What is knowledge representation?

Humans are best at understanding, reasoning, and interpreting knowledge. Human knows
things, which is knowledge and as per their knowledge they perform various actions in
the real world. But how machines do all these things comes under knowledge
representation and reasoning. Hence we can describe Knowledge representation as
following:

o Knowledge representation is representing information about the real world so that

a computer can understand and can utilize this knowledge to solve the complex
real world problems such as diagnosis a medical condition or communicating with
humans in natural language.
o It is also a way which describes how we can represent knowledge in artificial
intelligence. Knowledge representation is not just storing data into some database,
but it also enables an intelligent machine to learn from that knowledge and
experiences so that it can behave intelligently like a human.

What to Represent:
Following are the kind of knowledge which needs to be represented in AI systems:

o Object: All the facts about objects in our world domain. E.g., Guitars contains
strings, trumpets are brass instruments.
o Events: Events are the actions which occur in our world.
o Performance: It describe behavior which involves knowledge about how to do
things.
o Meta-knowledge: It is knowledge about what we know.
o Facts: Facts are the truths about the real world and what we represent.
o Knowledge-Base: The central component of the knowledge-based agents is the
knowledge base. It is represented as KB. The Knowledgebase is a group of the
Sentences (Here, sentences are used as a technical term and not identical with the
English language).

Knowledge: Knowledge is awareness or familiarity gained by experiences of facts, data,

and situations. Following are the types of knowledge in artificial intelligence:
Types of knowledge
Following are the various types of knowledge:

1. Declarative Knowledge:

o Declarative knowledge is to know about something.

o It includes concepts, facts, and objects.
o It is also called descriptive knowledge and expressed in declarativesentences.
o It is simpler than procedural language.

2. Procedural Knowledge

o It is also known as imperative knowledge.

o Procedural knowledge is a type of knowledge which is responsible for knowing
how to do something.
o It can be directly applied to any task.
o It includes rules, strategies, procedures, agendas, etc.
 o Procedural knowledge depends on the task on which it can be applied.

3. Meta-knowledge:

o Knowledge about the other types of knowledge is called Meta-knowledge.

4. Heuristic knowledge:

o Heuristic knowledge is representing knowledge of some experts in a filed or

subject.
o Heuristic knowledge is rules of thumb based on previous experiences, awareness
of approaches, and which are good to work but not guaranteed.

5. Structural knowledge:

o Structural knowledge is basic knowledge to problem-solving.

o It describes relationships between various concepts such as kind of, part of, and
grouping of something.
o It describes the relationship that exists between concepts or objects.

The relation between knowledge and intelligence:

Knowledge of real-worlds plays a vital role in intelligence and same for creating artificial
intelligence. Knowledge plays an important role in demonstrating intelligent behavior in
AI agents. An agent is only able to accurately act on some input when he has some
knowledge or experience about that input.

Let's suppose if you met some person who is speaking in a language which you don't
know, then how you will able to act on that. The same thing applies to the intelligent
behavior of the agents.

As we can see in below diagram, there is one decision maker which act by sensing the
environment and using knowledge. But if the knowledge part will not present then, it
cannot display intelligent behavior.
AI knowledge cycle:
An Artificial intelligence system has the following components for displaying intelligent
behavior:

o Perception
o Learning
o Knowledge Representation and Reasoning
o Planning
o Execution
The above diagram is showing how an AI system can interact with the real world and what
components help it to show intelligence. AI system has Perception component by which
it retrieves information from its environment. It can be visual, audio or another form of
sensory input. The learning component is responsible for learning from data captured by
Perception comportment. In the complete cycle, the main components are knowledge
representation and Reasoning. These two components are involved in showing the
intelligence in machine-like humans. These two components are independent with each
other but also coupled together. The planning and execution depend on analysis of
Knowledge representation and reasoning.

Approaches to knowledge representation:

There are mainly four approaches to knowledge representation, which are givenbelow:

1. Simple relational knowledge:

o It is the simplest way of storing facts which uses the relational method, and each
fact about a set of the object is set out systematically in columns.
o This approach of knowledge representation is famous in database systems where
the relationship between different entities is represented.
o This approach has little opportunity for inference.

Example: The following is the simple relational knowledge representation.

