1 let us say here 𝑤11 , yeah this is the first rules of first rule is 𝑤1 and then for
the first antecedents. So, 𝑤11 and then here corresponding to 𝑦1 similarly we get let us say
𝑤12 and if we take the min composition here.

So, we take the min criteria when we take the min criteria. So, firing strength of the rule
comes out to be this is firing strengths of the rule which is coming out to be 𝑤1 and 𝑤1
here is 0.36. And corresponding to this 0.36 you see here this is 0.36 and corresponding to
this the through the output fuzzy set which is monotonically decreasing fuzzy set, we are
getting 𝑧1 here 5.19. So, now since we have only one rule again here in this case because,
this is single rule with multiple antecedent model Tsukamoto model.

So, the final output is a weighted average and weighted average is going to be the same.
So, 5.19 is the final output corresponding to 𝑥1 is equal to 6.5, 𝑦1 is equal to 8.5. Now, let
us go to the third case, where our Tsukamoto model is characterized by the multiple rules
with multiple antecedents.

(Refer Slide Time: 31:35)

So, we have here two rules for simplicity we have taken only two rules, but it can be any
number of rules say up to 𝑛. And similarly, we can have antecedents here any number of
antecedents here we have for simplicity only two antecedents.

954
So, two rules and two antecedents for simplicity we have taken. Now, if we supply the
inputs to this model to this Tsukamoto model which is characterized by the multiple rules
with multiple antecedents. Let us see how we get the output final output computed.

(Refer Slide Time: 32:29)

So, here we have the rule number 1, I write here as rule 1. So, rule 1 basically has 𝐴1 , 𝐵1,
𝐶1 fuzzy sets which are known to us the all the parameters of this rule are known. Similarly
we have rule 2 here and here also 𝐴2 , 𝐵2, 𝐶2 are known.

So, when we feed the input 𝑥 is 𝐴′, 𝐴′ is a fuzzy value, similarly 𝑦 is let us say 𝐵′ which
is again a fuzzy value. So, when these two fuzzy valued values are supplied to this model,
let us say let us see the how the output final output we can compute. So, 𝑥 is equal to 𝐴′ 𝑦
is equal to 𝐵′, when it goes to the when it applies to the first rule. So, as we have already
done, we superimpose this fuzzy value these two fuzzy values to the fuzzy sets that are
already present in the antecedents 𝐴1 and 𝐵1 respectively.

So, here this is the antecedent for 𝑥 this is the antecedent for 𝑦. So, we superimpose 𝐴′
with 𝐴1 and we superimpose 𝐵′ with 𝐵1, 𝐵1 is known 𝐴′ is known 𝐴1 is known 𝐵′ is known
𝐴1 is know all our the fuzzy values are known. So, now, we look for the intersection points.
So, here since this is the first rule I am writing 𝑤1 and then first antecedent we write 1
here. So, 𝑤11 here is 0.86, similarly for this antecedent the intersection point is 𝑤12 is equal
to 0.36 this is 0.36 alright.

955
So, now, if we apply the min of these two. So, the min is going to be here the rule strength
is going to be 0.36 and corresponding 𝑧1 here, you see that the corresponding 𝑧1 here is
5.19 for the first rule. Similarly, when we apply the same input to the rule 2 these points
are coming out to be the weights are coming out to be here 𝑤21 and then this is 𝑤22 .

So, 𝑤21 is 0.58 𝑤22 is 0.35 and if we take min of this we are getting here as 𝑤2 is equal to
0.35. So, this is rule strength of 1 and this is the rule is strength of 2 or I can say here the
firing strength it is more appropriate to write firing strength of rule 1. So, this can be written
as firing strength of rule 2 and here corresponding to 𝑤2 you can see here corresponding
to 𝑤2 we are getting 𝑧2 , that means the corresponding output 4.79.

So, please note that here we are getting two outputs 𝑧1 and 𝑧2 . So, we are getting 𝑧1 and
𝑧2 output, 𝑧1 is corresponding to the first rule 𝑧2 is corresponding to the second rule. And
both these outputs are the rule outputs are corresponding to the input fuzzy input 𝐴′(𝑥) is
equal to 𝐴′ 𝑦 is equal to 𝐵′.

Now, the next step is to since we have two outputs two outputs from the rules two rules.
So, we go for the weighted average of it. So, weights we already know here, this is our 𝑤1
this is our 𝑤2 and here this is our 𝑧1 , 5.19 is 𝑧1 this and 4.79 is 𝑧2 . So, when we take
weighted average, we already know the formula of the weighted average I can once again
write it here.

𝑤1 𝑧1 +𝑤2 𝑧2
So, 𝑧 ∗ = . So, this way we compute the final output as 4.9928 corresponding to
𝑤1 +𝑤2

the fuzzy input 𝑥 is equal to 𝐴′ and 𝑦 is equal to 𝐵′. And please note that this model this
the outcome of this is the outcome of the model Tsukamoto model with multiple rows with
multiple antecedents.

956
(Refer Slide Time: 39:41)

Now, what happens when we supply the crisp inputs? So, here everything remains, the
same the input becomes crisp inputs 𝑥 is equal to 𝑥1 and 𝑦 is equal to 𝑦1 .

(Refer Slide Time: 40:07)

So, now, let us go ahead and see here we have the rule 1, here we have the rule 1 and in
this rule 1 we have the crisp input. So, when we apply this input crisp input 𝑥1 is equal to
7 and 𝑦1 is equal to 6.5. So, corresponding to this 𝑥1 is equal to 7 and 𝑦1 is equal to 6.5 we
get two points of intersection here.

957
So, for the first antecedent where we have 𝐴1 , for 𝐴1 we are getting this value this 𝑥1 is
equal to 7 is giving us the intersection point at 0.28. So, this I can write as 𝑤11 similarly
here why 𝑤1 one because this is for the first incident and antecedent and first rule, similarly
here this is first rule and first antecedent.

So, sorry this is second antecedent; first rule second antecedent. So, 𝑤12 is 0.97. Now, if
we use min so, min of these two is going to give us 𝑤1 which is as I mentioned the firing
strength here, firing strength of the rule of the rule 1. So, corresponding to this w here,
corresponding to this 𝑤1 rather corresponding to this 𝑤1 here we are getting the output 𝑧1
and 𝑧1 is here point 𝑧1 is 5.31.

So, the outcome of the rule 1 is 𝑧1 which is 5.31. Similarly now, we go to rule 2. So, here
we have rule 2. Now, the rule 2 here gives us the intersection points like this the 𝑤1, since
this is the second rule, so we have we will write 𝑤21 and then here we will write the 𝑤22 the
second rule second antecedent. So, corresponding to the 𝑦1 here which is crisp 6.5 we are
getting two values two intersection points. And when we take min we get the firing min of
these two will give us 𝑤2 and 𝑤2 is nothing but the firing strength of the rule 2 firing
strength of rule 2.

Now, we have corresponding to 𝑤1 𝑧2 here and 𝑧2 is 4.4. So, now, these two inputs we
already have 𝑧1 and 𝑧2 . Now, let us go ahead and find the weighted average. So, weighted
average is giving us here 𝑧 ∗ which is a final output and 𝑧 ∗ is equal to 5.0067 and we already
know that six we have our 𝑤1 0.28 and 𝑤2 0.14 and 5.31 as 𝑧1 , 4.4 as 𝑧2 .

