UNIT III: Fuzzy Logic:

Introduction

Fuzzy sets and Fuzzy reasoning

Basic functions on fuzzy sets,

Relations

Rule based models and linguistic variables,

Fuzzy controls,

Fuzzy decision making

Applications of fuzzy logic.

Introduction:
The term fuzzy refers to things that are not clear or are vague. In the real world many
times we encounter a situation when we can't determine whether the state is true or false, their
fuzzy logic provides very valuable flexibility for reasoning. In this way, we can consider the
inaccuracies and uncertainties of any situation.

Fuzzy Logic is a form of many-valued logic in which the truth values of variables may
be any real number between 0 and 1, instead of just the traditional values of true or false. It is
used to deal with imprecise or uncertain information and is a mathematical method for
representing vagueness and uncertainty in decision-making.
Fuzzy Logic is based on the idea that in many cases, the concept of true or false is too
restrictive, and that there are many shades of gray in between. It allows for partial truths,
where a statement can be partially true or false, rather than fully true or false.
Fuzzy Logic is used in a wide range of applications, such as control systems, image
processing, natural language processing, medical diagnosis, and artificial intelligence.
The fundamental concept of Fuzzy Logic is the membership function, which defines the
degree of membership of an input value to a certain set or category. The membership function
is a mapping from an input value to a membership degree between 0 and 1, where 0 represents
non-membership and 1 represents full membership.
Fuzzy Logic is implemented using Fuzzy Rules, which are if-then statements that
express the relationship between input variables and output variables in a fuzzy way. The
output of a Fuzzy Logic system is a fuzzy set, which is a set of membership degrees for each
possible output value.
In summary, Fuzzy Logic is a mathematical method for representing vagueness and
uncertainty in decision-making, it allows for partial truths, and it is used in a wide range of
applications. It is based on the concept of membership function and the implementation is done
using Fuzzy rules.
 In the boolean system truth value, 1.0 represents the absolute truth value and 0.0
represents the absolute false value. But in the fuzzy system, there is no logic for the absolute
truth and absolute false value. But in fuzzy logic, there is an intermediate value too present
which is partially true and partially false.

Fuzzy sets and Fuzzy reasoning

Introduction to Fuzzy Set In this chapter, the concept of fuzzy sets and the operations on the
fuzzy set are discussed. The concepts are the generalizations of crisp sets. Classical sets are also called
‘crisp’ sets so as to distinguish them from fuzzy sets. In fact, the Crisp sets can be taken as special cases
of fuzzy sets. Let A be a crisp set defined over the Universe X. Then for any element x in X, either x is a
member of A or not. In fuzzy set theory, this property is generalized. Therefore, in a fuzzy set, it is not
necessary that x is a full Member of the set or not a member. It can be a partial member of the sets.
The generalization is performed as follows: For any crisp set A, it is possible to define a
Characteristic function or membership function μA = {0, 1}.i.e. the characteristic function takes
either of the values 0 or1 in the classical set. For a fuzzy set, the characteristic function can take
any value between zero and one. Definition The membership function μA(x) of a fuzzy set A is a
function μA : X →[0,1] So, every element in x in X has membership degree: μA(x) ∈ [0,1] A is
completely determined by the set of tuples: A = {(x, μA(x)) x∈ X} Example: Suppose someone
wants to describe the class of cars having the property of being expensive by considering BMW,
Rolls Royce, Mercedes, Ferrari, Fiat, Honda and Renault. Some cars like Ferrari and Rolls
Royce are definitely expensive and some like Fiat and Renault are not expensive in comparison
and do not belong to the set. Using a fuzzy set, the fuzzy set of expensive cars can be described
as:

{(Ferrari, 1), (Rolls Royce, 1), (Mercedes, 0.8), (BMW, 0.7), (Honda,0.4)}. Obviously, Ferrari
and Rolls Royce have membership value of 1 whereas BMW, which is less expensive, has a
Membership value of 0.7 and Honda 0.4. The Fuzzy set is similar to the super set of the Boolean
logic with extra membership functions in between “true” and “false”. As its name suggests, it is
the logic underlying modes of reasoning which are approximate rather than exact. The
importance of fuzzy logic derives from the fact that most modes of human reasoning and
especially common sense reasoning are approximate in nature. The essential characteristics of
fuzzy logic are as follows.

In fuzzy logic, exact reasoning is viewed as a limiting case of approximate reasoning.

 In fuzzy logic everything is a matter of degree.

 Any logical system can be fuzzified

 In fuzzy logic, knowledge is interpreted as a collection of elastic or, equivalently , fuzzy

 Constraint on a collection of variables Inference is viewed as a process of propagation of

elastic constraints.