 Player Weight Age

Player1 65 23

Player2 58 18

Player3 75 24

2. Inheritable knowledge:

o In the inheritable knowledge approach, all data must be stored into a hierarchy of
classes.
o All classes should be arranged in a generalized form or a hierarchal manner.
o In this approach, we apply inheritance property.
o Elements inherit values from other members of a class.
o This approach contains inheritable knowledge which shows a relation between
instance and class, and it is called instance relation.
o Every individual frame can represent the collection of attributes and its value.
o In this approach, objects and values are represented in Boxed nodes.
o We use Arrows which point from objects to their values.
o Example:
3. Inferential knowledge:

o Inferential knowledge approach represents knowledge in the form of formal logics.

o This approach can be used to derive more facts.
o It guaranteed correctness.
o Example: Let's suppose there are two statements:
a. Marcus is a man
b. All men are mortal
Then it can represent as;

man(Marcus)
∀x = man (x) ----------> mortal (x)s

4. Procedural knowledge:

o Procedural knowledge approach uses small programs and codes which describes
how to do specific things, and how to proceed.
o In this approach, one important rule is used which is If-Then rule.
 o In this knowledge, we can use various coding languages such as LISP
language and Prolog language.
o We can easily represent heuristic or domain-specific knowledge using this
approach.
o But it is not necessary that we can represent all cases in this approach.

Requirements for knowledge Representation system:

A good knowledge representation system must possess the following properties.

1. 1. Representational Accuracy:
KR system should have the ability to represent all kind of required knowledge.
2. 2. Inferential Adequacy:
KR system should have ability to manipulate the representational structures to
produce new knowledge corresponding to existing structure.
3. 3. Inferential Efficiency:
The ability to direct the inferential knowledge mechanism into the most productive
directions by storing appropriate guides.
4. 4. Acquisitional efficiency- The ability to acquire the new knowledge easily using
automatic methods.

Techniques of knowledge representation

There are mainly four ways of knowledge representation which are given as follows:

1. Logical Representation
2. Semantic Network Representation
3. Frame Representation
4. Production Rules
1. Logical Representation
Logical representation is a language with some concrete rules which deals with
propositions and has no ambiguity in representation. Logical representation means
drawing a conclusion based on various conditions. This representation lays down some
important communication rules. It consists of precisely defined syntax and semantics
which supports the sound inference. Each sentence can be translated into logics using
syntax and semantics.

Syntax:

o Syntaxes are the rules which decide how we can construct legal sentences in the logic.
o It determines which symbol we can use in knowledge representation.
o How to write those symbols.

Semantics:

o Semantics are the rules by which we can interpret the sentence in the logic.
o Semantic also involves assigning a meaning to each sentence.

Logical representation can be categorised into mainly two logics:

a. Propositional Logics
b. Predicate logics

Note: We will discuss Prepositional Logics and Predicate logics in later chapters.

Advantages of logical representation:

1. Logical representation enables us to do logical reasoning.

2. Logical representation is the basis for the programming languages.

Disadvantages of logical Representation:

1. Logical representations have some restrictions and are challenging to work with.
2. Logical representation technique may not be very natural, and inference may not be so
efficient.

2. Semantic Network Representation

Semantic networks are alternative of predicate logic for knowledge representation. In
Semantic networks, we can represent our knowledge in the form of graphical networks.
This network consists of nodes representing objects and arcs which describe the
relationship between those objects. Semantic networks can categorize the object in
different forms and can also link those objects. Semantic networks are easy to understand
and can be easily extended.

This representation consist of mainly two types of relations:

a. IS-A relation (Inheritance)

b. Kind-of-relation

Example: Following are some statements which we need to represent in the form of
nodes and arcs.

Statements:

a. Jerry is a cat.
b. Jerry is a mammal
c. Jerry is owned by Priya.
 d. Jerry is brown colored.
e. All Mammals are animal.

In the above diagram, we have represented the different type of knowledge in the form
of nodes and arcs. Each object is connected with another object by some relation.

Drawbacks in Semantic representation:

1. Semantic networks take more computational time at runtime as we need to traverse the
complete network tree to answer some questions. It might be possible in the worst case
scenario that after traversing the entire tree, we find that the solution does not exist in this
network.
2. Semantic networks try to model human-like memory (Which has 1015 neurons and links)
to store the information, but in practice, it is not possible to build such a vast semantic
network.
3. These types of representations are inadequate as they do not have any equivalent
quantifier, e.g., for all, for some, none, etc.
4. Semantic networks do not have any standard definition for the link names.
5. These networks are not intelligent and depend on the creator of the system.

