Sets, Relations & Functions
Mathematical Reasoning / Contrapositive
Grade 11
Question:
<p>The contrapositive of the statement "If you are born in India, then you are a citizen of India", is:</p>
<p>If you are not a citizen of India, then you are not born in India.</p>
<p>If you are a citizen of India, then you are born in India.</p>
<p>If you are born in India, then you are not a citizen of India.</p>
<p>If you are not born in India, then you are not a citizen of India.</p>
Step-by-Step Solution
Key Concept: The contrapositive of 'If P then Q' is 'If not Q then not P'. It reverses and negates both the hypothesis and conclusion while maintaining logical equivalence.
<p><strong>Step 1:</strong> Identify the original statement structure.</p><p>Original: "If you are born in India, then you are a citizen of India"</p><p>Let P = "You are born in India"</p><p>Let Q = "You are a citizen of India"</p><p>Original form: P → Q</p><p><strong>Step 2:</strong> Apply contrapositive rule.</p><p>Contrapositive of (P → Q) is (¬Q → ¬P)</p><p><strong>Step 3:</strong> Write the contrapositive statement.</p><p>Negate both parts and reverse their order:</p><p>"If you are NOT a citizen of India, then you are NOT born in India"</p><p><strong>Step 4:</strong> Verify equivalence.</p><p>The contrapositive is logically equivalent to the original statement. If the original is true, the contrapositive must also be true.</p><p>∴ Answer: A (The contrapositive is: "If you are not a citizen of India, then you are not born in India")</p>
Correct Answer: A