<p>The Boolean expression \(\sim(p \wedge q) \wedge (p \vee q)\) is equivalent to</p>
Step-by-Step Solution
Key Concept: Distribute negation using De Morgan's Law: ~(p ∧ q) = (~p ∨ ~q), then apply the distributive property to simplify the resulting expression with (p ∨ q).
<p><strong>Step 1:</strong> Apply De Morgan's Law to ~(p ∧ q)</p><p>~(p ∧ q) = (~p ∨ ~q)</p><p><strong>Step 2:</strong> Substitute into the original expression</p><p>(~p ∨ ~q) ∧ (p ∨ q)</p><p><strong>Step 3:</strong> Distribute using the property (A ∨ B) ∧ (C ∨ D) = (A ∧ C) ∨ (A ∧ D) ∨ (B ∧ C) ∨ (B ∧ D)</p><p>(~p ∧ p) ∨ (~p ∧ q) ∨ (~q ∧ p) ∨ (~q ∧ q)</p><p><strong>Step 4:</strong> Simplify using A ∧ ~A = F (Contradiction)</p><p>F ∨ (~p ∧ q) ∨ (p ∧ ~q) ∨ F</p><p><strong>Step 5:</strong> Final simplification</p><p>(~p ∧ q) ∨ (p ∧ ~q)</p><p>This is the exclusive OR (XOR): p ⊕ q</p><p>∴ Answer: <strong>(~p ∧ q) ∨ (p ∧ ~q)</strong> or equivalently <strong>p ⊕ q</strong></p>
Correct Answer: D