Sets, Relations & Functions
Mathematical Reasoning - Contrapositive
Grade 11
Question:
<p>Contrapositive of the statement 'If two numbers are not equal, then their squares are not equal' is:</p>
<p>If the squares of two numbers are not equal, then the numbers are equal.</p>
<p>If the squares of two numbers are equal, then the numbers are not equal.</p>
<p>If the squares of two numbers are equal, then the numbers are equal.</p>
<p>If the squares of two numbers are not equal, then the numbers are not equal.</p>
Step-by-Step Solution
Key Concept: The contrapositive of 'If P then Q' is 'If not Q then not P'. You must negate both the hypothesis and conclusion, then reverse their order—this is logically equivalent to the original statement.
<p><strong>Step 1:</strong> Identify the original statement structure.</p><p>Original: 'If two numbers are not equal, then their squares are not equal'</p><p>Let P = 'two numbers are not equal'</p><p>Let Q = 'their squares are not equal'</p><p>Statement form: If P then Q</p><p><strong>Step 2:</strong> Form the contrapositive by negating both parts and reversing.</p><p>Contrapositive: If (not Q) then (not P)</p><p>If ¬Q then ¬P</p><p><strong>Step 3:</strong> Translate back to words.</p><p>¬Q = 'squares are equal' (negation of 'squares are not equal')</p><p>¬P = 'two numbers are equal' (negation of 'two numbers are not equal')</p><p><strong>Step 4:</strong> Write the final contrapositive statement.</p><p><strong>Contrapositive:</strong> 'If the squares of two numbers are equal, then the two numbers are equal'</p><p>∴ Answer: C</p>
Correct Answer: C