Sets, Relations & Functions
Mathematical Reasoning
Grade 11
Question:
<p>Which of the following is the negation of the statement \(p \rightarrow (\sim p \vee \sim q)\)?</p>
<p>\(p \rightarrow (p \vee q)\)</p>
<p>\(p \wedge (p \wedge q)\)</p>
<p>\(p \vee (p \wedge q)\)</p>
<p>\(\sim p \vee (\sim p \vee \sim q)\)</p>
Step-by-Step Solution
Key Concept: The negation of an implication (p → r) is (p ∧ ¬r). Here we must negate p → (¬p ∨ ¬q) to get p ∧ ¬(¬p ∨ ¬q), then simplify using De Morgan's law.
<p><strong>Step 1:</strong> Recall that the negation of an implication is: ¬(p → r) ≡ p ∧ ¬r</p><p><strong>Step 2:</strong> Apply this rule to ¬[p → (¬p ∨ ¬q)]</p><p>¬[p → (¬p ∨ ¬q)] ≡ p ∧ ¬(¬p ∨ ¬q)</p><p><strong>Step 3:</strong> Apply De Morgan's law to ¬(¬p ∨ ¬q):</p><p>¬(¬p ∨ ¬q) ≡ p ∧ q</p><p><strong>Step 4:</strong> Combine the results:</p><p>¬[p → (¬p ∨ ¬q)] ≡ p ∧ (p ∧ q) ≡ p ∧ q</p><p>∴ <strong>Answer: p ∧ q</strong> (or equivalent form)</p>
Correct Answer: B