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LECTURE NOTES

ON

MATHEMATICAL FOUNDATIONS OF COMPUTER SCIENCE

MCA I YEAR I SEMESTER (JNTUA-R09)

Mrs.B.Vijaya

Asst.Professor

DEPARTMENT OF MASTER OF COMPUTER APPLICATIONS

CHADALAWADA RAMANAMMA ENGINEERING COLLEGE CHADALAWADA NAGAR, RENIGUNTA ROAD, TIRUPATI (A.P) – 517506

Dept.of MCA,CREC

JAWAHARLAL NEHRU TECHNOLOGICAL UNIVERSITY ANANTAPUR

MCA Semester-I T

4

(9F00104) MATHEMATICAL FOUNDATIONS OF COMPUTER SCIENCE

UNIT-I

Mathematical Logic: Statements and notations, Connectives, Well formed formulas, Truth

Tables, tautology, equivalence implication, Normal forms, Quantifiers, universal quantifiers

UNIT-II

Predicates: Predicative logic, Free & Bound variables, Rules of inference, Consistency, proof of

contradiction, Automatic Theorem Proving.

UNIT-III

Relations: Properties of Binary Relations, equivalence, transitive closure, compatibility and

partial ordering relations, Lattices, Hasse diagram. Functions: Inverse Function, Composition of

functions, recursive Functions, Lattice and its Properties,

UNIT-IV

Algebraic structures: Algebraic systems examples and general properties, Semi groups and

monads, groups, sub groups homomorphism, Isomorphism.

UNIT-V

Elementary Combinatorics: Basis of counting, Enumerating Combinations & Permutations,

with repetitions, Constrained repetitions, Binomial Coefficients, Binomial Multinomial

theorems, the principles of Inclusion – Exclusion. Pigeon hole principles and its application

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UNIT-VI

Recurrence Relation: Generating Functions & Sequences , Calculating Coefficient of

generating function, Recurrence relations, Solving Recurrence relation by substitution and

Generating functions. Characteristic roots, solution of Inhomogeneous Recurrence Relation.

UNIT-VII

Graph Theory: Representation of Graph, DFS, BFS, Spanning Trees, planar Graphs

UNIT-VIII

Graph Theory Applications: Basic Concepts Isomorphism and Sub graphs, Multi graphs and

Euler circuits, Hamiltonian graphs, Chromatic Numbers

REFERENCE:

1. Elements of Discrete Mathematics- A Computer Oriented Approach,C.L.Liu,

D.P. Mohapatra, 3/e,TMH.

2. Discrete Mathematics for Computer Scientists & Mathematicians, 2/e,

J.L.Mott, A. Kandel, T.P. Baker, PHI

3. Discrete Mathematical Structures with Application to Computer Science, Tremblay,

Manohar McGraw Hill Publication

4. Discrete and Combinatorial Mathematics- An Applied Introduction,Ralph.

P.Grimaldi, 5/e,Pearson Education.

5. Discrete Mathematics and its applications, 6th edition, K.H.Rosen, TMH.

6. Discrete Mathematical Structures, Mallik and Sen, Cengage Learning.

7. Discrete Mathematical Structures, Bernand Kolman, Robert C. Busby, Sharon Cutler

Ross, PHI/ Pearson Education.

8. Discrete Mathematics with Applications,Thomas Koshy,Elsevier.

9. Discrete Mathematics, Lovasz, Springer.

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UNIT-I

Mathematical Logics

Propositional Logic

The rules of mathematical logic specify methods of reasoning mathematical statements. Greek

philosopher, Aristotle, was the pioneer of logical reasoning. Logical reasoning provides the

theoretical base for many areas of mathematics and consequently computer science. It has many

practical applications in computer science like design of computing machines, artificial

intelligence, definition of data structures for programming languages etc.

Propositional Logic is concerned with statements to which the truth values, “true” and “false”,

can be assigned. The purpose is to analyze these statements either individually or in a composite

manner.