Advantages of Semantic network:

 1. Semantic networks are a natural representation of knowledge.
2. Semantic networks convey meaning in a transparent manner.
3. These networks are simple and easily understandable.

3. Frame Representation
A frame is a record like structure which consists of a collection of attributes and its values
to describe an entity in the world. Frames are the AI data structure which divides
knowledge into substructures by representing stereotypes situations. It consists of a
collection of slots and slot values. These slots may be of any type and sizes. Slots have
names and values which are called facets.

Facets: The various aspects of a slot is known as Facets. Facets are features of frames
which enable us to put constraints on the frames. Example: IF-NEEDED facts are called
when data of any particular slot is needed. A frame may consist of any number of slots,
and a slot may include any number of facets and facets may have any number of values.
A frame is also known as slot-filter knowledge representation in artificial intelligence.

Frames are derived from semantic networks and later evolved into our modern-day
classes and objects. A single frame is not much useful. Frames system consist of a
collection of frames which are connected. In the frame, knowledge about an object or
event can be stored together in the knowledge base. The frame is a type of technology
which is widely used in various applications including Natural language processing and
machine visions.

Example: 1
Let's take an example of a frame for a book

Slots Filters

Title Artificial Intelligence

Genre Computer Science

Author Peter Norvig

 Edition Third Edition

Year 1996

Page 1152

Example 2:
Let's suppose we are taking an entity, Peter. Peter is an engineer as a profession, and his
age is 25, he lives in city London, and the country is England. So following is the frame
representation for this:

Slots Filter

Name Peter

Profession Doctor

Age 25

Marital status Single

Weight 78

Advantages of frame representation:

1. The frame knowledge representation makes the programming easier by grouping the
related data.
2. The frame representation is comparably flexible and used by many applications in AI.
3. It is very easy to add slots for new attribute and relations.
4. It is easy to include default data and to search for missing values.
5. Frame representation is easy to understand and visualize.
Disadvantages of frame representation:

1. In frame system inference mechanism is not be easily processed.

2. Inference mechanism cannot be smoothly proceeded by frame representation.
3. Frame representation has a much generalized approach.

4. Production Rules
Production rules system consist of (condition, action) pairs which mean, "If condition
then action". It has mainly three parts:

o The set of production rules

o Working Memory
o The recognize-act-cycle

In production rules agent checks for the condition and if the condition exists then
production rule fires and corresponding action is carried out. The condition part of the
rule determines which rule may be applied to a problem. And the action part carries out
the associated problem-solving steps. This complete process is called a recognize-act
cycle.

The working memory contains the description of the current state of problems-solving
and rule can write knowledge to the working memory. This knowledge match and may
fire other rules.

If there is a new situation (state) generates, then multiple production rules will be fired
together, this is called conflict set. In this situation, the agent needs to select a rule from
these sets, and it is called a conflict resolution.

Example:

o IF (at bus stop AND bus arrives) THEN action (get into the bus)
o IF (on the bus AND paid AND empty seat) THEN action (sit down).
o IF (on bus AND unpaid) THEN action (pay charges).
o IF (bus arrives at destination) THEN action (get down from the bus).

Advantages of Production rule:

 1. The production rules are expressed in natural language.
2. The production rules are highly modular, so we can easily remove, add or modify an
individual rule.

Disadvantages of Production rule:

1. Production rule system does not exhibit any learning capabilities, as it does not store the
result of the problem for the future uses.
2. During the execution of the program, many rules may be active hence rule-based
production systems are inefficient.

Propositional logic in Artificial intelligence

Propositional logic (PL) is the simplest form of logic where all the statements are made by
propositions. A proposition is a declarative statement which is either true or false. It is a
technique of knowledge representation in logical and mathematical form.

Example:

1. a) It is Sunday.
2. b) The Sun rises from West (False proposition)
3. c) 3+3= 7(False proposition)
4. d) 5 is a prime number.

Following are some basic facts about propositional logic:

o Propositional logic is also called Boolean logic as it works on 0 and 1.

o In propositional logic, we use symbolic variables to represent the logic, and we can use
any symbol for a representing a proposition, such A, B, C, P, Q, R, etc.
o Propositions can be either true or false, but it cannot be both.
o Propositional logic consists of an object, relations or function, and logical connectives.
o These connectives are also called logical operators.
o The propositions and connectives are the basic elements of the propositional logic.
o Connectives can be said as a logical operator which connects two sentences.
 o A proposition formula which is always true is called tautology, and it is also called a valid
sentence.
o A proposition formula which is always false is called Contradiction.
o A proposition formula which has both true and false values is called
o Statements which are questions, commands, or opinions are not propositions such as
"Where is Rohini", "How are you", "What is your name", are not propositions.