𝑤1 𝑧1 +𝑤2 𝑧2
So, if we substitute all these values here 𝑧 ∗ = . So, this is giving us 𝑧 ∗ is equal
𝑤1 +𝑤2

to the final output corresponding to 𝑥1 is equal to 7 , 𝑦1 is equal to 6.5 then getting 𝑧 ∗ is

equal to 5.0067. So, this is how corresponding to the crisp inputs with multiple rules and
multiple antecedents which is describing the Tsukamoto model we are getting the output.

958
(Refer Slide Time: 45:12)

Now, let us take a very simple example here of Tsukamoto fuzzy model, where we have
three rules. So, this fuzzy rule is defined by this fuzzy model is defined by three fuzzy
rules and this rule as we see here these three rules are of the type of multiple antecedents.
So, we see the previous part we have two antecedents in all these three rules. So, this is of
the kind of model fuzzy model Tsukamoto fuzzy model which is the multiple antecedents
and multiple rules. And the membership functions here are already known as since the
model is known.

So, membership functions have been given here 𝜇𝐴𝑙𝑜𝑤 ,

𝜇𝐴𝑚𝑒𝑑𝑖𝑢𝑚 , 𝜇𝐴ℎ𝑖𝑔ℎ , 𝜇𝐵𝑙𝑜𝑤 , 𝜇𝐵𝑚𝑒𝑑𝑖𝑢𝑚 , 𝜇𝐵ℎ𝑖𝑔ℎ . So, here the we have two this model has two

generic variables as inputs 𝑥 and 𝑦 are input generic variables. 𝑧 here is the output generic
variable output generic variable. And please understand that, the 𝑥 the complete 𝑥 is
divided into three fuzzy regions low, medium, high. So, this means that these fuzzy regions
have three fuzzy these fuzzy regions are characterized by three fuzzy sets.

And the membership functions corresponding membership functions here are mentioned
you can see here for the input and similarly for the input 𝑦. So, for input 𝑥 here input 𝑥
and then here we have input 𝑦 generic variable. So, we see that in both the inputs input
generic variables we have 3-3 fuzzy regions and in the output here 𝑧 we have only two
fuzzy regions and both the fuzzy regions are represented by fuzzy sets the here the sigmoid
fuzzy set low and high.

959
The input 𝑥 region is has three regions represented by three fuzzy sets all the fuzzy sets
are trapezoidal type. Similarly, here input y also all the fuzzy sets all the fuzzy sets of the
three fuzzy regions are trapezoidal type. So, now, when we give the input the crisp input
here crisp input. So, for this crisp input what will be the corresponding output and that too
in crisp form. So let us now, see as to how we can find it and the universe of discourses of
the inputs and outputs are mentioned here.

(Refer Slide Time: 49:37)

So, the fuzzy regions the membership functions that are given, if we plot the fuzzy sets
will look like this low medium high, similarly here also this will look like this. So, all of
these are given these are known. So, we do not have to worry for the parameters of these
fuzzy set these are already given here you can see. These are the membership functions
which are characterizing the corresponding fuzzy sets.

960
(Refer Slide Time: 50:21)

So, let us apply the input and what is the input here is 𝑥 is equal to 65, 𝑦 is equal to 55. So,
this two this input has two parameters 𝑥 is equal to 65, 𝑦 is equal to 55 both these values
are now, applied and since we have the three fuzzy regions.

So, the first step here is to find where it is intersecting where this 𝑥 65 is intersecting. So,
this line when we see is intersecting at here two point, 0.25 and then 0.75 you can see. So,
this is the first step, first step is to find the corresponding to the input that is here 65. So,
corresponding to 65 what are the fuzzy regions which are applicable what are the fuzzy
sets that applicable?

So, please understand that the 𝑥 is equal to 65 is intersecting medium and high only not
low in this case. So, low is not relevant only medium and high are relevant for 𝑥 is equal
to 65, this is the first step that we checked while we are computing the output. So, step
number 1 I can write here. Next is similarly we check for the other antecedent like 𝑦 is
equal to 55.

So, 𝑦 is equal to 55 we see that this corresponding to 𝑦 is equal to 55 we are getting high
and medium, medium is not intersected; because it is when we see where 𝑦 is equal to 55
is cutting at medium, medium is at 0. So, we see that 𝑦 we see that medium is not relevant
only high is relevant high fuzzy set fuzzy set for high is relevant.

961
So, 𝑦 is equal to 55 only one fuzzy set is applicable. Now, or in other words we can say
that low and medium are not at all applicable. So, we have for 𝑥 is equal to 65 two
intersection points, whereas for 𝑦 we have only one intersection points 𝑦 is equal to 55.
Now, we multiply these two means two intersection points and then the one intersection
point for the other antecedent. So, 2 into 1 means we have 2 into 1 is 2 here. So, only two
combinations are happening here what is that combination? The combination is when my
𝑥 is medium and high or what is the other combination when my 𝑥 is high and high.

So, we have two combinations I can write here the medium, I can write 𝑥 here and y here;
I can write medium by M and I can write high by H. So, first combination could be the
medium high and then other combination the next one is high because the 𝑥 is 65 is cutting
high fuzzy set also so, high and high. So, two combinations we are getting. Now,
corresponding to these two combinations what will be my output that is 𝑧. So, this we can
find by this we can find by the rules that are given.

(Refer Slide Time: 54:54)

So, let us now, check here the given rules are here. So, first rule here is the rule number 1
here is when my input is 𝐴 when my input is low input 𝐴 is low or I can say input 1 is low
and then my input 2 is medium then the output or I can write here 𝑥, 𝑦. So, input 𝑥 is input
𝑦 and output 𝑧 here. So, my input 𝑥 is low, the output 𝑦 is medium then the output 𝑧 is
going to be low this is 𝑧.

962
So, this rule 1 is not applicable because we see that we have first combination medium and
high the second combination is high and high, here we see that low and medium is not
applicable. So, we are not seeing any combination which is low and medium so, this rule
will not apply. Now, next we look for medium and high which is here. So, rule 2 is
applicable where we have medium high 𝑥 is equal to medium, 𝑦 is equal to high I mean
when 𝑥 falls in the medium, 𝑦 falls in high then the output also falls in high we can see
here this one.

So, this way from this rule we find we can very easily find the output is going to be the 𝑧
is going to fall in high region. So, I can write here high similarly what happens when 𝑥
falls in H and 𝑦 falls in H. So, when 𝑥 falls in H here and then 𝑦 falls in H output also falls
in H. So, this rule is already there. So, rule number 3 also applies and with this combination
we can write here high.

So, it means both the outputs are from high fuzzy region in the output. And these are the
computations that we do for getting the corresponding membership values which are
nothing but the intersection points here for 𝑥 is equal to 65 here. For low, for medium, for
high and similarly for the other antecedent the second antecedent here 𝑦 is equal to 55, 𝑦
is equal to 55 here.

So, for low medium and high. So, this way we get the points of intersection. Alright so
now, when we know these rules which are applicable these two rules are applicable here
for the input that we are supplying. Now, let us compute the corresponding outputs.

963
(Refer Slide Time: 58:25)

Now, this is very simple we have already done this rule number 2 we have medium high
and high. So, when we have the input 65 we see the points of intersection see here. So, this
I can write 𝑤11 and similarly corresponding to 𝑦 input we get the 𝑤11 𝑤12 .