Propositional Logic – Definition

A proposition is a collection of declarative statements that has either a truth value "true” or a

truth value "false". A proposition consists of propositional variables and connectives. We denote

the propositional variables by capital letters (A, B, etc). The connectives connect the

propositional variables.

Some examples of Propositions are given below −

"Man is Mortal", it returns truth value “TRUE”

"12 + 9 = 3 − 2", it returns truth value “FALSE”

The following is not a Proposition −

"A is less than 2". It is because unless we give a specific value of A, we cannot say whether the

statement is true or false.

Connectives

In propositional logic generally we use five connectives which are − OR (∨), AND (∧),

Negation/ NOT (¬), Implication / if-then (→), If and only if (⇔).

OR (∨) − The OR operation of two propositions A and B (written as A ∨ B) is true if at least any

of the propositional variable A or B is true.

The truth table is as follows −

A B A ∨ B

True True True

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True False True

False True True

False False False

AND (∧)− The AND operation of two propositions A and B (written as A ∧ B) is true if both the

propositional variable A and B is true.


A B A ∧ B

True True True

True False False

False True False

False False False

Negation (¬) − The negation of a proposition A (written as ¬A) is false when A is true and is

true when A is false.


A ¬A

True False

False True

Implication / if-then (→) − An implication A→B is False if A is true and B is false. The rest

cases are true.


A B A → B

True True True

True False False

False True True

False False True

If and only if (⇔) − A⇔B is bi-conditional logical connective which is true when p and q are

both false or both are true.


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A B A ⇔ B

True True True

True False False

False True False

False False True

Tautologies

A Tautology is a formula which is always true for every value of its propositional variables.

Example − Prove [(A → B) ∧ A] → B is a tautology


A B A → B (A → B) ∧ A [(A → B) ∧ A] → B

True True True True True

True False False False True

False True True False True

False False True False True

As we can see every value of [(A → B) ∧ A] → B is “True”, it is a tautology.

Contradictions

A Contradiction is a formula which is always false for every value of its propositional variables.

Example − Prove (A ∨ B) ∧ [(¬A) ∧ (¬B)] is a contradiction


A B A ∨ B ¬A ¬B (¬A) ∧ (¬B) (A ∨ B) ∧ [(¬A) ∧ (¬B)]

True True True False False False False

True False True False True False False

False True True True False False False

False False False True True True False

As we can see every value of (A ∨ B) ∧ [(¬A) ∧ (¬B)] is “False”, it is a contradiction.

Contingency

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A Contingency is a formula which has both some true and some false values for every value of

its propositional variables.

Example − Prove (A ∨ B) ∧ (¬A) a contingency


A B A ∨ B ¬A (A ∨ B) ∧ (¬A)

True True True False False

True False True False False

False True True True True

False False False True False

As we can see every value of (A ∨ B) ∧ (¬A) has both “True” and “False”, it is a

contingency.

Well Formed Formula

Well Formed Formula (wff) is a predicate holding any of the following −

All propositional constants and propositional variables are wffs

If x is a variable and Y is a wff, ∀x Y and ∃x Y are also wff

Truth value and false values are wffs

Each atomic formula is a wff

All connectives connecting wffs are wffs

Propositional Equivalences

Two statements X and Y are logically equivalent if any of the following two conditions −

The truth tables of each statement have the same truth values.

The bi-conditional statement X ⇔ Y is a tautology.

Example − Prove ¬(A ∨ B) and [(¬A) ∧ (¬B)] are equivalent

Testing by 1st method (Matching truth table)

A B A ∨ B ¬ (A ∨ B) ¬A ¬B [(¬A) ∧ (¬B)]

True True True False False False False

True False True False False True False

False True True False True False False

False False False True True True True

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Here, we can see the truth values of ¬ (A ∨ B) and [(¬A) ∧ (¬B)] are same, hence the statements

are equivalent.

Testing by 2nd method (Bi-conditionality)

A B ¬ (A ∨ B) [(¬A) ∧ (¬B)] [¬ (A ∨ B)] ⇔ [(¬A) ∧ (¬B)]

True True False False True

True False False False True

False True False False True

False False True True True

As [¬ (A ∨ B)] ⇔ [(¬A) ∧ (¬B)] is a tautology, the statements are equivalent.