Fuzzy set:

1. Fuzzy set is a set having degrees of membership between 1 and 0. Fuzzy sets are
represented with tilde character(~). For example, Number of cars following traffic signals
at a particular time out of all cars present will have membership value between [0,1].
2. Partial membership exists when member of one fuzzy set can also be a part of other fuzzy
sets in the same universe.
3. The degree of membership or truth is not same as probability, fuzzy truth represents
membership in vaguely defined sets.
4. A fuzzy set A~ in the universe of discourse, U, can be defined as a set of ordered pairs and
it is given by

Properties of Fuzzy sets:

Fuzzy sets follow the same properties as crisp sets. Since membership values of crisp sets are a subset of
the interval [0,1], classical sets can be thought of as generalization of fuzzy sets.

OperationsonFuzzysets

The well-known operations which can be performed on fuzzy sets are the operations of
union, intersection, complement, algebraic product and algebraic sum. Much research concerning
fuzzy sets and their applications to automata theory, logic, control, game, topology, pattern
recognition, integral, linguistics, taxonomy,system,decision making, information retrieval andso
on, has been earnestly undertaken by using these operations for fuzzy sets.
In addition to these operations, new operations called "bounded-sum" and In addition to
these operations, new operations called "bounded-sum" and "bounded-difference" are introduced
by Zadeh (1975) to investigate the fuzzy reasoning which provides a way of dealing with the
reasoning problems which are too complex for precise solution.

Types of operators
1. Equality
2. Complement
3. Intersection
 4. Union
5. Algebraic product
6. Multiplication of fuzzy set with crisp number
7. Power of fuzzy set
8. Algebraic sum
9. Algebraic difference
10. Bounded sum
11. Bounded difference
12. Cartesian product
13. Composition

1. Equal fuzzysets
Two fuzzy sets A(x) and B(x) are said to be equal, if µA(x) = µB(x) for all x ∈X. It is expressed as
follows

A(x)=B(x), if µA(x)=µB(x)

Note:Two fuzzy sets A(x) and B(x) are said to be unequal,if µA(x)≠µ B(x) for atleast x∈X.

Example:
A(x)={(x1,0.1),(x2,0.2),(x3,0.3),(x4,0.4)}
B(x)={(x1,0.1),(x2,0.5),(x3,0.3),(x4,0.6)}

As µA(x)≠µB(x) for different x∈X, A(x)≠B(x)

2. ComplementoffuzzysetA(x)

The complement is the opposite of the set. The complement of a fuzzy set is denoted by Ā(x) and
is defined with respect to the universal set X as follows:

Ā(x)=1-A(x) for all xϵX

 Figure4:Example of complement operation on a fuzzyset

3. Intersections of fuzzy sets:

Inter section of a fuzzy sets define how much of the element belongs to both sets. May have
different degrees of membership in each set. The degree of membership is the lower
membership in both sets of each element. Let A(x) and B(x) are two fuzzy sets, the intersection of is
denoted by (A∩B)(x) and the membership function value is given as follows

µ(A∩B)(x)=min{µA(x),µB(x)}

Intersection is analogous to logical AND operation

Example A(x)={(x1,0.7),(x2,0.3),(x3,0.9),(x4,0.1)}
B(x)={(x1,0.2),(x2,0.5),(x3,0.7),(x4,0.4)}

µ(A∩B)(x1)=min{µA(x1),µB(x1)}=min{0.7,0.2}=0.2
µ(A∩B)(x2)=min{µA(x2),µB(x2)}=min{0.3,0.5}=0.3
µ(A∩B)(x3)=min{µA(x3),µB(x3)}=min{0.9,0.7}=0.7
µ(A∩B)(x4)=min{µA(x4),µB(x4)}=min{0.1,0.4}=0.1

The graphical representation of the intersection operator is given below

 Figure5:Example of intersection operation on a fuzzy set
4. Union of fuzzy sets

Union of fuzzy sets consists of every element that falls in to either set.The value of the membership
value is will be the largest membership value of the element in either set

Let A(x) and B(x) are two fuzzy sets for all x ∈X, Union of fuzzy sets is denoted by (AUB)(x)
and the membership function value is determined as follows

µ (AUB)(x)= max{µA(x),µB(x)}

Example:
A(x)={(x1,0.7),(x2,0.3),(x3,0.9),(x4,0.1)}
B(x)={(x1,0.2),(x2,0.5),(x3,0.7),(x4,0.4)}

µ(AUB)(x 1)=max{µA(x1),µB(x1)}=max{0.7,0.2}=0.7
µ(AUB)(x 2)=max{µA(x2),µB(x2)}=max{0.3,0.5}=0.5
µ(AUB)(x 3)=max{µA(x3),µB(x3)}=max{0.9,0.7}=0.9
µ(AUB)(x 4)=max{µA(x4),µB(x4)}=max{0.1,0.4}=0.4

Note:Union is analogous to logical OR operation.

