Sets, Relations & Functions
Mathematical Reasoning
Grade 11
Question:
<p>\(\sim(p \vee (\sim q))\) is equal to</p>
<p>\(\sim p \vee q\)</p>
<p>\((\sim p) \wedge q\)</p>
<p>\(\sim p \vee \sim p\)</p>
<p>\(\sim p \wedge \sim q\)</p>
Step-by-Step Solution
Key Concept: Apply De Morgan's Law to distribute negation over disjunction: ~(A ∨ B) = (~A) ∧ (~B). Then simplify the double negation ~(~q) = 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> Simplify the double negation ~(~q) = q</p><p>(~p) ∧ (~(~q)) = (~p) ∧ q</p><p><strong>Step 3:</strong> Write in standard form</p><p>∴ Answer: <strong>~p ∧ q</strong> (or equivalently <strong>(~p) ∧ q</strong>)</p>
Correct Answer: B