So, our first step was let me just recap the first step was to get the points of intersection in
the fuzzy regions, means we need to first find all the combinations corresponding to the
input that we are supplying. So, we need to know what are the fuzzy sets that we are getting
intersected that are applicable and that is how we make the combinations and these
combinations we check with the given rules. So, then from these rules we find that what
are the corresponding outputs.

And, so here in the given rules we find we check what are the applicable rules. So, these
are the applicable rules here. So, the given rules are normally more than the applicable
rules alright.

964
(Refer Slide Time: 60:03)

So, this way when we have come to know the what are the rules that are going to be
applicable. So, this I can write as the step number 2, next is same as what we have done in
lecture like for particular inputs how do we get the output for the case multiple antecedents
and multiple rules of Tsukamoto fuzzy model.

So, now, when we apply 𝑥 is equal to 65, 𝑦 is equal to 55. So, since rule 1 is not applicable.
So, we have not written here only rule 2 is written here. So, we are applying this in these
two inputs in rule 2 and then we find 𝑤11 , 𝑤12 and corresponding to this we have here the
firing strength of the rule 1, which is nothing but the 𝑤1 is equal to 0.75 will be take min.

Similarly, here this is 𝑤21 this intersection is 𝑤22 and if we take min of these two we are
getting here 𝑤2 which is the firing strength of the rule 2. Now, corresponding to 𝑤1 here
the 𝑧1 that is coming out to be 57.197 corresponding to 𝑤2 the firing strength of the rule 2
we are getting 𝑧2 is equal to 52.803.

Now, when we have gotten these two values, these two outputs which are coming from
the rule number 2, rule number 3 respectively; now, we take the weighted average of this.
So, weighted average will be 𝑧 ∗ let us say we are interested in final output. So, final output
𝑤1 𝑧1 +𝑤2 𝑧2
will be ; let us now, quickly here we are here.
𝑤1 +𝑤2

965
(Refer Slide Time: 62:23)

So, let us now, quickly substitute these values and our z star becomes here finally, 𝑧 ∗ is
56.0985. So, this is the output corresponding to the input that we have supplied. Here one
thing that needs to be noted here is that even if we apply even if we use rule number 1 our
inputs that are here 𝑥 is equal to 65, 𝑦 is equal to 55 it is not cutting it is not giving any
intersection point. So, that is why we have through that step 2 through step 1 in step 2 we
have already filtered that. So, we are not going to get any point of intersection even when
we use rule number 1.

So, in this process right from the beginning we have found out what are the rules that are
applicable, where we are getting the points of intersection. So, this way using rule number
2 rule number 3, we are getting our z star is equal to the final output is equal to 56.0985
corresponding to the crisp inputs 𝑥 is equal to here 65 and 55.

966
(Refer Slide Time: 63:56)

So, this way we have seen that as to how we can use Tsukamoto fuzzy model to find out
the output corresponding to the crisp input or fuzzy input. So, with this I would like to stop
here. And in the next lecture I will discuss the TSK fuzzy model, TSK fuzzy model is
nothing but the, Takagi-Sugeno and Kang fuzzy model.

Thank you.

967
 Fuzzy Sets, Logic and Systems and Applications
Prof. Nishchal K. Verma
Department of Electrical Engineering
Indian Institute of Technology, Kanpur

Lecture - 60
TSK Fuzzy Model

(Refer Slide Time: 00:25)

Hi, welcome to the lecture number 60 of Fuzzy Sets, Logic and Systems and Applications.
Today in this lecture I will discuss Takagi Sugeno and Kang fuzzy model, this model is in
short known as TSK fuzzy model or TS fuzzy model. So, let us first understand the TS
fuzzy model or TSK fuzzy model as it was introduced by Takagi Sugeno and Kang in
1985. And a typical fuzzy rule in a TSK fuzzy model has the form of IF 𝑥 is 𝐴 and then
we have some connective either AND or OR and then next antecedent 𝑦 is 𝐵.

So, here we have the premise and then we have the consequent. So, the fuzzy rule for TSK
fuzzy model is of the form of it has the premise part and the consequent part. So, premise
part remains as it is or as it was in Mamdani, Larsen, Tsukamoto. So, here also the same
means the premise part is fuzzy. It may have either single antecedent or multiple
antecedent. Whereas when it comes to the consequent part here there is a significant
difference. What is this difference? Difference is this.

So, in TSK fuzzy model we have the output in terms of a polynomial, the polynomial is
here and this polynomial is of the form of 𝑓 (𝑥, 𝑦) means function of 𝑥, 𝑦 if 𝑥, 𝑦 are the

968
inputs. So, let us say we have two inputs here first input is 𝑥, the second input is 𝑦. So,
𝑓(𝑥, 𝑦) = 𝑝0 + 𝑝1 𝑥 + 𝑝2 𝑥 2 + ⋯ + 𝑝𝑚 𝑥 𝑚 + 𝑞1 𝑦 + 𝑞2 𝑦 2 + ⋯ + 𝑞𝑚 𝑦 𝑚 .

So, here in this polynomial we have basically we have a constant which is 𝑝0 and then we
have the 𝑥 terms. So, we have first degree terms of 𝑥 and 𝑦 like 𝑝1 𝑥, 𝑞1 𝑦. Then we have
second degree terms 𝑝2 𝑥 2 , 𝑞2 𝑦 2 and so on. So, here in this polynomial we have the mx
degree or m degree terms. So, in general we are writing 𝑓(𝑥, 𝑦) as the m degree
polynomial. So, the rule that is used here is a two input rule or I would say two antecedent
rule, whereas, in general it can be any number of it can have any number of antecedents.

So, what we have to remember here is that in TSK fuzzy model the consequent part has
the output which is always in the form of a polynomial and this polynomial is in the terms
of the inputs that are fed in the premise part. So, here says 𝑥 and 𝑦 are the inputs which
you can see in the premise part here. So, the polynomial is always a function of 𝑥 and 𝑦
which is nothing but the function of the generic variables 𝑥 and 𝑦.

Now, here this is very important to note that a TSK fuzzy model takes comparatively lesser
computation time because here the 𝑧 can be obtained very quickly just by substituting the
values of 𝑥 and 𝑦. So, this takes very less time of computation and then TSK fuzzy model
takes only crisp values as the input. So, please note here this is very important point like
in Mamdani model, Tsukamoto model, Larsen model all of the models we have had the
inputs in the form of fuzzy or crisp, but here in TSK fuzzy model we always have the
inputs that means, the 𝑥 and 𝑦 only crisp values.

969
(Refer Slide Time: 06:28)

So, then we have the each rule has a crisp output in terms of the 𝑓(𝑥, 𝑦) as I have just
explained and then since every rule is generating some output. So, the overall output of
the TSK fuzzy model which is characterized by multiple rules will be the weighted average
of these outputs.

So, here we can have 𝑓 the function the polynomial as a constant and if we have such a
case then we see we say this as the zero order TSK fuzzy model we will discuss this in
detail in coming slides. We have explained this and then when 𝑦 when 𝑓(𝑥, 𝑦) is a first
order polynomial the resulting fuzzy inference system is called for first order TSK fuzzy
model.

The zero order TSK fuzzy model is the special case of the mamdani fuzzy model in which
consequent part of each rule is a specified by a constant value which can be represented
by a fuzzy singleton as we already have seen in the beginning of this course we understood
what is a singleton fuzzy singleton. So, constant can also be represented in terms of fuzzy
singleton and then the special case of Tsukamoto fuzzy model in which the consequent of
each rule is a specified by a membership function of a step function centered at the
constant.