 Figure6:Example of union operation on a fuzzyset

5. Algebraicproductoffuzzysets

The Algebraic product of two fuzzy sets A(x) and B(x) forall x∈X, is denoted by A(x).B(x)
anddefinedasfollows

A(x).B(x)={(x,µA(x).µB(x)),xϵX}

Example
A(x)={(x1,0.1),(x2,0.2),(x3,0.3),(x4,0.4)}
B(x)={(x1,0.5),(x2,0.7),(x3,0.8),(x4,0.9)}

A(x).B(x)={(x1,0.05),(x2,0.14),(x3,0.24),(x4,0.36)}

6. Multiplication of fuzzy sets by a crisp number

The product of fuzzy set A(x) and a crisp number‘d’ is expressed as follows

A(x).B(x)= {(x, d . µA(x)), x ϵ X }

Example
Let us consider a fuzzy set A(x) such that
A(x)={(x1,0.1),(x2,0.2),(x3,0.3),(x4,0.4)}
d = 0.2
thend.A(x)={(x1,0.02),(x2,0.04),(x3,0.06),(x4,0.08)}

7. Power ofafuzzyset
The p-th power of a fuzzy set A(x) yields another fuzzy set Aᴾ(x), whose membership value can
be determined as follows
 µAᴾ(x)={µA(x)}ᴾ,x∈X}

p >= 1Aᴾ(x) is called concentration p

<1 Aᴾ(x) is called dilation
Example:

Let us consider a fuzzy set A(x)

A(x)={(x1,0.1),(x2,0.2),(x3,0.3),(x4,0.4)}

P=2
ThenA²(x) ={(x1,0.01),(x2,0.04),(x3,0.09),(x4,0.16)}

8. Algebraic sum of two fuzzy sets

The Algebraic sum of two fuzzy sets A(x) and B(x) for all x ∈X, is denoted by A(x)+B(x)and
defined as follows

A(x)+B(x)={(x,µA+B(x),xϵX}

Where µA+B(x) = µA(x)+µB(x) - µA(x).µB(x)

Example:
A(x)={(x1,0.1),(x2,0.2),(x3,0.3),(x4,0.4)}
B(x)={(x1,0.5),(x2,0.7),(x3,0.8),(x4,0.9)}

Now(x)+B(x)={(x1,0.55),(x2,0.76),(x3,0.86),(x4,0.94)}

9. Bounded sum of two fuzzy sets

The bounded sum of two fuzzy sets A(x) and B(x) forall x ∈X, is denoted by
A(x)⊕B(x)and defined as follows

A(x) ⊕ B(x) = {(x,µA ⊕ B(x), x ∈X }

Where µ A ⊕ B(x)= min{1,µA(x)+µB(x)}

Example:
A(x)={(x1,0.1),(x2,0.2),(x3,0.3),(x4,0.4)}
B(x)={(x1,0.5),(x2,0.7),(x3,0.8),(x4,0.9)}
 A(x)⊕B(x)={(x1,0.6),(x 2,0.9),(x3,1.0),(x4,1.0)}

10. Algebraic deference of two fuzzy sets

The Algebraic deference of two fuzzy sets A(x) and B(x) for all x ∈X, is denoted byA(x)+B(x)
And defined as follows
A(x)-B(x)={(x,µA-B(x),xϵX}

WhereµA-B(x)=µA∩B(x) Example:

A(x)={(x1,0.1),(x2,0.2),(x3,0.3),(x4,0.4)}
B(x)={(x1,0.5),(x2,0.7),(x3,0.8),(x4,0.9)}

B̅(x)={(x1,0.5),(x2,0.3),(x3,0.2),(x4,0.1)}

A(x)-B(x)={(x1,0.1),(x2,0..2),(x3,0.2),(x4,0.1)}

11. Cartesian product of two fuzzy sets

Let us consider two fuzzy sets A(x) and B(y) defined on the Universal sets X and Y, respectively.
The Cartesian product of fuzzy sets A(x) and B(y), is denoted by A(x) X B(x), such that x ∈X, y∈
Y. It is determined, so that the following conditions satisfy
µ(AXB)(x,y)=min{µA(x),µB(y)}

Example:

