Question

The negation of the statement ~p ∧ (p ∨ q) is :

• A. ~ P ∨ q
• B. ~ P ∧ q
• C. P ∧ ~ q
• D. P ∨ ~ q

Hint:

De Morgan's Laws: $\sim(A \wedge B) \equiv \sim A \vee \sim B$ and $\sim(A \vee B) \equiv \sim A \wedge \sim B$.

Answer 1

The Correct Option is D

Step 1: Simplify the expression first: $\sim p \wedge (p \vee q) \equiv (\sim p \wedge p) \vee (\sim p \wedge q)$.
Step 2: Since $(\sim p \wedge p)$ is a contradiction ($F$), the expression is $F \vee (\sim p \wedge q) \equiv \sim p \wedge q$.
Step 3: Find the negation: $\sim (\sim p \wedge q)$.
Step 4: By De Morgan's Law: $\sim (\sim p) \vee \sim q \equiv p \vee \sim q$.