Chapter 5.4, Problem 8CYU

To determine

The solution of statement assumption of the indirect proof.

Explanation of Solution

Given information:

Write an indirect proof to show that if $5x-2$ is an odd integer, then $x$ is an odd integer.

Calculation:

Given: $5x-2$ is an odd integer Prove: $x$ is an odd integer

Assume that $x$ is not an odd integer. That is, assume that $x$ is an even integer.

Let $x=2k$ for some integer $k$ .

  $\begin{array}{l}5x-2=5(2k)-2\\ \text{=10}k-2\\ =2(5k-1)\end{array}$

Since $k$ is an integer, $5k-1$ is also an integer. Let $p$ represent the integer $5k-1$ . So, $5k-2$ can be represented by $2p$ , where $p$ is an integer. This means that $5x-2$ is an even integer, but this contradicts the given that $5x-2$ is an odd integer.

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