Key Concepts and Formulas

• Conditional Statement: A statement of the form "If p, then q," denoted as $p \rightarrow q$.
• Contrapositive: The contrapositive of the statement $p \rightarrow q$ is $\sim q \rightarrow \sim p$, which is logically equivalent to the original statement.
• Negation: The negation of a statement reverses its truth value. For example, the negation of "n is odd" is "n is not odd," which is equivalent to "n is even." The negation of "n³ - 1 is even" is "n³ - 1 is not even," which is equivalent to "n³ - 1 is odd."

Step-by-Step Solution

Step 1: Identify the hypothesis and conclusion.

The given statement is "If n³ – 1 is even, then n is odd." Let $p$ be the statement "n³ – 1 is even" and $q$ be the statement "n is odd." So, the given statement can be written as $p \rightarrow q$.

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

We need to find $\sim p$ and $\sim q$. $\sim p$ is the negation of "n³ – 1 is even," which is "n³ – 1 is not even," or equivalently, "n³ – 1 is odd." $\sim q$ is the negation of "n is odd," which is "n is not odd," or equivalently, "n is even."

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 n is even, then n³ – 1 is odd."

Step 4: Compare the result with the given options.

The contrapositive statement we found is "For an integer n, if n is even, then n³ – 1 is odd." This matches option (D). However, the "Correct Answer" provided states that the correct answer is A, "For an integer n, if n is even, then n 3 – 1 is even." This is incorrect. The contrapositive must negate both the hypothesis and conclusion.

The given solution incorrectly concludes with "For an integer n, if n is even, then n 3 – 1 is odd." This matches option (D). However, the "Correct Answer" provided states that the correct answer is A, "For an integer n, if n is even, then n 3 – 1 is even." This is incorrect based on the definition of contrapositive.

Step 5: Re-evaluate the options.

The question asks for the contrapositive of the statement: "For an integer n, if n³ – 1 is even, then n is odd." Let p: n³ – 1 is even. Let q: n is odd. The statement is p -> q. The contrapositive is ~q -> ~p. ~q: n is not odd, i.e., n is even. ~p: n³ – 1 is not even, i.e., n³ – 1 is odd. Therefore, the contrapositive is: "For an integer n, if n is even, then n³ – 1 is odd." This matches option (D). There must be an error in the provided correct answer.

Let's test option (A). The original statement is "For an integer n, if n³ – 1 is even, then n is odd." Let's pick an even n, say n=2. Then n³ - 1 = 2³ - 1 = 8 - 1 = 7, which is odd. So, option (A) "For an integer n, if n is even, then n³ – 1 is even" is false for n=2. This confirms option (D) is the contrapositive.

Common Mistakes & Tips

• Remember to negate both the hypothesis and the conclusion when forming the contrapositive.
• Be careful with negating statements involving "even" and "odd." The negation of "even" is "not even," which is equivalent to "odd," and vice versa.
• Double-check your work to ensure you have correctly identified the hypothesis, conclusion, and their negations.

Summary

To find the contrapositive of the given statement, we first identified the hypothesis and conclusion. Then, we found the negations of both. Finally, we formed the contrapositive statement by reversing the order and negating both parts. The contrapositive of "For an integer n, if n³ – 1 is even, then n is odd" is "For an integer n, if n is even, then n³ – 1 is odd," which corresponds to option (D). However, the given "Correct Answer" is option (A) which is incorrect.

Final Answer

The final answer is \boxed{D}.