Question

Prove the following statements with contrapositive proof. (In each case, think about how a direct proof would work. In most cases contrapositive is easier.) Suppose \(n \in \mathbb{Z}\). If \(n^{2}\) is odd, then \(n\) is odd.

Short answer

The statement 'if \(n^{2}\) is odd, then \(n\) is odd' is proven to be true by showing that its contrapositive 'if \(n\) is even, then \(n^{2}\) is even' is true.

Step-by-step solution

Understand the Statement

The initial statement is 'if \(n^{2}\) is odd, then \(n\) is odd'. The contrapositive of a statement 'if P then Q' is 'if not Q then not P'. Thus, the contrapositive of the given statement is: 'if \(n\) is not odd (which means \(n\) is even), then \(n^{2}\) is not odd (meaning it's even)'.

Prove the Contrapositive Statement

To prove this, let's assume that \(n\) is even. This means that \(n\) can be written in the form \(2k\), where \(k\) is an integer. If \(n\) is even, then \(n^{2}\) is \( (2k)^{2} = 4k^{2} = 2(2k^{2})\), is also even, since it's a multiple of 2.

Final Step: Assertion

Therefore, the contrapositive statement: 'if \(n\) is even then \(n^{2}\) is even', was proven true. This shows that the original statement 'if \(n^{2}\) is odd, then \(n\) is odd' is also true, because a statement and its contrapositive are logically equivalent.