Sets, Relations & Functions
Mathematical Reasoning / Contrapositive
Grade 11
Question:
<p>Contrapositive of the statement, "If a number is divisible by 9, then it is divisible by 3" is</p>
<p>If a number is not divisible by 3, then it is not divisible by 9.</p>
<p>If a number is not divisible by 3, then it is divisible by 9.</p>
<p>If a number is not divisible by 9, then it is not divisible by 3.</p>
<p>None of these</p>
Step-by-Step Solution
Key Concept: The contrapositive of 'If P then Q' is 'If not Q then not P'. Both statements are logically equivalent, so the contrapositive is always true when the original is true.
<p><strong>Step 1:</strong> Identify the original statement structure.</p><p>Original: 'If a number is divisible by 9, then it is divisible by 3'</p><p>Let P = 'a number is divisible by 9'</p><p>Let Q = 'a number is divisible by 3'</p><p>Original form: P → Q</p><p><strong>Step 2:</strong> Form the contrapositive using the rule: Contrapositive of (P → Q) is (¬Q → ¬P)</p><p>¬Q = 'a number is NOT divisible by 3'</p><p>¬P = 'a number is NOT divisible by 9'</p><p><strong>Step 3:</strong> Write the contrapositive statement.</p><p>Contrapositive: 'If a number is not divisible by 3, then it is not divisible by 9'</p><p>∴ Answer: A</p>
Correct Answer: A