Sets, Relations & Functions
Mathematical Reasoning
Grade 11
Question:
<p>\(\sim(p \vee (\sim p \vee q))\) is equal to</p>
<p>\(\sim p \wedge (p \wedge \sim q)\)</p>
<p>\((p \wedge \sim q) \vee \sim p\)</p>
<p>\((p \vee \sim q) \vee \sim p\)</p>
<p>None of these</p>
Step-by-Step Solution
Key Concept: Apply De Morgan's laws systematically from outside to inside, recognizing that ∼p ∨ q simplifies the inner expression before negating the entire disjunction.
<p><strong>Step 1:</strong> Start with ∼(p ∨ (∼p ∨ q))</p><p><strong>Step 2:</strong> Apply De Morgan's law to the outer negation: ∼(p ∨ (∼p ∨ q)) = ∼p ∧ ∼(∼p ∨ q)</p><p><strong>Step 3:</strong> Apply De Morgan's law to ∼(∼p ∨ q): ∼(∼p ∨ q) = ∼(∼p) ∧ ∼q = p ∧ ∼q</p><p><strong>Step 4:</strong> Substitute back: ∼p ∧ (p ∧ ∼q)</p><p><strong>Step 5:</strong> Rearrange using associativity: (∼p ∧ p) ∧ ∼q</p><p><strong>Step 6:</strong> Since ∼p ∧ p = F (contradiction), we get: F ∧ ∼q = F</p><p>∴ Answer: <strong>F (False)</strong> or <strong>⊥ (Contradiction)</strong>, which is option D</p>
Correct Answer: D