Key Concepts and Formulas

• Conditional Statement: A statement of the form "If p, then q," denoted as $p \to q$, where p is the hypothesis and q is the conclusion.
• Contrapositive: The contrapositive of the conditional statement $p \to q$ is $\sim q \to \sim p$, where $\sim$ denotes negation.
• Negation: The negation of a statement reverses its truth value.

Step-by-Step Solution

Step 1: Identify the hypothesis and conclusion.

The given statement is "If the side of a square doubles, then its area increases four times."

Let p be the statement: "The side of a square doubles." Let q be the statement: "The area of a square increases four times."

Thus, the given statement can be written as $p \to q$.

Step 2: Find the negation of the hypothesis and the conclusion.

The negation of p ($\sim p$) is: "The side of a square does not double." The negation of q ($\sim q$) is: "The area of a square does not increase four times."

Step 3: Form the contrapositive statement.

The contrapositive of $p \to q$ is $\sim q \to \sim p$. Substituting the negations, we get: "If the area of a square does not increase four times, then its side is not doubled."

Common Mistakes & Tips

• Remember that the contrapositive involves negating both the hypothesis and the conclusion and reversing their order.
• Be careful with the wording of the negations. Ensure they accurately represent the opposite of the original statements.
• The contrapositive of a statement is logically equivalent to the original statement. This means they have the same truth value.

Summary

We were given the statement "If the side of a square doubles, then its area increases four times" and asked to find its contrapositive. By identifying the hypothesis (p) and conclusion (q), negating them, and then reversing the order to form $\sim q \to \sim p$, we found the contrapositive to be "If the area of a square does not increase four times, then its side is not doubled," which corresponds to option (D).

Final Answer

The final answer is \boxed{D}, which corresponds to option (D).