Inverse, Converse, and Contra-positive

A conditional statement has two parts − Hypothesis and Conclusion.

Example of Conditional Statement − “If you do your homework, you will not be punished.”

Here, "you do your homework" is the hypothesis and "you will not be punished" is the

conclusion.

Inverse − An inverse of the conditional statement is the negation of both the hypothesis and the

conclusion. If the statement is “If p, then q”, the inverse will be “If not p, then not q”. The

inverse of “If you do your homework, you will not be punished” is “If you do not do your

homework, you will be punished.”

Converse − The converse of the conditional statement is computed by interchanging the

hypothesis and the conclusion. If the statement is “If p, then q”, the inverse will be “If q, then p”.

The converse of "If you do your homework, you will not be punished" is "If you will not be

punished, you do not do your homework”.

Contra-positive − The contra-positive of the conditional is computed by interchanging the

hypothesis and the conclusion of the inverse statement. If the statement is “If p, then q”, the

inverse will be “If not q, then not p”. The Contra-positive of "If you do your homework, you will

not be punished” is "If you will be punished, you do your homework”.

Duality Principle

Duality principle set states that for any true statement, the dual statement obtained by

interchanging unions into intersections (and vice versa) and interchanging Universal set into Null

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set (and vice versa) is also true. If dual of any statement is the statement itself, it is said self-dual

statement.

Example − The dual of (A ∩ B) ∪ C is (A ∪ B) ∩ C

NAND Connective

A NAND is a logic connective that gives a False output only if all its inputs are True, otherwise

it gives True output.

NOR Connective

A B ~ (A.B)

F F T

F T T

T F T

F F F

A B ~(A+B)

0 0 1

0 1 0

1 0 0

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NOR is a logic Connective that gives True output if both the inputs are False, otherwise it gives

low output.

XOR (Exclusive OR) connective

An EX-NOR is a logic connective that gives True output if the inputs are same, otherwise it

gives False output.

X-

NOR (Exclusive NOR)

1 1 0

A B A⊕B

0 0 0

0 1 1

1 0 1

1 1 0

A B A X-NOR B

0 0 1

0 1 0

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An EX-NOR is a logic connective that gives True output if the inputs are same, otherwise it

gives False output.

Boolean algebra is algebra of logic. It deals with variables that can have two discrete values, 0

(False) and 1 (True); and operations that have logical significance. The earliest method of

manipulating symbolic logic was invented by George Boole and subsequently came to be

known as Boolean Algebra.

Boolean algebra has now become an indispensable tool in computer science for its wide

applicability in switching theory, building basic electronic circuits and design of digital

computers.

Boolean Functions

A Boolean function is a special kind of mathematical function f: Xn → X of degree n, where X

= {0, 1} is a Boolean domain and n is a non-negative integer. It describes the way how to derive

Boolean output from Boolean inputs.

Example − Let, F(A, B) = A’B’. This is a function of degree 2 from the set of ordered pairs of

Boolean variables to the set {0, 1} where F(0, 0) = 1, F(0, 1) = 0, F(1, 0) = 0 and F(1, 1) = 0

Boolean Expressions

A Boolean expression always produces a Boolean value. A Boolean expression is composed of

a combination of the Boolean constants (True or False), Boolean variables and logical

connectives. Each Boolean expression represents a Boolean function.

