This document discusses discrete mathematical structures and propositional logic. It introduces topics like normal forms, negation normal form, disjunctive normal form, and conjunctive normal form. These normal forms are syntactic restrictions on logical formulas. The document provides examples of converting formulas to different normal forms using truth tables. It also describes how to derive the disjunctive and conjunctive normal forms of compound propositions.

1. DISCRETE MATHEMATICAL STRUCTRES
Mr Ch Viswanatha Sarma
M.Sc,M.Phil,M.Tech (Ph.D.)
Asst.Prof
Department of Computer Science and Engineering
Department of Computer Science and Engineering
Course: Discrete Mathematical Structures

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Course: Discrete Mathematical Structures
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❖ Logic
❖ Mathematical Logic
➢ Propositional Logic
■ Syntax of PL
● Statements
● Logical Connectivities
● Truth Tables
◆ Tautology
◆ Contradiction
■ Semantics of PL
■ Logical Equivalences
Summary of Previous Lecture

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Course: Discrete Mathematical Structures
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Normal Forms
❖ A normal form of formulae is a syntactic restriction
such that for every formula of logic, there is an
equivalent formula in the restricted form.
❖ Each formula must use only ¬, ∨, and ∧

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Course: Discrete Mathematical Structures
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Normal Forms
Disjunctive Normal
Form
Negation Normal
Form
Conjunctive Normal
Form
Types

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Terminology used in normal forms
❖ Propositional variables are also referred as atom
❖ A literal is either an atom or its negation
❖ A clause is a disjunction of literals.

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Negation Normal Form (NNF)
❖ A formula is in NNF if ¬ appears only in front of the
propositional variables.
Example : In NNF: ¬A ∧ ¬B
Not in NNF: ¬(A ∨ B)

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Transformation into NNF
Any formula can be transformed into an equivalent
formula in NNF by pushing ¬ inwards.
● ¬¬F ≡ F
● ¬(F ∧ G) ≡ (¬F ∨ ¬G)
● ¬(F ∨ G) ≡ (¬F ∧ ¬G)

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Examples of NNF
1. Convert (¬(A ∧ ¬B) ∧ C) in NNF
2. Convert ¬(r V¬(¬p ∧q)) in NNF

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Disjunctive Normal Form (DNF)
❖ A minterm is a conjunction of literals in which each
variable is represented exactly once.
Example : Given two simple propositions p and q,
p∧ q, p∧ ¬q, ¬p∧ ¬q, ¬p∧ q are minterms of p and q
❖ Each minterm is true for exactly one assignment.

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Disjunctive Normal Form (DNF)
A disjunction of minterms is called Disjunctive
Normal Form.
i.e.; (p∧ q) ∨ (¬p ∧ q ∧ r) ∨ (¬ p∧r )
A disjunction of minterms is true only if at least one
of its constituents minterms is true
Definition

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From truth table to DNF
❖ If a function, e.g. F, is given by a truth table,
determine exactly for which assignments it is true.
❖ We can select the minterms that make the function
true and form the disjunction of these minterms.

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From truth table to DNF
F is true for three assignments:
DNF of F :
( p∧ q ∧ r ) ∨ ( p ∧ ¬q ∧ r ) ∨
(¬ p ∧ ¬q ∧ r)
p q r F
1 1 1 1
1 1 0 0
1 0 1 1
1 0 0 0
0 1 1 0
0 1 0 0
0 0 1 1
0 0 0 0

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Convert the compound proposition p → (q ∧ r ) into DNFExample
p q r q ∧ r p → (q ∧ r )
1 1 1 1 1
1 1 0 0 0
1 0 1 0 0
1 0 0 0 0
0 1 1 1 1
0 1 0 0 1
0 0 1 0 1
0 0 0 0 1

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From the above table, we getExample

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Find the DNF of the compound proposition ¬ ( p→(q ∧ r))Example

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Conjunctive Normal Form (CNF)
❖ A clause is a disjunction of literals in which each
variable is represented exactly once.
Example : Given two simple propositions p and q,
pV q, pV ¬q, ¬pV ¬q, ¬pV q are clauses of p and q
.

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Conjunctive Normal Form (CNF)
A conjunction of one or more clauses is called
conjunctive Normal Form.
i.e.; (p ∨q) ∧ (¬p V q V r) ∧ (¬ p V r )
Definition

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From truth table to CNF
❖ If a function, e.g. F, is given by a truth table,
determine exactly for which assignments it is False.
❖ We can select the minterms that make the function
false and form the disjunction of these minterms.
❖ Then take the complement of disjunction of these
minterms.

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From truth table to CNF
p q r F
1 1 1 1
1 1 0 0
1 0 1 1
1 0 0 0
0 1 1 0
0 1 0 0
0 0 1 1
0 0 0 0

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Example:
Find the Conjunctive normal form of compound
proposition (p→q)→(¬r∧q)
Sol :

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Example:
Find the Conjunctive normal form of compound
proposition ((p ∧ q) ∨ (r ∧ s)) ∨ (¬q ∧ (p ∨ t))
Sol :

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Example:

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Example:

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Example: