Question

The logical statement [~(~p ∨ q) ∨ (p ∧ r)] ∧ (~q ∧ r) is equivalent to:
1. (~p ∧ ~q) ∧ r
2. ~p ∨ r
3. (p ∧ r) ∧ ~q
4. (p ∧ ~q) ∨ r

Answer 1

Correct Answer - Option 3 : (p ∧ r) ∧ ~q

Logical statement given is,

[~(~p ∨ q) ∨ (p ∧ r)] ∧ (~q ∧ r)

⇒ [(p ∧ ~q) ∨ (p ∧ r)] ∧ (~q ∧ r)

⇒ [(p ∧ ~q) ∧ (~q ∧ r)] ∨ [(p ∧ r) ∧ (~q ∧ r)]

⇒ [p ∧ ~q ∧ r] ∨ [p ∧ r ∧ ~q]

⇒ (p ∧ ~q) ∧ r

⇒ (p ∧ r) ∧ ~q

Hence, [~(~p ∨ q) ∨ (p ∧ r)] ∧ (~q ∧ r) = (p ∧ r) ∧ ~q