Example − AB’C is a Boolean expression

Boolean Identities

Double Complement Law

~(~A) = A

1 0 0

1 1 1

https://en.wikipedia.org/wiki/Boolean_domain

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Complement Law

A + ~A = 1 (OR Form)

A · ~A = 0 (AND Form)

Idempotent Law

A + A = A (OR Form)

A · A = A (AND Form)

Identity Law

A + 0 = A (OR Form)

A · 1 = A (AND Form)

Dominance Law

A + 1 = 1 (OR Form)

A · 0 = 0 (AND Form)

Commutative Law

A + B = B + A (OR Form)

A· B = B · A (AND Form)

Associative Law

A + (B + C) = (A + B) + C (OR Form)

A· (B · C) = (A · B) · C (AND Form)

Absorption Law

A· (A + B) = A

A + (A · B) = A

Simplification Law

A · (~A + B) = A · B

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A + (~A · B) = A + B

A · B + A · C + ~B · C = A · B + ~B ·C

Distributive Law

A + (B · C) = (A + B) · (A + C)

A · (B + C) = (A · B) + (A · C)

De-Morgan's Law

~(A · B) = ~A + ~B

~(A+ B) = ~A · ~B

Canonical Forms

For a Boolean expression there are two kinds of canonical forms −

The sum of minterms (SOM) form

The product of maxterms (POM) form

A B C Minterm

0 0 0 m0

0 0 1 m1

0 1 0 m2

0 1 1 m3

1 0 0 m4

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The Sum of Minterms (SOM) or Sum of Products (SOP) form

A minterm is a product of all variables taken either in their direct or complemented form. Any

Boolean function can be expressed as a sum of its 1-minterms and the inverse of the function

can be expressed as a sum of its 0-minterms. Hence,

F (list of variables) = ∑ (list of 1-minterm indices)

And F’ (list of variables) = ∑ (list of 0-minterm indices)

Example

Let, F(x, y, z) = x’ y’ z’ + x y’ z + x y z’ + x y z

Or, F(x, y, z) = m0 + m5 + m6 + m7

Hence,

F(x, y, z) = ∑ (0, 5, 6, 7)

Now we will find the complement of F(x, y, z)

F’ (x, y, z) = x’ y z + x’ y’ z + x’ y z’ + x y’ z’

Or, F’(x, y, z) = m3 + m1 + m2 + m4

Hence,

F’(x, y, z) = ∑ (3, 1, 2, 4) = ∑ (1, 2, 3, 4)

The Product of Maxterms (POM) or Product of Sums (POS) form

1 0 1 m5

1 1 0 m6

1 1 1 m7

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A maxterm is addition of all variables taken either in their direct or complemented form. Any

Boolean function can be expressed as a product of its 0-maxterms and the inverse of the

function can be expressed as a product of its 1-maxterms. Hence,

F (list of variables) = π (list of 0-maxterm indices)

and

F’(list of variables) = π (list of 1-maxterm indices).

Example

Let, F(x, y, z) = (x+y+z) • (x+y+z’) • (x+y’+z) • (x’+y+z)

Or, F(x, y, z) = M0 • M1 • M2 • M4

A B C Maxterm

0 0 0 M0

0 0 1 M1

0 1 0 M2

0 1 1 M3

1 0 0 M4

1 0 1 M5

1 1 0 M6

1 1 1 M7

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Hence,

F (x, y, z) = π(0, 1, 2, 4)

F’'(x, y, z) = (x+y’+z’) • (x’+y+z’) • (x’+y’+z) • (x’+y’+z’)

Or, F(x, y, z) = M3 • M5 • M6 • M7

Hence,

F ' (x, y, z) = π(3, 5, 6, 7)

Normal Forms

Let A(P1, P2, P3, …, Pn) be a statement formula where P1, P2, P3, …, Pn are the atomic variables.

If A has truth value T for all possible assignments of the truth values to the variables P1, P2, P3,

…, Pn , then A is said to be a tautology. If A has truth value F, then A is said to be identically

false or a contradiction.

Disjunctive Normal Forms

A product of the variables and their negations in a formula is called an elementary product. A

sum of the variables and their negations is called an elementary sum. That is, a sum of

elementary products is called a disjunctive normal form of the given formula.

Example:

The disjunctive normal form of

1. P (P Q) P ( P Q) (P P) (P Q).

2. (P Q) (P Q) ( (P Q) (P Q)) ((P Q) (P Q))

( P Q P Q) (P P) (Q P) (P Q) (Q Q).

