Key Concepts and Formulas

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

Step-by-Step Solution

Step 1: Identify the hypothesis and conclusion of the given statement.

The statement is "If two numbers are not equal, then their squares are not equal." Let's define:

• $p$: two numbers are not equal
• $q$: their squares are not equal

Therefore, the given statement is in the form $p \rightarrow q$.

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

• $\sim p$: two numbers are equal.
• $\sim q$: their squares are equal.

Step 3: Form the contrapositive statement.

The contrapositive of $p \rightarrow q$ is $\sim q \rightarrow \sim p$. Substituting the negations we found in Step 2, we get:

"If the squares of two numbers are equal, then the numbers are equal."

Step 4: Compare the contrapositive statement to the given options.

The contrapositive we found matches option (B): "If the squares of two numbers are equal, then the numbers are equal."

Common Mistakes & Tips

• Remember the order of the contrapositive. It's $\sim q \rightarrow \sim p$, not $\sim p \rightarrow \sim q$.
• Be careful when negating statements, especially those involving "equal" or "not equal." The negation of "not equal" is "equal," and vice versa.
• The contrapositive of a statement is logically equivalent to the original statement. They always have the same truth value.

Summary

The problem asks for the contrapositive of the statement "If two numbers are not equal, then their squares are not equal." We identified the hypothesis (p) and conclusion (q), negated both, and then formed the contrapositive statement ($\sim q \rightarrow \sim p$). This resulted in the statement "If the squares of two numbers are equal, then the numbers are equal," which corresponds to option (B).

Final Answer

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