Syntax of propositional logic:

The syntax of propositional logic defines the allowable sentences for the knowledge
representation. There are two types of Propositions:

a. Atomic Propositions
b. Compound propositions

o Atomic Proposition: Atomic propositions are the simple propositions. It consists of a

single proposition symbol. These are the sentences which must be either true or false.

Example:

1. a) 2+2 is 4, it is an atomic proposition as it is a true fact.

2. b) "The Sun is cold" is also a proposition as it is a false fact.
o Compound proposition: Compound propositions are constructed by combining simpler
or atomic propositions, using parenthesis and logical connectives.

Example:

1. a) "It is raining today, and street is wet."

2. b) "Ankit is a doctor, and his clinic is in Mumbai."

Logical Connectives:
Logical connectives are used to connect two simpler propositions or representing a
sentence logically. We can create compound propositions with the help of logical
connectives. There are mainly five connectives, which are given as follows:
 1. Negation: A sentence such as ¬ P is called negation of P. A literal can be either Positive
literal or negative literal.
2. Conjunction: A sentence which has ∧ connective such as, P ∧ Q is called a conjunction.
Example: Rohan is intelligent and hardworking. It can be written as,
P= Rohan is intelligent,
Q= Rohan is hardworking. → P∧ Q.
3. Disjunction: A sentence which has ∨ connective, such as P ∨ Q. is called disjunction,
where P and Q are the propositions.
Example: "Ritika is a doctor or Engineer",
Here P= Ritika is Doctor. Q= Ritika is Doctor, so we can write it as P ∨ Q.
4. Implication: A sentence such as P → Q, is called an implication. Implications are also
known as if-then rules. It can be represented as
If it is raining, then the street is wet.
Let P= It is raining, and Q= Street is wet, so it is represented as P → Q
5. Biconditional: A sentence such as P⇔ Q is a Biconditional sentence, example If I am
breathing, then I am alive
P= I am breathing, Q= I am alive, it can be represented as P ⇔ Q.

Following is the summarized table for Propositional Logic

Connectives:

Truth Table:
In propositional logic, we need to know the truth values of propositions in all possible
scenarios. We can combine all the possible combination with logical connectives, and the
representation of these combinations in a tabular format is called Truth table. Following
are the truth table for all logical connectives:
Truth table with three propositions:
We can build a proposition composing three propositions P, Q, and R. This truth table is
made-up of 8n Tuples as we have taken three proposition symbols.

Precedence of connectives:
Just like arithmetic operators, there is a precedence order for propositional connectors or
logical operators. This order should be followed while evaluating a propositional problem.
Following is the list of the precedence order for operators:

Precedence Operators

First Precedence Parenthesis

Second Precedence Negation

Third Precedence Conjunction(AND)

Fourth Precedence Disjunction(OR)

Fifth Precedence Implication

Six Precedence Biconditional

Note: For better understanding use parenthesis to make sure of the correct
interpretations. Such as ¬R∨ Q, It can be interpreted as (¬R) ∨ Q.

Logical equivalence:
Logical equivalence is one of the features of propositional logic. Two propositions are said
to be logically equivalent if and only if the columns in the truth table are identical to each
other.

Let's take two propositions A and B, so for logical equivalence, we can write it as A⇔B. In
below truth table we can see that column for ¬A∨ B and A→B, are identical hence A is
Equivalent to B

Properties of Operators:

o Commutativity:
o P∧ Q= Q ∧ P, or
o P ∨ Q = Q ∨ P.
o Associativity:
o (P ∧ Q) ∧ R= P ∧ (Q ∧ R),
o (P ∨ Q) ∨ R= P ∨ (Q ∨ R)
o Identity element:
o P ∧ True = P,
o P ∨ True= True.
o Distributive:
o P∧ (Q ∨ R) = (P ∧ Q) ∨ (P ∧ R).
o P ∨ (Q ∧ R) = (P ∨ Q) ∧ (P ∨ R).
o DE Morgan's Law:
o ¬ (P ∧ Q) = (¬P) ∨ (¬Q)
o ¬ (P ∨ Q) = (¬ P) ∧ (¬Q).
o Double-negation elimination:
o ¬ (¬P) = P.
Limitations of Propositional logic:

o We cannot represent relations like ALL, some, or none with propositional logic. Example:
a. All the girls are intelligent.
b. Some apples are sweet.
o Propositional logic has limited expressive power.
o In propositional logic, we cannot describe statements in terms of their properties or logical
relationships.

