<p>\((\sim(p \vee q)) \vee (\sim p \wedge q\) is logically equivalent to</p>
Step-by-Step Solution
Key Concept: Apply De Morgan's laws systematically: ~(p ∨ q) = (~p ∧ ~q), then use the absorption property to simplify disjunctions with common terms.
<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 back into the original expression</p><p>(~p ∧ ~q) ∨ (~p ∧ q)</p><p><strong>Step 3:</strong> Factor out ~p using the distributive law</p><p>= ~p ∧ (~q ∨ q)</p><p><strong>Step 4:</strong> Simplify using the law of excluded middle: (~q ∨ q) = T (True)</p><p>= ~p ∧ T</p><p><strong>Step 5:</strong> Apply identity law: X ∧ T = X</p><p>= ~p</p><p><strong>Step 6:</strong> Therefore, the expression is logically equivalent to <strong>~p</strong></p><p>∴ Answer: B (~p)</p>
Correct Answer: B