A(x)={(x1,0.2),(x2,0.3),(x3,0.5),(x4,0.6)}
B(y)={(y1,0.8),(y2,0.6),(y3,0.3)}

min{µA(x1),µB(y1)} = min{0.2,0.8} = 0.2 min{µA(x1),µB(y2)} = min{0.2,0.6} = 0.2

min{µA(x1),µB(y3)} = min{0.2,0.3} = 0.2

min{µA(x2),µB(y1)} = min{0.3,0.8} = 0.3 min{µA(x2),µB(y2)} = min{0.3,0.6} = 0.3

min{µA(x2),µB(y3)} = min{0.3,0.3} = 0.3

min{µA(x3),µB(y1)} = min{0.5,0.8} = 0.5 min{µA(x3),µB(y2)} = min{0.5,0.6} = 0.5

min{µA(x3),µB(y3)} = min{0.5,0.3} = 0.3

min{µA(x4),µB(y1)} = min{0.6,0.8} = 0.6 min{µA(x4),µB(y2)} = min{0.6,0.6} = 0.6

min{µA(x4),µB(y3)} = min{0.6,0.3} = 0.3

0.2 0.2 0.2

 0.3 0.2 0.3
AXB=[ ]
0.5 0.5 0.3
0.6 0.6 0.3
Fuzzy relations

Fuzzy relation is a fuzzy set defined on the Cartesian product of crisp set X1, X2, ..., Xn Here, n-
tuples (x1, x2, ..., xn) may have varying degree of memberships within the relationship.

The membership values indicate the strength of the relation between the tuples.

Example: X = { typhoid, viral, cold } and Y = { running nose, high temp, shivering }

The fuzzy relation R is defined as

Suppose A is a fuzzy set on the universe of discourse X with µA(x)|x ∈ X B is a fuzzy set on the
universe of discourse Y with µB(y)|y ∈ Y Then R = A × B ⊂ X × Y ;

where R has its membership function given by

µR(x, y) = µA×B(x, y) = min{µA(x), µB(y)}

Example : A = {(a1, 0.2),(a2, 0.7),(a3, 0.4)}and B = {(b1, 0.5),(b2, 0.6)}

Operations on Fuzzy relations

Let R and S be two fuzzy relations on A × B.

Union: µR∪S(a, b) = max{µR(a, b), µS(a, b)}

Intersection: µR∩S(a, b) = min{µR(a, b), µS(a, b)}

Complement: µR (a, b) = 1 − µR(a, b)

Composition T = R ◦ S µR◦S = max y∈Y {min(µR(x, y), µS(y, z))}

Fuzzy relation : An example

Consider the following two sets P and D, which represent a set of paddy plants and a set of plant
diseases. More precisely P = {P1, P2, P3, P4} a set of four varieties of paddy plants D = {D1, D2,
D3, D4} of the four various diseases affecting the plants In addition to these, also consider another
set S = {S1, S2, S3, S4} be the common symptoms of the diseases. Let, R be a relation on P × D,
representing which plant is susceptible to which diseases, then R can be stated as

Also, consider T be the another relation on D × S, which is given by

Obtain the association of plants with the different symptoms of the disease using max-min
composition.

Hint: Find R ◦ T, and verify that

Fuzzy rule-based models and linguistic variables

Fuzzy rule-based models and linguistic variables are core components of fuzzy logic systems,
enabling the representation and manipulation of imprecise or uncertain information. Linguistic
variables represent concepts using words or phrases rather than precise numerical values, while fuzzy
rules combine these variables to express relationships and make inferences.
:Representation of Uncertainty:
Fuzzy rule-based models are designed to handle situations where information is vague, ambiguous,
or incomplete.

Fuzzy Rules:

These models use fuzzy rules, which are IF-THEN statements that incorporate fuzzy sets and
linguistic variables. For example, a rule might be: "IF temperature is high AND humidity is low
THEN air conditioning is high."

Knowledge Representation:

Fuzzy rules provide a way to encode expert knowledge and human reasoning into a system.

Inference Engine:

Fuzzy inference systems use a fuzzy inference engine to process the fuzzy rules and input data to
derive conclusions.

Fuzzification and Defuzzification:

Fuzzification converts crisp (numerical) inputs into fuzzy sets, while defuzzification converts fuzzy
outputs back into crisp values.

Linguistic Variables:

Words as Values:

Linguistic variables represent concepts using words or phrases from a natural language, rather than
numerical values.

Fuzzy Sets:

Linguistic variables are associated with fuzzy sets, which define the degree to which a particular
value belongs to a linguistic term. For example, a linguistic variable "temperature" might have fuzzy
sets like "cold," "warm," and "hot," each with a corresponding membership function.