Conjunctive Normal Forms

A formula which is equivalent to a given formula and which consists of a product of elementary

sums is called a conjunctive normal form of a given formula.

Example 1:

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The conjunctive normal form of

(a) P (P Q) P ( P Q).

(b) (P Q) P Q.

(c) (P⇔Q) ((P Q) (Q P))

(( P Q) ( Q P))

( P Q) ( Q P)

(P Q) (Q P)

(P Q) (Q Q) (P P) ( P Q).

(d) (P Q) (P Q) ( (P Q) (P Q)) ((P Q) (P Q))

((P Q) (P Q)) ( (P Q) ( P Q)

(P Q P) (P Q Q) ( P Q P) ( P Q Q).

Example 2:

(a) Show that the formula B ( A B) (A B) is a tautology.

B ( A B) (A B) B (( A A) B)

( B A A) ( B B)

T T

T.

Therefore, it is a tautology.

(b) Show that the formula ( B A) B is a contradiction.

( B A) B A (B B)

A F

F.

Therefore, it is a contradiction.

(c) Show that (Q P) ( P Q) is a contradiction.

(Q P) ( P Q) ( Q P) ( P Q)

( P Q Q) (P P Q)

F F

F.

Therefore, it is a contradiction.

Principal Disjunctive Normal Form (PDNF)

Let us assume A and B be two statement variables. All possible formulas by using conjunction

are given as follows. The total number of formulas for two variables A and B are 22 formulas.

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They are A B, A B,

A B and A B.

These are called minterms or Boolean conjunctions of A and B. The minterms (2n terms) are

denoted by M0, M1, … ,M2n-1.

A formula equivalent to a given formula consisting of the disjunction of minterms only is

called PDNF of the given formula.

Example 1:

Obtain the PDNF of ( P Q) (P⇔ Q)

P Q P Q P ⇔Q ( P Q) (P⇔ Q)

T T F F T

T F T T T

F T T T T

F F T F F

From the above table

( P Q) (P⇔ Q) (P Q) (P Q) ( P Q)

( P Q) (P Q) (P Q)

Example 2:

Obtain the PDNF of

P Q P Q

T T T

T F F

F T T

F F T

P Q (P Q) ( P Q) ( P Q)

( P Q) ( P Q) (P Q)

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Procedure for obtaining PDNF

Step1 : Get rid of ⇔, .

Step2 : Use De Morgan’s law.

Step3 : Use distributive law.

Step4 : Introduce missing factor in the disjunction.

Step5 : If there is elementary product of the form.

(P P) ( ) …… , delete P P.

For every truth value T in the truth table of the given formula, select minterm which also has the

value T for the same combination of the truth values of P and Q. The disjunction of these

minterms will be equivalent to the given formula.

To find a particular minterm Mi, the subscript i is expressed in the binary representation and

suitable number of 0’s are added to the left.

Example :

Obtain PDNF for P ((P Q ( Q P))).

Solution:

P ((P Q ( Q P))) P ((P Q (P Q)))

P ((P P Q))

P ( P (P Q))

P ( P (P Q))

P (P Q)

( P (Q Q)) (P Q)

( P Q) ( P Q) (P Q)

( P Q) ( P Q) (P Q)

.

1.2.4 Principal Conjunctive Normal Forms (PCNF)

The duals of minterms are called maxterms. For a given number of variables the maxterm

consists of disjunctions in which each variable or its negation, but not both, appears only once.

For a given formula, an equivalent formula consisting of conjunctions of maxterms only is

known as its principal conjunctive normal form. This is also called the product of

sums canonical form.

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Example 1:

Obtain the PCNF of ( P R) (Q⇆P).

Solution:

( P R) (Q⇆P) (P R) ((Q P) (P Q))

(P R) (( Q P) ( P Q))

(P R (Q Q)) ((P Q (R R)) ( P Q (R R))

(P Q R) (P Q R) (P Q R) ( P Q R) ( P Q R)

S (0,2,3,4,5).

S consisting of missing maxterms

M1 M6 M7

(P Q R) ( P Q R) ( P Q R).