970
(Refer Slide Time: 08:55)

So, that way the zero order TSK fuzzy model can be understood and here in zero order
TSK fuzzy model if we go ahead to understand we see that the 𝑓(𝑥, 𝑦) is a constant value
that is 𝑝0 here. Then so since we have let us say the 𝑖 𝑡ℎ rule in TSK model, let us say we
have a set of rules and 𝑖 𝑡ℎ rule is here is 𝑖 𝑡ℎ rule.

So, 𝑖 𝑡ℎ rule will have the rule of this kind if 𝑥 is 𝐴𝑖 , 𝑥 is the input 𝐴𝑖 is the fuzzy region
where this 𝑥 is falling into and then we have the connective here AND or OR, and then we
have another input 𝑦 and which is again falling into 𝐵𝑖 , 𝑖 here signifies that the fuzzy region
which is relevant for 𝑖 𝑡ℎ rule and then we have the premise part which is 𝑧𝑖 for 𝑖 𝑡ℎ rule the
output for 𝑖 𝑡ℎ rule and this is nothing but a polynomial which is 𝑓(𝑥, 𝑦).

So, here we may have a TSK fuzzy model which can have n number of fuzzy rule small n
number of fuzzy rules. So, if this is the case then for 𝐴𝑖 to be a fuzzy set here 𝐵𝑖 again to
be a fuzzy set continuous fuzzy sets and if this is the case then for certain input 𝑥 and 𝑦,
we can have the 𝑤𝑖 which is nothing, but the firing strength of the particular 𝑖 𝑡ℎ rule. So,
I can write here the firing strengths of 𝑖 𝑡ℎ rule and this we can obtain by either taking min
of 𝜇𝐴𝑖 (𝑥) and 𝜇𝐵𝑖 (𝑦) or we can multiply these two if we are interested in the product.

So, here since these firing strengths are of each rules are known then we can go ahead for
the final output by taking the weighted average of the each rule. So, this way we find the
final output of the of the TSK fuzzy model.

971
(Refer Slide Time: 11:48)

So, let us now start with first zero order TSK fuzzy model. So, as I have already mentioned
that the consequent part here in TSK fuzzy model the fuzzy rules will have the consequent
part of this kind. If the output basically the 𝑧 will be defined in terms of 𝑓(𝑥, 𝑦) as I have
already mentioned this 𝑥 and 𝑦 will be the corresponding generic variable generic input
variables and when we talk of zero order zero order TSK fuzzy model then it means we
will have only the constant that is 𝑝0 . So, here we have the 𝑝0 only.

So, 𝑓 𝑥, 𝑦 or I can say 𝑓(𝑥, 𝑦) will be equal to only the constant part that means, the 𝑝0 .
So, this is the constant part. So, no 𝑥 term, no 𝑦 term. So, this means that the zero order or
zero order TSK fuzzy model output will not depend on the generic input variables here in
this case 𝑥 and 𝑦.

972
(Refer Slide Time: 13:29)

So, let us now go for the fuzzy reasoning of zero order TSK fuzzy model and we have
three cases as we have had earlier for other fuzzy models.

(Refer Slide Time: 14:05)

So, we have the first case where the single rule with single antecedent type of TSK fuzzy
model we have and let us now first go with this and understand as to how we are going to
get the output computed if we have a TSK fuzzy model with single rule with single
antecedent.

973
So, as I have already mentioned that we can have only the crisp input. So, here 𝑥 is equal
to 𝑥1 we are supplying. So, the fact or the input we can have as 𝑥 is equal to 𝑥1 and here
is the rule we have only one rule. So, only one rule is mentioned here and rule has the
single premise, single antecedent. So, single antecedent is here 𝑥 is a and then we have the
consequent part and consequent part is 𝑓(𝑥, 𝑦) which is the 𝑝0 here and which is a constant
value means it is not depending upon the input generic variables that means, here in this
case we have 𝑥.

(Refer Slide Time: 15:13)

So, let us move ahead and see how can we get the output computed with respect to the
certain input that we feed to the output. So, let us assume that we have we have 𝑥1 which
is the input variable input generic variable here 2.0202. Let us say we have this value and
we are feeding this value to this model to the TSK fuzzy model characterized by the single
rule with single antecedent rule fuzzy rule which is mentioned here.

974
(Refer Slide Time: 16:00)

If 𝑥 is 𝐴 so 𝑥 is here. So, I can write here the rule that is characterizing is characterizing
this model is if 𝑥 is a then the output here in this case let us say 𝑦. So, 𝑦 is 𝑓(𝑥, 𝑦) or
maybe since we are using 𝑦. So, I can write here 𝑧. So, 𝑧 is equal to the output is 𝑝0 or if
you use 𝑦 it is better. So, let us say 𝑦 which is the function of 𝑓𝑥 because we have only 𝑥
in the premise part as a single antecedent.

So, here we have a constant and constant is 𝑝0 . So, this is what is the rule which is
characterizing, which is representing the fuzzy the TSK fuzzy model. So, if this is the case
now and please note that we have 𝐴 and 𝑝0 both are known for this model when we say
we have a depth we have a model it means all these parameters of 𝐴 and 𝑝0 are known.
So, let us now apply the input here.

So, when we apply the input we see that corresponding to 𝑥1 we have the membership
value here which is coming out to be 0.5204. So, here. So, this is the intersection point and
this I can write as 𝑤1. So, since we have only one antecedent min is not applicable here.
So, the same value will be transferred to the output here for the weighted average. So, 𝑤1
is known now how we will have the 𝑦1 corresponding to this rule.

So, let us say 𝑦1 if we are supplying 𝑥1 corresponding output is 𝑦1 . So, 𝑦1 since the, so, 𝑦1
will be 3.737 which is same as p 0 because this output is not depending upon the input.
So, we have the 𝑦1 and now the final output will be the same even when we do the weighted
average. So, you see here we have done the weighted average since we have single rule.

975
So, the weighted average is going to remain the same. So, this way the output
corresponding to 𝑥1 is equal to 2.0020 will be 3.737 all right. So, now the next case here
for the zero order TSK model would be where we can have a single rule, but with the
multiple antecedent.

(Refer Slide Time: 19:41)

So, let us now go ahead with this case here. So, here also we have you see the zero order
and zero order means the output is a constant the every the consequent part is a constant
consequent part means the output of every rule is a constant and which is not depending
upon the input generic variables.

So, here also we have 𝑝0 so, but what is the difference here is that we have two antecedents
one is the 𝑥 is A and the other one is here, the other one is 𝑦 is B and these two antecedents
can be joined by a connective here we have AND. So, this is the connective. Please note
that this connective can be OR as well. So, we can have any other connective. So, here in
this case we have AND connective. So, now, since we have two antecedents. So obviously,
here we have two input generic variables 𝑥 and 𝑦.

976
(Refer Slide Time: 21:09)

Now when we have the input and we are now applying this input and we are interested in
finding the corresponding output, now let us see how to proceed with this. So, let us assume
that we have 𝑥1 the input value which is a crisp value 𝑥1 is equal to point a 4.444 and 𝑦 is
equal to 𝑦1 is equal to 6.2626. So, let us assume these two values to crisp values as the
input values.