Universe of Discourse:

The universe of discourse defines the range of possible values for a linguistic variable.

Example:

The linguistic variable "speed" might have a universe of discourse from 0 to 220 km/h and fuzzy sets
like "very slow," "slow," "medium," "fast," and "very fast."
Fuzzy controls:

Fuzzy control is a control system design approach that utilizes fuzzy logic, a form of many-valued
logic, to handle imprecise or uncertain information in control systems. It allows for the
implementation of human expertise in controller design by using linguistic terms and fuzzy rules,
rather than relying solely on precise mathematical models and crisp values. The diagram of Fuzzy
Logic Control System is as given below.
Fuzzy Logic:

Unlike traditional (crisp) logic which uses binary values (0 or 1), fuzzy logic allows for degrees of
truth, represented by values between 0 and 1.

Fuzzy Sets:

Fuzzy sets are used to represent linguistic terms like "hot," "cold," or "medium." Each element in a
fuzzy set has a membership value between 0 and 1, indicating the degree to which it belongs to that
set.

Fuzzy Rules:

Fuzzy control systems use "if-then" rules to describe control actions. For example, "If temperature is
hot, then reduce the heating power slightly."

Fuzzy Inference:

This process involves evaluating the fuzzy rules based on the input values and determining the
appropriate control action. It often involves techniques like fuzzification (converting crisp inputs to
fuzzy sets), inference (applying fuzzy rules), and defuzzification (converting fuzzy output to a crisp
control signal).

How it Works:

1. Input Measurement:

The system measures the relevant parameters of the process to be controlled (e.g., temperature,
pressure, speed).

2. Fuzzification:

The measured values are converted into fuzzy sets using membership functions.

3. Rule Evaluation:

The fuzzy rules in the knowledge base are evaluated based on the fuzzified input values.

4. Inference:
The results of the rule evaluation are combined using fuzzy inference methods (e.g., Mamdani or
Sugeno).

5. Defuzzification:

The resulting fuzzy output is converted back into a crisp control signal that can be used to adjust the
process.

Advantages of Fuzzy Control:

Handles Uncertainty and Imprecision:

Fuzzy logic is well-suited for systems with uncertain or vaguely defined characteristics.

Human-like Reasoning:

Fuzzy control can mimic human decision-making processes, making it easier to design controllers for
complex systems.

Non-linear Control:

Fuzzy logic can handle non-linearities in the system, which can be difficult to model with traditional
control methods.

Simple Implementation:

Fuzzy control systems can be relatively simple to implement, especially for complex systems where
traditional modeling is difficult.

Applications:

Fuzzy control is used in a wide range of applications, including:

Industrial process control

Consumer electronics

Automatic train operation

Traffic control

Power electronic systems

Biomedical instrumentation

Security systems

Fuzzy Decision Making:

Fuzzy decision-making is a process that deals with situations where information is imprecise, vague,
or uncertain. It uses fuzzy logic, which allows for degrees of truth rather than absolute true or false
values, to model and analyze complex decision problems. This approach is particularly useful when
dealing with human subjectivity and preferences in decision-making processes.

Fuzzy Decision Making:

This involves applying fuzzy logic and fuzzy sets to analyze decision problems, evaluate alternatives,
and make choices in situations where information is imprecise or uncertain.

How it Works:

1. Define Fuzzy Sets:

Identify the relevant variables and define fuzzy sets to represent their ranges of values (e.g., "high
temperature," "low risk," "good quality").

2. Formulate Fuzzy Rules:

Create rules that link the fuzzy sets to the desired outcomes or decisions. These rules are often
expressed in an "if-then" format (e.g., "if temperature is high and risk is low, then the decision is
safe").

3. Evaluate Alternatives:

Input the available information into the fuzzy system, which then uses the fuzzy rules to assess the
degree to which each alternative satisfies the defined goals and constraints.

4. Make a Decision:

Based on the evaluation, the system can suggest the most suitable alternative or provide a ranking of
alternatives.

Applications:

Fuzzy decision-making is used in various fields, including:

Expert Systems:

Building systems that can make decisions based on expert knowledge, even when that knowledge is
imprecise or uncertain.

Control Systems:

Designing controllers for complex systems, such as industrial processes or robots, where precise
mathematical models are difficult to obtain.

Decision Support Systems:

Developing tools that help decision-makers analyze complex situations and make better choices.

Supply Chain Management:

Optimizing supplier selection and other decisions related to the flow of goods and information.

Engineering Design:

Evaluating design alternatives and selecting the best option based on multiple criteria.

Medical Diagnosis:

Aiding in the diagnosis of diseases by providing early detection and preventing the progression of
complex conditions.