Sets, Relations & Functions
Mathematical Reasoning - Contrapositive
Grade 11
Question:
<p>Consider the following two statements:<br/><b>P:</b> If 7 is an odd number, then 7 is divisible by 2.<br/><b>Q:</b> If 7 is a prime number, then 7 is an odd number.<br/>If \(V_1\) is the truth value of the contrapositive of \(P\) and \(V_2\) is the truth value of contrapositive of \(Q\), then the ordered pair \((V_1, V_2)\) equals</p>
<p>(F, F)</p>
<p>(F, T)</p>
<p>(T, F)</p>
<p>(T, T)</p>
Step-by-Step Solution
Key Concept: The contrapositive of (A → B) is (¬B → ¬A), and a statement and its contrapositive always have the same truth value. Evaluate the original statements' truth values, then their contrapositives will share those same values.
<p><strong>Step 1: Analyze Statement P</strong></p><p>P: If 7 is odd, then 7 is divisible by 2.</p><p>• Hypothesis (7 is odd): TRUE</p><p>• Conclusion (7 is divisible by 2): FALSE</p><p>• P is (T → F) = FALSE</p><p></p><p><strong>Step 2: Find contrapositive of P</strong></p><p>Contrapositive of P: If 7 is not divisible by 2, then 7 is not odd.</p><p>• Hypothesis (¬divisible by 2): TRUE</p><p>• Conclusion (¬odd): FALSE</p><p>• Contrapositive of P is (T → F) = FALSE</p><p>• V₁ = FALSE</p><p></p><p><strong>Step 3: Analyze Statement Q</strong></p><p>Q: If 7 is prime, then 7 is odd.</p><p>• Hypothesis (7 is prime): TRUE</p><p>• Conclusion (7 is odd): TRUE</p><p>• Q is (T → T) = TRUE</p><p></p><p><strong>Step 4: Find contrapositive of Q</strong></p><p>Contrapositive of Q: If 7 is not odd, then 7 is not prime.</p><p>• Hypothesis (¬odd): FALSE</p><p>• Conclusion (¬prime): FALSE</p><p>• Contrapositive of Q is (F → F) = TRUE</p><p>• V₂ = TRUE</p><p></p><p><strong>Step 5: Form ordered pair</strong></p><p>(V₁, V₂) = (FALSE, TRUE) or (F, T)</p><p>∴ Answer: D</p>
Correct Answer: D