Rules of Inference in Artificial intelligence

Inference:
In artificial intelligence, we need intelligent computers which can create new logic from
old logic or by evidence, so generating the conclusions from evidence and facts is
termed as Inference.

Inference rules:
Inference rules are the templates for generating valid arguments. Inference rules are
applied to derive proofs in artificial intelligence, and the proof is a sequence of the
conclusion that leads to the desired goal.

In inference rules, the implication among all the connectives plays an important role.
Following are some terminologies related to inference rules:

o Implication: It is one of the logical connectives which can be represented as P →

Q. It is a Boolean expression.
o Converse: The converse of implication, which means the right-hand side
proposition goes to the left-hand side and vice-versa. It can be written as Q → P.
o Contrapositive: The negation of converse is termed as contrapositive, and it can
be represented as ¬ Q → ¬ P.
o Inverse: The negation of implication is called inverse. It can be represented as ¬ P
→ ¬ Q.
From the above term some of the compound statements are equivalent to each other,
which we can prove using truth table:

Hence from the above truth table, we can prove that P → Q is equivalent to ¬ Q → ¬ P,
and Q→ P is equivalent to ¬ P → ¬ Q.

Types of Inference rules:

1. Modus Ponens:
The Modus Ponens rule is one of the most important rules of inference, and it states that
if P and P → Q is true, then we can infer that Q will be true. It can be represented as:

Example:

Statement-1: "If I am sleepy then I go to bed" ==> P→ Q

Statement-2: "I am sleepy" ==> P
Conclusion: "I go to bed." ==> Q.
Hence, we can say that, if P→ Q is true and P is true then Q will be true.

Proof by Truth table:

2. Modus Tollens:
The Modus Tollens rule state that if P→ Q is true and ¬ Q is true, then ¬ P will also true.
It can be represented as:

Statement-1: "If I am sleepy then I go to bed" ==> P→ Q

Statement-2: "I do not go to the bed."==> ~Q
Statement-3: Which infers that "I am not sleepy" => ~P

Proof by Truth table:

First-Order Logic in Artificial intelligence

In the topic of Propositional logic, we have seen that how to represent statements using
propositional logic. But unfortunately, in propositional logic, we can only represent the
facts, which are either true or false. PL is not sufficient to represent the complex sentences
or natural language statements. The propositional logic has very limited expressive power.
Consider the following sentence, which we cannot represent using PL logic.

o "Some humans are intelligent", or

o "Sachin likes cricket."

To represent the above statements, PL logic is not sufficient, so we required some more
powerful logic, such as first-order logic.

First-Order logic:
o First-order logic is another way of knowledge representation in artificial intelligence. It is
an extension to propositional logic.
 o FOL is sufficiently expressive to represent the natural language statements in a concise
way.
o First-order logic is also known as Predicate logic or First-order predicate logic. First-
order logic is a powerful language that develops information about the objects in a more
easy way and can also express the relationship between those objects.
o First-order logic (like natural language) does not only assume that the world contains facts
like propositional logic but also assumes the following things in the world:
o Objects: A, B, people, numbers, colors, wars, theories, squares, pits, wumpus, ......
o Relations: It can be unary relation such as: red, round, is adjacent, or n-any
relation such as: the sister of, brother of, has color, comes between
o Function: Father of, best friend, third inning of, end of, ......
o As a natural language, first-order logic also has two main parts:
a. Syntax
b. Semantics

Syntax of First-Order logic:

The syntax of FOL determines which collection of symbols is a logical expression in first-
order logic. The basic syntactic elements of first-order logic are symbols. We write
statements in short-hand notation in FOL.

Basic Elements of First-order logic:

Following are the basic elements of FOL syntax:

Constant 1, 2, A, John, Mumbai, cat,....

Variables x, y, z, a, b,....

Predicates Brother, Father, >,....

Function sqrt, LeftLegOf, ....

