Sets, Relations & Functions
Mathematical Reasoning / Biconditional
Grade 11
Question:
<p>The statement \(\sim(p \leftrightarrow \sim q)\) is</p>
<p>a tautology.</p>
<p>a fallacy.</p>
<p>equivalent to \(p \leftrightarrow q\).</p>
<p>equivalent to \(\sim p \leftrightarrow q\).</p>
Step-by-Step Solution
Key Concept: Apply De Morgan's laws and logical equivalences to biconditional statements. Recognize that ∼(p ↔ q) is logically equivalent to (p ⊕ q), the exclusive or, which means p and q have opposite truth values.
<p><strong>Step 1:</strong> Recall that p ↔ q ≡ (p ∧ q) ∨ (∼p ∧ ∼q) (both have same truth value)</p><p><strong>Step 2:</strong> So p ↔ ∼q ≡ (p ∧ ∼q) ∨ (∼p ∧ q)</p><p><strong>Step 3:</strong> Therefore ∼(p ↔ ∼q) ≡ ∼[(p ∧ ∼q) ∨ (∼p ∧ q)]</p><p><strong>Step 4:</strong> Apply De Morgan's law: ∼(p ∧ ∼q) ∧ ∼(∼p ∧ q) ≡ (∼p ∨ q) ∧ (p ∨ ∼q)</p><p><strong>Step 5:</strong> Expand: (∼p ∧ p) ∨ (∼p ∧ ∼q) ∨ (q ∧ p) ∨ (q ∧ ∼q) ≡ (p ∧ q) ∨ (∼p ∧ ∼q)</p><p><strong>Step 6:</strong> This is equivalent to p ↔ q (biconditional: p and q have same truth value)</p><p>∴ Answer: C (Statement is equivalent to p ↔ q)</p>
Correct Answer: C