Chapter 5.4, Problem 44HP

To determine

The statement of an indirect proof.

Explanation of Solution

Given information:

Write the contrapositive of the statement and write a direct proof of the contrapositive of the statement and the indirect proof of the statement related?

Calculation:

To prove that a statement is false, using an indirect proof, prove that a contradiction exists if the hypothesis is assumed to be true. Let assume that $x$ is not an odd integer, or in other words, an even integer, then $5x-2$ must also be even. If no contradiction exists, then the original statement must be true.

If $x$ is not an odd integer, then $5x-2$ is also even, because two subtracted from an even number is even. Therefore, the statement if $x$ is not an odd integer, then $5x-2$ is not an odd integer is true; The direct proof of the contrapositive of the statement and the indirect proof of the statement make the same assumption and reach the same conclusions.

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, consider choosing something like a definition or a theorem that can be easily proven true by contradiction.