<p>The proposition \(\sim(p \vee \sim q) \vee \sim(p \vee q)\) is logically equivalent to</p>
Step-by-Step Solution
Key Concept: Use De Morgan's Laws systematically: ∼(A ∨ B) ≡ (∼A ∧ ∼B). Apply this twice to simplify the compound negations, then recognize the resulting logical form.
<p><strong>Step 1: Apply De Morgan's Law to first term</strong></p><p>∼(p ∨ ∼q) ≡ (∼p ∧ q)</p><p><strong>Step 2: Apply De Morgan's Law to second term</strong></p><p>∼(p ∨ q) ≡ (∼p ∧ ∼q)</p><p><strong>Step 3: Substitute into original expression</strong></p><p>∼(p ∨ ∼q) ∨ ∼(p ∨ q) ≡ (∼p ∧ q) ∨ (∼p ∧ ∼q)</p><p><strong>Step 4: Factor out common term ∼p</strong></p><p>(∼p ∧ q) ∨ (∼p ∧ ∼q) ≡ ∼p ∧ (q ∨ ∼q)</p><p><strong>Step 5: Apply Law of Excluded Middle</strong></p><p>Since (q ∨ ∼q) ≡ T (tautology), we have:</p><p>∼p ∧ T ≡ <strong>∼p</strong></p><p>∴ Answer: C (which represents ∼p or equivalent statement)</p>
Correct Answer: C