So, this is arbitrary value you can have any other value and please remember that we take
only those values which are going to intersect the fuzzy sets 𝐴 and 𝐵. Why because
otherwise these inputs these fuzzy reasons may not be relevant, and this rule not be
applicable. So, this is the rule in this case here where we have if 𝑥 is 𝐴 and 𝑦 is 𝐵. So, in
this case we have the output which is 𝑝0 . So, here also if we apply this these two inputs.

So, let us see now how to get the 𝑤11 and then 𝑤12 as the intersection points corresponding
to 𝑥1 and 𝑦1 respectively. And here we are taking min why we are taking min because we
have the connective AND. So, connective of the connective in between the two antecedents
we have AND. So, that is why we are taking min here if we would have or connective then
we would have taken max. So, for and connective we could have also taken gone for
product. So, it is up to us what we want or what as what is needed to be what is the
preference.

So, if we take the minimum of here the 𝑤11 and then 𝑤12 . So, when we take minimum we
are going to get the 𝑤1 is equal to here 𝑤1 is equal to 0.1868 which is the lowest value and

977
since we have 𝑤1 we have 𝑧1 here and 𝑧1 is nothing but the here the 𝑧1 is 𝑓(𝑥, 𝑦) and
which is nothing, but 𝑝0 . So, we can very easily say that 𝑧1 is 𝑝0 and 𝑧1 since 𝑧1 is 𝑝0 we
have now 𝑧1 is equal to 3.737. So, we now know the 𝑤1 and 𝑧1 .

So, now since we have only one fuzzy rule here only one rule that is in this case we have.
So, the weighted average is going to give us the same output here because we have only
one rule. Had had it being many rules or more than one rules then the output would have
been accordingly computed. So, let us now go for the third case where we have multiple
rules with multiple antecedents.

(Refer Slide Time: 25:12)

So, here you see for simplicity we have two rules first rule second rule and then we have
multiple antecedents here we have only two antecedents we have taken two antecedents
for simplicity. So, otherwise we could have taken any number of antecedents rest other
things remain the same.

978
(Refer Slide Time: 25:43)

So, we have two antecedents and the connective is and. So, let us now proceed with the
this model having two rules and two antecedents.

(Refer Slide Time: 25:52)

So, let us apply. So, we have the rule number 1 here and we have the rule number 2 here.
And we are now giving the input to the model. So, and please remember that this model
this input is the crisp input you can see. So, any arbitrary input which falls within the fuzzy
region can be fuzzy regions of 𝐴1 and 𝐵1 can be taken 𝐴1 , 𝐵1 , 𝐴2 , 𝐵2 can be taken or else
it may be given as the unknown input suitably.

979
So, for 𝑥1 is equal to 6 𝑦1 is equal 8.2. Let us now find first using first rule the 𝑤11 which
is the intersection point. So, intersection point let us call this as 𝑤11 and this is coming out
to be corresponding to 𝑥1 is equal to 6 it is cutting 𝐴1 at 0.93 membership value. So, this
becomes our 𝑤11 .

Similarly, let us now find the corresponding to 𝑦1 8.2 value the intersection point on 𝐵1.
So, let us call this as 𝑤12 , so this is our 𝑤12 . Now, we have 𝑤11 𝑤12 both now let us take min
because the connective is AND. So, when we take min our value is going to be 𝑤1 which
is also called as the rule strength or I or I can write here or I can say here the firing is
strengths of rule 1.

So, firing a strength of rule 1 is 𝑤1 is also called weight of the rule. So, this is 0.78. Now
𝑝01 we already have as 8 this is a constant value. So, we we write it by 𝑝0 and 𝑝0 is the
constant value and 1 we are writing here for rule number 1 for rule 1. So, that is why we
have also used one similarly here for rule number 2, 𝑝02 all right.

So, now we have obtained 𝑤1 and 𝑧1 . So, 𝑧1 we have very quickly found this is 𝑥, 𝑦 which
is 𝑝01 , here in this case let us say this is here in this case 𝑧1 . So, 𝑧1 is 𝑝01 similarly now on
the same lines when we use rule number 2 we find the intersection point that is let us say
𝑤21 for the first second rule first antecedent. And similarly here for 𝑦1 is equal to 8.2 we
write here 𝑤22 . So, 𝑤22 means second rule second antecedents now both of these values are
0.99 and 0.13.

And when we take min of the two we are getting the firing a strength firing a strength of
rule number 2. So, we have firing a strength of rule number 2 which is coming out to be
𝑤2 is equal to 0.13 and second rule gives us 𝑝02 . So, 𝑝02 is giving us 6 already. So, 𝑧2 will
become 6 here. So, I can write here that our 𝑧1 𝑧2 basically is 𝑓 2 (𝑥, 𝑦) is equal to 𝑝02 . So,
this way we get 𝑧2 is equal to 6. So, now, we have 𝑤1 𝑧1 , 𝑤2 𝑧2.

Now, we can very easily find the weighted average. So, let us find the weighted average
of the outputs of each rule and when we use this formula for weighted average the z is star
which is the final outcome the weighted average of the output. We are getting 7.7143. So,
this way we can find the complete the final output corresponding to 𝑥1 is equal to 6, 𝑦1 is
equal to 8.2 using the TSK fuzzy model, which is characterized by 2 rules with 2
antecedents and this comes under the category of multiple rules with multiple antecedents.

980
(Refer Slide Time: 32:06)

Now, let us move to the first order TSK fuzzy model. So, as I have already mentioned the
ordered basically is defined here with this with the if when we talk of first order TSK
model. It means the input generic variable at least the maximum the maximum power of
these input cannot exceed 1. So, this means what? So, the output of each rule will be of
this form only. So, 𝑝0 the constant can be there 𝑝0 and then 𝑝1 𝑥 this 𝑝2 𝑥 cannot be there,
similarly no higher terms cannot be there no higher terms can be there or the higher terms
cannot be higher terms of the input generic variables are not there.

Similarly, for y also only since this is first order TSK fuzzy models only y will be there y
raise to power 1. So, more than 1 cannot be there. So, what is coming out to be here is that
𝑓(𝑥, 𝑦) which is the form of the output of each rule. So, 𝑓(𝑥, 𝑦) let us say for 2 input
generic variables. So, we can have 𝑝0 + 𝑝1 𝑥 and then we write here 𝑞1 𝑦, but since there is
no other variable. So, I can write here 𝑝2 𝑦. So, that is how this is of this form. So, first
order TSK fuzzy model will have the output the consequent part, that means the output of
this form for 2 inputs if we have only 1 input then; obviously, this 𝑦 term will go away.

So, along with the constant we will have one more term and this term will be the one more
term for the single antecedent, but similarly if we have multiple antecedents we will have
multiple antecedent, but the power of the degree of I mean the power of each generic
variables the input generic variables cannot exceed more than 1.

981
(Refer Slide Time: 35:42)

So, let us quickly go ahead and understand. So, here this is mentioned this is the general
form of the polynomial of the output here first order TSK fuzzy model, but here m cannot
be, m can be only 1.

So, m value of m can be only 1. So, I can write here the when we say first order TSK fuzzy
model it means m is equal to 1. The value of m is equal to 1. Similarly, when we say 0 this
is for first order, similarly 𝑚 is equal to 0 for 0 order. So, 𝑚 is equal to 0 0 means 0 order
what does this mean this means that only constant will be there. So, now, as I mentioned
then the output will be like this similarly all others terms.

