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

To solve this problem, we need to find the negation of the compound statement ~p \wedge (p \vee q).

1. First, identify the main logical structure of the statement: ~p \wedge (p \vee q). Here, ~p represents "not p" and p \vee q represents "p or q".
2. In the expression ~p \wedge (p \vee q), the main operator is conjunction \wedge, which stands for "and".
3. To negate the entire expression, we apply De Morgan's laws which state that the negation of a conjunction is the disjunction of the negations. Specifically: \neg (A \wedge B) = \neg A \vee \neg B.
4. Apply De Morgan's law to negate ~p \wedge (p \vee q):
   • Negate each part: \neg (~p) becomes p, and \neg (p \vee q) becomes ~p \wedge ~q.
   • So, the negation is: p \vee (~p \wedge ~q).
5. Further simplify the expression:
   • Notice that p \vee (~p \wedge ~q) can be simplified using distribution: p \vee ~p \vee ~q.
   • Since p \vee ~p is a tautology (always true), the expression simplifies to: P \vee ~q.
6. Check that p \vee ~q corresponds to the given answer options, confirming it matches the correct answer
   P ∨ ~ q
   .

Therefore, the negation of the statement ~p \wedge (p \vee q) is P ∨ ~ q. This step-by-step deduction justifies the provided correct answer.