Sets, Relations & Functions
Mathematical Reasoning - Negation
Grade 11
Question:
<p>For any two statements \(p\) and \(q\), the negation of the expression \(p \vee (\sim p \wedge q)\) is:</p>
<p>\(\sim p \wedge \sim q\)</p>
<p>\(p \wedge q\)</p>
<p>\(p \leftrightarrow q\)</p>
<p>\(\sim p \vee \sim q\)</p>
Step-by-Step Solution
Key Concept: Apply De Morgan's laws systematically: negate the disjunction by negating each part and flipping OR to AND, then simplify using the fact that p ∧ (~p) is always false.
<p><strong>Step 1:</strong> Apply De Morgan's law to negate p ∨ (~p ∧ q)</p><p>~[p ∨ (~p ∧ q)] = ~p ∧ ~(~p ∧ q)</p><p><strong>Step 2:</strong> Apply De Morgan's law again to ~(~p ∧ q)</p><p>~(~p ∧ q) = ~~p ∨ ~q = p ∨ ~q</p><p><strong>Step 3:</strong> Substitute back</p><p>~p ∧ (p ∨ ~q)</p><p><strong>Step 4:</strong> Distribute ~p over the disjunction using distributive law</p><p>(~p ∧ p) ∨ (~p ∧ ~q)</p><p><strong>Step 5:</strong> Simplify using ~p ∧ p ≡ F (contradiction)</p><p>F ∨ (~p ∧ ~q) = ~p ∧ ~q</p><p>∴ Answer: <strong>~p ∧ ~q</strong> (or equivalently ~(p ∨ q))</p>
Correct Answer: A