Chapter 5.4, Problem 23PPS

To determine

The solution of the statement of indirect proof.

Explanation of Solution

Given information:

Given: $xy$ is an odd integer.

Prove: $x$ and $y$ are both odd 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 even integer.

Given: $xy$ is an odd integer.

Prove: $x$ and $y$ are both odd integer.

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

Firstly, assume that $x$ and $y$ are not both odd integers. That is, assume that either $x$ or $y$ is an even integer.

Secondly, show that the assumption that $x$ is an even integer leads to a contradiction, since the argument for $y$ is an even integer follows the same reasoning. So, assume that $x$ is an even integer and $y$ is an odd integer. This means that $x=2k$ for some integer $k$ and $y=2m+1$ for some integer $m$ .

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

Since $k$ and $m$ are integers, $km+k$ is also an integer. Let $p$ represents the integer $km+k$ . So $xy$ is an odd integer.

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