<p>The statement <strong>~</strong>\((p \leftrightarrow\ \sim q)\) is</p>
Step-by-Step Solution
Key Concept: A biconditional p ↔ ~q is true when both sides have the same truth value. Its negation ~(p ↔ ~q) is true when p and ~q have opposite truth values, which means p and q have the same truth value.
<p><strong>Step 1:</strong> Recall that p ↔ q is true iff p and q have the same truth value (both T or both F).</p><p><strong>Step 2:</strong> So p ↔ ~q is true when p and ~q have the same truth value, i.e., when p is T and q is F, or p is F and q is T.</p><p><strong>Step 3:</strong> Therefore ~(p ↔ ~q) is true when p ↔ ~q is FALSE, which happens when p and ~q have opposite truth values.</p><p><strong>Step 4:</strong> This occurs when: (p is T and q is T) OR (p is F and q is F). This is equivalent to <strong>p ↔ q</strong>.</p><p><strong>Step 5:</strong> Alternatively: ~(p ↔ ~q) ≡ p ↔ ~~q ≡ <strong>p ↔ q</strong></p><p>∴ Answer: C (The statement is equivalent to p ↔ q)</p>
Correct Answer: C