And conditions will remain the same means we have the min if it is needed for the
connective AND or we will take max in case of the OR connective and similarly we will
have the weighted average we have already done.

982
(Refer Slide Time: 36:25)

So, this way we proceed and now let us under this first order TSK fuzzy model let us
discuss all those three cases we just have discussed. So, the first case again is the single
rule with single antecedent.

(Refer Slide Time: 36:44)

So, when we have a single rule with single antecedent we have the single antecedent here
and please note that here for the first order first order TSK, first order TSK we have 𝑚 is
equal to 1 only this means that we have 𝑓(𝑥) = 𝑝0 + 𝑝1 𝑥.

983
(Refer Slide Time: 37:11)

And now, if we have the rule of this kind let us say we have a TSK fuzzy model which has
the polynomial the 𝑓(𝑥) like this is let us say 𝑧 = 1 + 0.5𝑥.

(Refer Slide Time: 37:36)

So, if this is the case then let us apply some input and let us apply the crisp input 𝑥1 =
2.020. So, this is the same input which we have applied in the previous case previous
category that was the zeroth order.

984
And that was again for single rule with single antecedent. So, we get the 𝑤1 here and since
there is no connective only 1 antecedent is there single antecedent is there. So, this
becomes the 𝑤1. So, the this becomes the 𝑤1. So, this is the weight of the rule or firing
strength of the rule so 𝑤 is known. And then our 𝑧1 = 1 + 0.5𝑥 we already know because
we have applied this.

So, 1 + 0.5 × 2.0202. So, this way we get the total here is 2.0101. So, we have now 𝑧1
and since we have only 1 rule we have single rule. So, the weighted average is not needed
or even if you go for this that is going to remain the same. So, this way we get the output
of the TSK fuzzy model which is characterized by a single rule with single antecedent and
when we apply the input 𝑥1 ,2.0202 we are getting the output here we are getting the output
2.0101.

Alright so now, let us go to the second case of first order TSK fuzzy model where we have
a single rule, but with multiple antecedent.

(Refer Slide Time: 40:08)

So, here also for simplicity we have taken only two antecedents 𝑥 is 𝐴 and 𝑦 is 𝐵, but these
two antecedents are connected by the connective AND and again I would like to mention
that this connective can be any connective can be OR as well or, but or whatever. So, here
since we have AND, AND will have the composition of min. So, we will use min if you
would have.

985
Or here then max would apply as the composition. So, AND is the connective here. So,
rule says if 𝑥 is 𝐴 the first antecedent and then we have the connective and then we have
the second antecedent 𝑦 is 𝐵. So, then here the output is of the same type which we have
just discussed and this category is the first order TSK fuzzy model.

(Refer Slide Time: 41:15)

Now when we and before this I would like to mention that the here for simplicity we have
taken the 2 antecedents, but we can have any number of antecedents now let us go ahead
and apply some crisp inputs 𝑥1 and 𝑦1 and since we have here the single rule. So, let us
apply this to a TSK model which is characterized by a single rule with 2 antecedents 𝑥 is
𝐴, 𝑦 is 𝐵.

So, we see that here we are applying the same inputs as we have done in the zero order
tsk. So, 𝑥1 is equal to 4.4444 and 𝑦1 is equal to 6.2626. So, we see the corresponding points
of intersection and this is designated as the 𝑤11 and then this is also designated as 𝑤12 . Now,
within these we take the min because we use the connective AND if we would have used
or connective then here you would have taken max. So, this gives us the again the firing a
strengths of rule is also called weight.

So, we have the 𝑧1 because we have the first order polynomial equation here which is the
output and please note that since we have here two antecedents. So, that is why the 𝑧 is in
terms of 𝑥 and 𝑦. So, we see here the 𝑥, 𝑦 which is nothing, but the output let us say 𝑧 and
this is of this form 𝑝0 + 𝑝1 𝑥 + 𝑝2 𝑦. So, this is how we are getting the z calculated for this.

986
So, and this way this is nothing, but the 𝑧1 . So, here what is 𝑧1 , 𝑧1 is this. So, 𝑧1
corresponding to 𝑥1 𝑦1 and 𝑝1 is known 𝑝0 is known 𝑝0 is known 𝑝1 known 𝑝2 known.

So, and 𝑤1 is also known. So, we can quickly get 𝑧1 corresponding to 𝑥1 and 𝑦1 . And since
we have again here the single rule. So, the output is going to remain the same. So, the
weighted average even if we take here is going to be 3.5747. So, this is equal to 𝑧1 now
the third case is multiple rules with multiple antecedent.

(Refer Slide Time: 44:43)

So, let us now take two rules under this case for simplicity rule number 1, rule number 2
and both the rules have 2 antecedents again for the simplicity and is the connective as I
have mentioned.

So, here the output also remains the same as we have seen in this the previous cases of
first order TSK fuzzy model the only difference that we have here is because we have 2
rules. So, we have differentiated we have a written 𝑧1 and 𝑧2 for the corresponding rules
as the outputs.

987
(Refer Slide Time: 45:30)

Now, let us use these two rules rule number 1 and rule number 2 to generate the outputs
corresponding to the inputs, crisp inputs supplied. So, let us apply the input 𝑥1 and 𝑦1 , 𝑥1
is equal to 6 𝑦1 is equal to 8.2 and let us see the corresponding outputs. So, corresponding
to rule number 1 the 𝑤1 is the points of intersection corresponding to 𝑥1 the first antecedent
first input.

So, we find here the 𝑤11 and then we have here in rule number 1 the points of intersection
here for the second antecedent for 𝑦1 is equal to 8.2 is 𝑤12 . Similarly when we use rule 2
we get this as this intersection point corresponding to 𝑥1 is equal to 6 as 𝑤21 here for
corresponding to 𝑦1 in rule number 2 for the second antecedent we get the point of
intersection as the membership value and which represents the 𝑤22 .

Now, let us use rule number one output here. So, let me first write the z is of the form of
the output is of the form of 𝑥, 𝑦 and then we have 𝑝0 + 𝑝1 𝑥 + 𝑝2 𝑦. And since the 𝑧1 is
given for the first rule, 𝑧2 is given for the second rule. So now, if we use the 𝑥1 and 𝑦1 this
is 𝑥1 , 𝑦1 . So, if we use 𝑥1 , 𝑦1 values which is the input set of inputs which is the input 𝑥1 , 𝑦1
so then we get 𝑧1 . So, we see that we get the 𝑧1 here which is 4.74. So, when we use 𝑥1 is
equal to 6 and 𝑦1 is equal to 8.2 using the first rule output we get 𝑧1 is equal to 4.74 here.

So, similarly now we use rule number 2 and find the 𝑧2 . So, here on the same lines when
we use the input here. So, 𝑥1 and 𝑦1 we use as the input. So, 𝑧2 for the same 𝑥1 , 𝑦1 we are
getting as the 𝑧2 is equal to here 2.34 can see here and how do we get this we already have

988
𝑧2 = 0.02 + 0.25𝑥1 + 0.1𝑦1 . So, please understand here and that this z 1 is like this, so
𝑧1 is here 𝑧1 = 𝑓 1 (𝑥, 𝑦).