Example2:

Obtain PCNF for A : ( P R) ((Q P) (P Q)).

Solution:

A (P R) (( Q P) ( P Q))

(P R (Q Q)) (P Q (R R)) ( P Q (R R))

(P Q R) (P Q R) (P Q R) (P Q R) ( P Q R) ( P Q R)

(P Q R) (P Q R) (P Q R) ( P Q R) ( P Q R)

(0,2,3,4,5).

Example 3:

From the given truth table formula S, determine its PDNF and PCNF

Table 1.2.4

A B C S

T T T T

T T F F

T F T F

T F F F

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F T T F

F T F F

F F T F

F F F T

By choosing the minterms corresponding to each T value of S,

the PDNF of S .

By choosing the maxterms corresponding to each F value of S,

the PCNF of

S ( A B C) ( A B C) ( A B C) (A B C) (A B C)

(A B C) .

Example 4:

Form the given truth table formula S, determine its PDNF and PCNF

A B C S

T T T F

T T F F

T F T T

T F F F

F T T T

F T F T

F F T F

F F F T

By choosing minterms corresponding to each T value of S,

the PDNF of S (A B C) ( A B C) ( A B C) ( A B C).

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By choosing maxterms corresponding to each F value of S,

the PCNF of S ( A B C) ( A B C) ( A B C) (A B C)

(1,4,6,7).

Example 5:

Obtain the product of sums canonical form of the formula A which is given by

(P Q R) ( P Q R) ( P Q R)

Solution:

A ( P Q R) (P Q R) (P Q R)

(0,3,7).

( A) consisting of missing maxterms

(1,2,4,5,6)

(P Q R) (P Q R) ( P Q R) ( P Q R) ( P Q R) .

Example 6:

Obtain the product-of-sums canonical form of the formula A, which is given by

( P Q R S) (P Q R S) (P Q R S) ( P Q R S) (P Q

R S).

Solution:

A (P Q R S) ( P Q R S) ( P Q R S) (P Q R S)

( P Q R S)

(P Q R S) (P Q R S) ( P Q R S) ( P Q R S)

( P Q R S)

(5, 6, 9, 10, 12).


(0,1,2,3,4,7,8,11,13,14,15)

M0 M1 M2 M3 M4 M7 M8 M11 M13 M14 M15

(P Q R S) (P Q R S) (P Q R S) (P Q R S) (P Q

R S)

(P Q R S) ( P Q R S) ( P Q R S) ( P Q R S)

( P Q R S) ( P Q R S).

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Example 7:

Obtain the product of sums canonical form of (P Q) ( P Q) (P Q).

Solution:

A ( P Q) (P Q) ( P Q)

(P Q) ( P Q) ( P Q)

(1,2,3).


(0)

M0

P Q.

QUANTIFIERS

The variable of predicates is quantified by quantifiers. There are two types of quantifier in

predicate logic − Universal Quantifier and Existential Quantifier.

Universal Quantifier

Universal quantifier states that the statements within its scope are true for every value of the

specific variable. It is denoted by the symbol ∀.

∀x P(x) is read as for every value of x, P(x) is true.

Example − "Man is mortal" can be transformed into the propositional form ∀x P(x) where P(x)

is the predicate which denotes x is mortal and the universe of discourse is all men.

Existential Quantifier

Existential quantifier states that the statements within its scope are true for some values of the

specific variable. It is denoted by the symbol ∃.

∃x P(x) is read as for some values of x, P(x) is true.

Example − "Some people are dishonest" can be transformed into the propositional form ∃x P(x)

where P(x) is the predicate which denotes x is dishonest and the universe of discourse is some

people.

Nested Quantifiers

If we use a quantifier that appears within the scope of another quantifier, it is called nested

quantifier.

Example

∀a ∃b P (x, y) where P (a, b) denotes a + b = 0

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∀a ∀b ∀c P (a, b, c) where P (a, b) denotes a + (b+c) = (a+b) +c

Note − ∀a ∃b P (x, y) ≠ ∃a ∀b P (x, y)

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