Connectives ∧, ∨, ¬, ⇒, ⇔
 Equality ==

Quantifier ∀, ∃

Atomic sentences:

o Atomic sentences are the most basic sentences of first-order logic. These sentences are
formed from a predicate symbol followed by a parenthesis with a sequence of terms.
o We can represent atomic sentences as Predicate (term1, term2, ......, term n).

Example: Ravi and Ajay are brothers: => Brothers(Ravi, Ajay).

Chinky is a cat: => cat (Chinky).

Complex Sentences:

o Complex sentences are made by combining atomic sentences using connectives.

First-order logic statements can be divided into two parts:

o Subject: Subject is the main part of the statement.

o Predicate: A predicate can be defined as a relation, which binds two atoms together in a
statement.

Consider the statement: "x is an integer.", it consists of two parts, the first part x is the
subject of the statement and second part "is an integer," is known as a predicate.

Quantifiers in First-order logic:

o A quantifier is a language element which generates quantification, and quantification
specifies the quantity of specimen in the universe of discourse.
 o These are the symbols that permit to determine or identify the range and scope of the
variable in the logical expression. There are two types of quantifier:
a. Universal Quantifier, (for all, everyone, everything)
b. Existential quantifier, (for some, at least one).

Universal Quantifier:
Universal quantifier is a symbol of logical representation, which specifies that the
statement within its range is true for everything or every instance of a particular thing.

The Universal quantifier is represented by a symbol ∀, which resembles an inverted A.

Note: In universal quantifier we use implication "→".

If x is a variable, then ∀x is read as:

o For all x
o For each x
o For every x.

Example:
All man drink coffee.

Let a variable x which refers to a cat so all x can be represented in UOD as below:
∀x man(x) → drink (x, coffee).

It will be read as: There are all x where x is a man who drink coffee.

Existential Quantifier:
Existential quantifiers are the type of quantifiers, which express that the statement within
its scope is true for at least one instance of something.

It is denoted by the logical operator ∃, which resembles as inverted E. When it is used with
a predicate variable then it is called as an existential quantifier.

Note: In Existential quantifier we always use AND or Conjunction symbol (∧).

If x is a variable, then existential quantifier will be ∃x or ∃(x). And it will be read as:

o There exists a 'x.'

o For some 'x.'
o For at least one 'x.'

Example:
Some boys are intelligent.

∃x: boys(x) ∧ intelligent(x)

It will be read as: There are some x where x is a boy who is intelligent.

Points to remember:
o The main connective for universal quantifier ∀ is implication →.
o The main connective for existential quantifier ∃ is and ∧.

Properties of Quantifiers:
o In universal quantifier, ∀x∀y is similar to ∀y∀x.
o In Existential quantifier, ∃x∃y is similar to ∃y∃x.
o ∃x∀y is not similar to ∀y∃x.

Some Examples of FOL using quantifier:

1. All birds fly.

In this question the predicate is "fly(bird)."
And since there are all birds who fly so it will be represented as follows.
∀x bird(x) →fly(x).

2. Every man respects his parent.

In this question, the predicate is "respect(x, y)," where x=man, and y= parent.
Since there is every man so will use ∀, and it will be represented as follows:
∀x man(x) → respects (x, parent).

3. Some boys play cricket.

In this question, the predicate is "play(x, y)," where x= boys, and y= game. Since there
are some boys so we will use ∃, and it will be represented as:
∃x boys(x) ∧ play(x, cricket).

4. Not all students like both Mathematics and Science.

In this question, the predicate is "like(x, y)," where x= student, and y= subject.
Since there are not all students, so we will use ∀ with negation, so following
representation for this:
¬∀ (x) [ student(x) → like(x, Mathematics) ∧ like(x, Science)].

5. Only one student failed in Mathematics.

In this question, the predicate is "failed(x, y)," where x= student, and y= subject.
Since there is only one student who failed in Mathematics, so we will use following
representation for this:
∃(x) [ student(x) → failed (x, Mathematics) ∧∀ (y) [¬(x==y) ∧ student(y) →
¬failed (x, Mathematics)].

Free and Bound Variables:

The quantifiers interact with variables which appear in a suitable way. There are two types
of variables in First-order logic which are given below:

Free Variable: A variable is said to be a free variable in a formula if it occurs outside the
scope of the quantifier.

Example: ∀x ∃(y)[P (x, y, z)], where z is a free variable.

Bound Variable: A variable is said to be a bound variable in a formula if it occurs within

the scope of the quantifier.

Example: ∀x [A (x) B( y)], here x and y are the bound variables.