And this is of this form 𝑝01 + 𝑝11 𝑥 + 𝑝21 𝑦 similarly our 𝑧1 𝑧2 is of this form our 𝑧2 is
𝑓 2 (𝑥, 𝑦) = 𝑝02 + 𝑝12 𝑥 + 𝑝22 𝑦. So, here we have 𝑧1 , 𝑧2 and we also have 𝑤1 and 𝑤2 by taking
the min of the corresponding weights corresponding membership values. Now since we
have 2 rules here and corresponding to each rule we have its outputs.

Now, we take the weighted average of it to get the final output. So, final output we use
here the weighted we take the weighted average and this way we get 𝑧 is equal to 4.3971.
So, this way if we have a TSK fuzzy model which has which is characterized by the set of
rules set of multiple rules with multiple antecedents then we will proceed with this kind
of.

(Refer Slide Time: 51:45)

And then here let us take an example to understand the TSK fuzzy model better and this
example is basically a first order TSK fuzzy model and which is having four rules you see
here.

So, we have rule number 1 we have rule number 2, 3, 4. So, all the rules have the outputs
that are different. So, I can write here this is the 𝑧1 , this is 𝑧2 , this is 𝑧3 this is giving us
𝑧1 , 𝑧2 , 𝑧3 and so on this giving us 𝑧4 . So, when we have a TSK fuzzy model and this is
characterized by 4 rules and we see that each and every rule is coming under the category

989
of the first order TSK fuzzy model. So, and then we see that the fuzzy regions that are
mentioned here in the rule 𝐴, 𝐵 they are basically having three fuzzy regions they are
having three fuzzy regions.

And all and each fuzzy regions are represented by the fuzzy sets which are represented
which are characterized by the membership functions 𝜇𝐴𝑙𝑜𝑤 , 𝜇𝐴𝑀𝐸𝐷𝐼𝑈𝑀 , 𝜇𝐴𝐻𝐼𝐺𝐻 and 𝜇𝐴𝐿𝑂𝑊
is Gaussian, 𝜇𝐴𝑀𝐸𝐷𝐼𝑈𝑀 is you know triangular, 𝜇𝐴𝐻𝐼𝐺𝐻 is trapezoidal.

Similarly, the input in the antecedent part is divided into three fuzzy regions and every
fuzzy region is defined by or represented by fuzzy set LOW, MEDIUM, HIGH and all
these fuzzy sets are characterized by the membership functions 𝜇𝐵𝐿𝑂𝑊 , 𝜇𝐵𝑀𝐸𝐷𝐼𝑈𝑀 , 𝜇𝐵𝐻𝐼𝐺𝐻
and you can see the 𝜇𝐵 is trapezoidal, 𝜇𝐵𝑀𝐸𝐷𝐼𝑈𝑀 , 𝜇𝐵𝐿𝑂𝑊 is trapezoidal, 𝜇𝐵𝑀𝐸𝐷𝐼𝑈𝑀 is also
trapezoidal, 𝜇𝐵𝐻𝐼𝐺𝐻 is also trapezoidal.

So, what is the what is that which we need to do is here we have to find the we have to
compute calculate the output for the inputs 𝑥 is equal to 70 and please understand here the
connective that is mentioned here is OR. So, here also you see the connective instead of
AND we have OR. So, if we have this kind of situation we know what to do. So, we have
the inputs 𝑥 is equal to 70 or 𝑦 is 60.

(Refer Slide Time: 55:27)

So, let us now proceed with this. So, first of all what we do we plot all these membership
functions and these membership functions basically designate the fuzzy sets LOW,

990
MEDIUM, HIGH and similarly for the output as well. Now, both the inputs so first input
here and then second input here means the first input means first input variable for first
input generic variable, second input generic variable 𝑦 here and both the inputs are now
plotted and we have the membership functions here.

And now, our task is to find the intersection points. So first of all corresponding to this
input the input that is given. So, our input that is given is 𝑥 is equal to 70 or 𝑦 is equal to
60. So, corresponding to 𝑥 is equal to 60 when we plot this we see that here this 𝑥 is equal
to 60 falls under two fuzzy regions.

First fuzzy region is the this one the HIGH region which is basically the HIGH fuzzy set
and the other one is this fuzzy region this is a Gaussian this is represented by Gaussian.
So, LOW fuzzy region is also coming into picture and LOW fuzzy region here is very less
because, this is represented by a Gaussian fuzzy set Gaussian fuzzy membership function.
So, there are two regions which are relevant for 𝑥 is equal to 70 similarly.

Now, let us check for the 𝑦 is equal to 60. So, for 𝑦 is equal to 60 we see that only one
fuzzy region is relevant that is high. So, for 𝑦 is equal to 60 we have only HIGH fuzzy set
which is relevant. So, we see that we have two points of intersection in the first antecedent,
that means the first input generic variable and the 𝑦 is equal to 60, we have only one fuzzy
region which is relevant. So, 2 into 1 is going to give us 2 combinations what are those 2
combinations first is HIGH-HIGH, HIGH and then HIGH.

So, both the inputs basically. So, when the input is falling in the I am writing here as the
when x is falling in 𝑥 is falling in HIGH and then 𝑦 is falling in HIGH then what is going
to be the 𝑧 the output. So, this we will get by inspecting the rules that are given to us the
fuzzy rules that are given to us, but here we have two combinations. So, another
combination is when 𝑥 is falling in LOW and 𝑦 is falling in HIGH because here only the
𝑦 is falling only in high. So, now, for these 2 combinations only we need 𝑧.

So, let us check here we have 4 rules. So, first is the when the input is falling in the 𝑥 is
falling in LOW this is basically 𝑥 this is when the input is this is 𝑦, this is 𝑦 this is 𝑥, 𝑥, 𝑦, 𝑦.
So, when 𝑥 is LOW OR 𝑦 is HIGH. So, this combination is this.

991
(Refer Slide Time: 60:04)

So, for this the output will be applicable is this 𝑧1 . So, we can say the rule number 1, that
means the 𝑧1 will apply and 𝑧1 is defined by the 10 plus the output is equal to 10 + 0.2𝑥 +
0.3𝑦 similarly the other case HIGH HIGH.

So, when 𝑥 is HIGH here the fourth ones. So, this is up this is relevant and this is a relevant
means rule number 1 and rule number 4 are relevant. So, HIGH-HIGH. So, output four
here will be mentioned. So, here we will write the 𝑧4 . So, this way we see that no other
rule is applicable because we get only 2 combinations and this can be written like this as I
just explained. So, rule number 1 is applicable and rule number 4 is applicable out of all
the 4 rules given this is not applicable this is not applicable. Now take these 2 rules and
find the output of each rule corresponding the inputs that are being supplied.

So, and then whatever output that we get we take the weighted average of it and that is
going to be the final output. So, let us go ahead and do that.

992
(Refer Slide Time: 61:42)

So, here as I have already mentioned that rule number 4 are rule number 1 and rule number
4 are applicable. So, let us take rule number 1. So, rule number 1 is already given if you
have any confusion you can refer here. So, rule number 1 is given 𝑥 is LOW 𝑦 is HIGH.
So, LOW we already know. So, LOW is here this fuzzy set is LOW for LOW.

So, we take this fuzzy set here LOW is for low. So, we have to plot it and then similarly
for HIGH we can see from there. So, HIGH is here in rule number 1 we have LOW and
HIGH combination in rule number 4 we have HIGH-HIGH combination. So, the
membership function are the fuzzy set for HIGH-HIGH we can obtain from here if you
have any doubt it is shown here the HIGH is here this is HIGH. So, this can be very easily
plotted. So, let us proceed with this. So, let us take rule number 1 and apply the input.

