Q.7) Obtain the conjunctive normal form of:

i) p ∧ (p ⇒ q)

ii) ¬p ⇒ [r ∧ (p ⇒ q)]

To obtain the conjunctive normal form (CNF) of a logical expression, we need to express it as a conjunction of disjunctions. Let's solve each part of the question step by step.
i) Expression: p ∧ ( p ⇒ q )
The implication p ⇒ q can be rewritten using the logical equivalence p ⇒ q ≡ ¬ p ∨ q . Substitute this into the expression:
p ∧ ( ¬ p ∨ q )
Now, distribute the ∧ over the ∨ :
( p ∧ ¬ p ) ∨ ( p ∧ q )
Since p ∧ ¬ p is always false (contradiction), the expression simplifies to:
p ∧ q
This is already in CNF as it is a conjunction of literals.
ii) Expression: ¬ p ⇒ [ r ∧ ( p ⇒ q )]
First, rewrite the implication ¬ p ⇒ [ r ∧ ( p ⇒ q )] :
¬ ( ¬ p ) ∨ ( r ∧ ( p ⇒ q ))
p ∨ [ r ∧ ( ¬ p ∨ q )]
Now, distribute p ∨ over the conjunction:
( p ∨ r ) ∧ ( p ∨ ( ¬ p ∨ q ))
Simplify p ∨ ( ¬ p ∨ q ) using associativity and distributive properties:
( p ∨ r ) ∧ (( p ∨ ¬ p ) ∨ q )
Since ( p ∨ ¬ p ) is always true (tautology), it simplifies to:
( p ∨ r ) ∧ True ∧ q
Finally, the CNF is:
( p ∨ r ) ∧ q
Both expressions are now in their conjunctive normal form. The CNF represents a way to express logical formulas as an intersection (AND) of unions (OR), which is useful in logical operations and computational logic.

Answered by EmmaGraceJohnson | 2025-07-06