Question

The negative of the statement $\sim p \wedge( p \vee q )$ is

• A. $\sim p \vee q$
• B. $p \vee \sim q$
• C. $\sim p \wedge q$
• D. $p \wedge \sim q$

Answer 1

The Correct Option is B

To solve this problem, we need to find the negation of the logical expression \sim p \wedge ( p \vee q ).

The expression can be broken down into:

• \sim p: This represents "not p".
• p \vee q: This represents "p or q".
• The entire expression \sim p \wedge ( p \vee q ) means "not p AND (p OR q)".

To find the negation of this expression, we apply De Morgan's laws. De Morgan’s laws state:

• The negation of an AND statement is equivalent to the OR of the negations: \sim (A \wedge B) \equiv \sim A \vee \sim B.
• The negation of an OR statement is equivalent to the AND of the negations: \sim (A \vee B) \equiv \sim A \wedge \sim B.

Applying these laws to our expression:

1. Negate the original expression: \sim (\sim p \wedge (p \vee q)).
2. Using De Morgan's first law: \sim (\sim p \wedge (p \vee q)) = \sim(\sim p) \vee \sim(p \vee q).
3. Simplify \sim(\sim p) to p.
4. Using De Morgan's second law, \sim(p \vee q) becomes \sim p \wedge \sim q.
5. Thus the entire expression simplifies to: p \vee (\sim p \wedge \sim q).
6. Applying distribution: p \vee \sim q.

Therefore, the negation of \sim p \wedge (p \vee q) is p \vee \sim q, which is option 2.

This explanation validates that the correct answer is p \vee \sim q.