Chapter 5.4, Problem 26PPS

To determine

The solution of the statement of indirect proof.

Explanation of Solution

Given information:

Given: $xy$ is an even integer.

Prove: $x$ and $y$ is an even integer.

Calculation:

In an indirect proof or proof by contradiction, assume that the conclusion is false. By showing the assumption to be logically impossible, prove the assumption is false and the original conclusion is true. For this problem, assume that $x$ or $y$ are odd integer.

Given: $xy$ is an even integer.

Prove: $x$ and $y$ is an even integer.

Following steps are used for the indirect proof are as shown below:

Firstly, assume that $x$ and $y$ are odd integers.

Secondly, let $x=2n+1$ and $y=2k+1$ for some integer $n$ and $k$ .

  $\begin{array}{l}xy=(2n+1)(2k+1)\\ =4nk+2n+2k+1\\ =2(2nk+n+k)+1\end{array}$

Since $k$ and $n$ are integers, $2nk+n+k$ is also an integer. Let $p$ represents the integer $2nk+n+k$ . So $xy$ can be represented by $2p+1$ , where $p$ is an integer. This means that $xy$ is an odd integer, but this contradicts the given that $xy$ is an odd integer, but this contradicts the given that $xy$ is an even integer.

Since the assumption that $x$ and $y$ are odd integer leads to a contradiction of the given, the original conclusion that $x$ or $y$ is an even integers must be true.