Sets, Relations & Functions
Mathematical Reasoning
Grade None
Question:
<p>The contrapositive of \((p \vee q) \rightarrow r\) is</p>
<p>\(r \rightarrow (p \vee q)\)</p>
<p>\(\sim r \rightarrow (p \vee q)\)</p>
<p>\(\sim r \rightarrow (\sim p \wedge \sim q)\)</p>
<p>\(p \rightarrow (q \vee r)\)</p>
Step-by-Step Solution
Key Concept: The contrapositive of (A → B) is (¬B → ¬A). Apply this rule to (p ∨ q) → r by negating both the conclusion and hypothesis, then reversing their positions.
<p><strong>Step 1:</strong> Recall that the contrapositive of (A → B) is (¬B → ¬A).</p><p><strong>Step 2:</strong> For (p ∨ q) → r, identify A = (p ∨ q) and B = r.</p><p><strong>Step 3:</strong> Form the contrapositive: ¬r → ¬(p ∨ q).</p><p><strong>Step 4:</strong> Apply De Morgan's Law: ¬(p ∨ q) = (¬p ∧ ¬q).</p><p><strong>Step 5:</strong> The contrapositive is <strong>¬r → (¬p ∧ ¬q)</strong>.</p><p>∴ Answer: C</p>
Correct Answer: C