Sets, Relations & Functions
Mathematical Reasoning
Grade 11
Question:
<p>The negation of \(q \vee \sim(p \wedge r)\) is</p>
<p>\(\sim q \wedge (p \wedge r)\)</p>
<p>\(\sim q \wedge \sim(p \wedge r)\)</p>
<p>\(\sim q \vee (p \wedge r)\)</p>
<p>None of these</p>
Step-by-Step Solution
Key Concept: Apply De Morgan's laws systematically: negate the disjunction to get a conjunction, then negate the negation to recover the inner expression, finally negate the conjunction inside.
<p><strong>Step 1:</strong> Start with the expression: q ∨ ~(p ∧ r)</p><p><strong>Step 2:</strong> Apply negation to the entire expression: ~[q ∨ ~(p ∧ r)]</p><p><strong>Step 3:</strong> Use De Morgan's Law on the disjunction: ~q ∧ ~[~(p ∧ r)]</p><p><strong>Step 4:</strong> Simplify ~[~(p ∧ r)] using double negation: ~q ∧ (p ∧ r)</p><p><strong>Step 5:</strong> Simplify using associativity: ~q ∧ p ∧ r (or equivalently p ∧ ~q ∧ r)</p><p>∴ Answer: <strong>~q ∧ (p ∧ r)</strong> or <strong>(p ∧ ~q ∧ r)</strong></p>
Correct Answer: A