Sets, Relations & Functions
Mathematical Reasoning
Grade 11
Question:
<p>\(\sim((\sim(\sim p)) \wedge q)\) is equal to</p>
<p>\(\sim p \wedge q\)</p>
<p>\(\sim p \vee \sim q\)</p>
<p>\(p \wedge \sim q\)</p>
<p>\(\sim p \wedge \sim q\)</p>
Step-by-Step Solution
Key Concept: Apply De Morgan's law systematically: negation of a conjunction equals the disjunction of negations, while double negation simplifies to the original proposition.
<p><strong>Step 1:</strong> Simplify the innermost negations in ∼(∼(∼p)) ∧ q</p><p>∼(∼(∼p)) = ∼p (since ∼(∼x) = x, we have three negations: ∼(∼(∼p)) = ∼p)</p><p><strong>Step 2:</strong> Substitute back into the expression</p><p>∼(∼p ∧ q)</p><p><strong>Step 3:</strong> Apply De Morgan's Law: ∼(A ∧ B) = ∼A ∨ ∼B</p><p>∼(∼p ∧ q) = ∼(∼p) ∨ ∼q = p ∨ ∼q</p><p><strong>Step 4:</strong> Verify by truth table if needed</p><p>∴ Answer: <strong>p ∨ ∼q</strong> (Option B)</p>
Correct Answer: B