Question

For any two statements \(p\) and \(q\), the statement \( \sim(p \vee q) \vee (\sim p \wedge q) \) is equivalent to

• A. \( \sim p \)
• B. \( p \)
• C. \( q \)
• D. \( \sim q \)
• E. \( p \vee q \)

Hint:

Always try factoring common logical terms to simplify expressions quickly.

Answer 1

The Correct Option is A

Understanding the Concept: Use logical identities:
• De Morgan's Law: \( \sim(p \vee q) = \sim p \wedge \sim q \)
• Distributive Law
• Absorption Law

Step 1: Apply De Morgan's Law
\[ \sim(p \vee q) = \sim p \wedge \sim q \] So expression becomes: \[ (\sim p \wedge \sim q) \vee (\sim p \wedge q) \]

Step 2: Factor common term
\[ = \sim p \wedge (\sim q \vee q) \]

Step 3: Use tautology
\[ \sim q \vee q = 1 \]

Step 4: Simplify
\[ = \sim p \wedge 1 = \sim p \]

Step 5: Final Answer
\[ \boxed{\sim p} \]