Sets, Relations & Functions
Mathematical Reasoning
Grade 11
Question:
<p>Which of the following is the inverse of the proposition 'If a number is a prime then it is odd'?</p>
<p>If a number is not odd then it is not a prime.</p>
<p>If a number is a prime then it is odd.</p>
<p>If a number is not odd then it is a prime.</p>
<p>If a number is not a prime then it is not odd.</p>
Step-by-Step Solution
Key Concept: The inverse of 'If P then Q' is 'If not P then not Q'. You must negate both the hypothesis and conclusion, which is fundamentally different from the converse (which swaps them).
<p><strong>Step 1:</strong> Identify the original proposition structure.</p><p>Original: 'If a number is a prime then it is odd'</p><p>Let P = 'a number is prime' and Q = 'it is odd'</p><p>Form: P → Q</p><p><strong>Step 2:</strong> Apply the definition of inverse.</p><p>The inverse of P → Q is: ¬P → ¬Q</p><p>This means: 'If not P then not Q'</p><p><strong>Step 3:</strong> Substitute back with negations.</p><p>¬P = 'a number is NOT prime'</p><p>¬Q = 'it is NOT odd'</p><p><strong>Step 4:</strong> Write the inverse proposition.</p><p>'If a number is not prime, then it is not odd'</p><p>∴ Answer: D</p>
Correct Answer: D