Sets, Relations & Functions
Mathematical Reasoning - Contrapositive
Grade None
Question:
<p>The contrapositive of the following statement,<br/>"If the side of a square doubles, then its area increases four times", is</p>
<p>If the area of a square increases four times, then its side is not doubled.</p>
<p>If the area of a square increases four times, then its side is doubled.</p>
<p>If the area of a square does not increase four times, then its side is not doubled.</p>
<p>If the side of a square is not doubled, then its area does not increase four times.</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.
<p><strong>Step 1:</strong> Identify the original statement structure.</p><p>Original: "If the side of a square doubles, then its area increases four times"</p><p>Let P = "the side of a square doubles"</p><p>Let Q = "its area increases four times"</p><p>Original form: If P then Q</p><p><strong>Step 2:</strong> Form the contrapositive by negating both parts and reversing.</p><p>Contrapositive form: If (not Q) then (not P)</p><p>¬Q = "the area does NOT increase four times"</p><p>¬P = "the side of the square does NOT double"</p><p><strong>Step 3:</strong> Write the contrapositive statement.</p><p>"If the area of a square does not increase four times, then its side does not double."</p><p><strong>Key Property:</strong> A statement and its contrapositive are logically equivalent (always have the same truth value).</p><p>∴ <strong>Answer: C</strong> (The contrapositive statement as formulated above)</p>
Correct Answer: C