993
(Refer Slide Time: 63:00)

So, we apply the input here 𝑥 is equal to 70, 𝑦 is equal to 60. So, corresponding to 𝑥 is
equal to 60 we get here the intersection point let us call this as 𝑤11 . Similarly corresponding
to 𝑦 60, 𝑦 is equal to 60 for rule number 1, let us call this as 𝑤12 , 𝑤12 . So, 𝑤11 is 0.0022, 𝑤12
is 1. And since here we have the connective OR that we have given that we have been
given. So, the we use instead of min we use max.

So, 𝑤1 will be the maximum of 𝑤11 and 𝑤12 and which is coming out to be 1. So, now we
have the rule strength the firing we have the firing strength of the rule 𝑤1 and similarly I
can write here the firing is strength of the rule 1, alright so and 𝑧 is already given. So, by
just substituting the value of 𝑥 and 𝑦 we get 𝑧1 here. So, 𝑧1 is 42 if we substitute the value
of 𝑥 and 𝑦. So, this is 𝑧1 and similarly when it comes to the rule number 4 we find here
the point of intersection as 𝑤21 and here 𝑤22 .

And here also we take max of these 2. So, both of these are 1, 1. So, we take max and max
is again going to be 1 and this is the output which is 𝑧2 . So, 𝑧2 by just substituting this we
are going to get here the these is 𝑧2 and when we substitute the values of 𝑥 and 𝑦 here we
are going to get 𝑧2 is equal to 8.8. So, now, we have what we have 𝑤1 we have 𝑤2 we have
𝑧1 we have 𝑧2 . Now we can very easily get the weighted average of these outputs
corresponding to the rule number 1 and rule number 4 see here.

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(Refer Slide Time: 65:50)

So, we have 𝑤1 is equal to 1 you can see here and then 𝑤2 is equal to 1, 𝑧1 is equal to 42,
𝑧2 is equal to 8.8 and when we take the weighted average we are getting the weighted
average as 25.4. So, this means that corresponding to the input 𝑥 is equal to 70 and 𝑦 is
equal to 60 we are getting the overall output as 25.4 when we use the TSK fuzzy model
with multiple antecedent and multiple rules.

(Refer Slide Time: 66:57)

So, this way we quickly get to know as to how we can manage to get the final output
corresponding to the inputs that are given for any TSK fuzzy model. So, let us take another

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TSK fuzzy model which is having 6 rule here 6 rules we have. So, any fuzzy model which
is characterized by 6 fuzzy rules and we have the rule number 1 as 𝑥1 is a we have first
rule as 𝑥 is LOW 𝑦 is LOW then the output is this let us say this is 𝑧1 .

And then the second rule here is 𝑥 is LOW 𝑦 is HIGH then the 𝑧2 is given by this
expression. Similarly 𝑥 is MEDIUM, then 𝑦 is LOW, and then this is 𝑧3 means 𝑥 is HIGH
and then 𝑦 is LOW then the output here is for the first fourth rule is 𝑧4 . Similarly rule
number 5 as 𝑥 is HIGH 𝑦 is MEDIUM then the output is 𝑧5 . Rule number 6 has 𝑥 is HIGH
𝑦 is HIGH then the output 𝑧6 is given by this expression, and the corresponding fuzzy
regions that are given for LOW, MEDIUM, HIGH for 𝑥 input variable is given here and
similarly for the 𝑦 input variable the corresponding LOW, MEDIUM, HIGH fuzzy regions
are characterized by these membership functions.

Now, then task is here to find the corresponding output to 𝑥 is equal to, the input 𝑥 is equal
to 70 and 𝑦 is equal to 60. So, here the input remains the same, but interesting thing that
we see here is that the connective is AND instead of OR. So, here we have this connective.
So, if we have this kind of example, this kind of problem. So, in the previous one we saw
that we have had or as the connective here we have and so obviously we are going to take
composition as min.

(Refer Slide Time: 70:16)

So, let us proceed with this we have already seen as to how we can get this plotted. So,
here also for 𝑥 is equal to 70 we have two intersection points first is the for LOW and

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second is for HIGH. Similarly the 𝑦 for 𝑦 is equal to 60 only one intersection points. So,
the combinations again will remain same as the previous example.

(Refer Slide Time: 70:54)

Now, here we have to look for the applicable rules. So, since we have 6 rules that are given
for the fuzzy model for the TSK fuzzy model. So, we have only two combinations that is
LOW-HIGH, HIGH-HIGH. So, rule number 2 will apply and 6 will apply this rule will
not apply because the combination wise we LOW-LOW does not exist only LOW-HIGH
exist here.

So, the rule number three MEDIUM-LOW does not exist HIGH-LOW does not exist
HIGH-MEDIUM does not exist and HIGH-HIGH exists. How do we get this I have
already explained in the previous example, but I can do it again. So, we see that 𝑥 is equal
to 70 cuts HIGH fuzzy region here HIGH fuzzy region here and then LOW fuzzy region
which is characterized by the membership Gaussian membership function.

So, 𝑥 and 𝑦 if we write and then 𝑧. So, 𝑥 is cutting two fuzzy regions LOW and HIGH
and 𝑦 is cutting only one fuzzy region here the HIGH for others it is 0. So, we can write
L-H and H-H. Because HIGH is for both the cases and this way we have to let look for the
corresponding 𝑧 and this we can find by looking at the set of rules and we have here two
rules that are applicable and then with this we will proceed same as we have done in the
previous example.

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(Refer Slide Time: 72:53)

So now, we see that here we get when we take 2 rules. So, we have to take only 2 rules
other rules rest of the rules will just discard. So, because here rule number 2 and rule
number 6 are only applicable. So, we will start with rule number 2 rule number 2 is here.
Now we apply the input x is equal to 70. So, 𝑤11 let us say similarly here this is 𝑤12 for the
second antecedent 𝑦 is equal to 60 and then corresponding output here is 𝑧1 .

And this 𝑧1 by just substituting the value of 𝑥 and 𝑦 we can have here this is coming out
to be 47 and now when it comes to 𝑤1 will have min why min because here the connective
is AND. So, AND connective is used. So, that is why we are taking min here. So, if we
take min of the two values 𝑤11 and 𝑤12 we get 0.0022. So, we have here 𝑤1 and 𝑧1 now
similarly when we apply the rule 6 for the input 𝑥 is equal to 70, 𝑦 is equal to 60 we have
𝑧2 like this 𝑧2 is equal to this is already given.

And when we use 𝑥 and 𝑦 the values of 𝑥 and 𝑦 we get 𝑧2 here computed like the 68.9 and
the 𝑤2 is here 𝑤21 and 𝑤21 and 𝑤22 . So, both are 1 here. So, even if we take min is going to
be the same. So, 𝑤2 is the firing strength of the second rule which is 1.

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(Refer Slide Time: 75:08)

So, now we use these values here. So, 𝑤1 𝑤2 , 𝑧1 𝑧2.

And then the weighted average is coming out to be 68.8519. So, this way here we can say
that the corresponding to 𝑥 is equal to 70 and 𝑦 is equal to 60 we are getting the output of
the TSK fuzzy model as 68.8519. So, I think we have understood now as to how we can
compute the outcomes of the TSK fuzzy model in various cases for various inputs and with
different kinds of sets of the rules and with this I would like to stop here.

Thank you very much.

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