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 find the negative of the statement \(\sim p \wedge (p \vee q)\), we need to apply logical negation and use De Morgan's Laws where applicable.

1. The original statement is \(\sim p \wedge (p \vee q)\). The symbol \(\sim\) represents the negation (NOT), \(\wedge\) represents the logical AND, and \(\vee\) represents the logical OR.
2. We are asked to find the negative of this whole statement, which is equivalent to saying: \(\sim(\sim p \wedge (p \vee q))\).
3. Apply De Morgan's Law for negations involving AND: \(\sim(a \wedge b) \equiv \sim a \vee \sim b\).
4. In our case, identify \(a = \sim p\) and \(b = (p \vee q)\). Hence, \(\sim(\sim p \wedge (p \vee q))\) becomes \((\sim(\sim p)) \vee \sim(p \vee q)\).
5. Simplify \(\sim(\sim p)\) to \(p\) because the double negation cancels out.
6. Apply De Morgan's Law again to \(\sim(p \vee q)\): \(\sim(p \vee q) \equiv \sim p \wedge \sim q\).
7. Thus, the expression becomes: \(p \vee (\sim p \wedge \sim q)\).
8. Applying the distributive law: \(p \vee (\sim p \wedge \sim q) \equiv (p \vee \sim p) \wedge (p \vee \sim q)\).
9. \(p \vee \sim p\) is a tautology (always true, no matter the truth value of \(p\)). Hence the expression simplifies to \(p \vee \sim q\).

The negative of the statement \(\sim p \wedge (p \vee q)\) is therefore \(\boldsymbol{p \vee \sim q}\).

Thus, the correct answer is: \(p \vee \sim q\).