Sets, Relations & Functions
Mathematical Reasoning - Contrapositive
Grade 11
Question:
<p>Contrapositive of the statement<br/>"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'. Here, you must negate both the hypothesis and conclusion, then reverse them. The contrapositive 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: P → Q</p><p><strong>Step 2:</strong> Form the contrapositive by reversing and negating both parts.</p><p>Contrapositive: ¬Q → ¬P</p><p>¬Q = negation of 'their squares are not equal' = 'their squares are equal'</p><p>¬P = negation of 'two numbers are not equal' = 'two numbers are equal'</p><p><strong>Step 3:</strong> Write the contrapositive statement.</p><p>Contrapositive: If their squares are equal, then the two numbers are equal.</p><p>∴ Answer: C</p>
Correct Answer: C