Chapter 5.4, Problem 21PPS

To determine

The solution of the statement of indirect reasoning.

Explanation of Solution

Given information:

Kwan-Yong bought two computer games for just over $\$80$ before tax. A few weeks later, his friend asked how much each game cost. Kwan-Yong could not remember the individual prices. Use indirect reasoning to show that at least one of the games cost more than $\$40$ .

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.

Let the cost of one game be $x$ and the other be $y$ .

Given: $x+y>80$

Prove: $x>40\text{ or }y>40$

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

Firstly, assume that $x\le 40$ and $y\le 40$ .

Secondly, if $x\le 40$ and $y\le 40$ , then $x+y\le 40+40\text{ or }x+y\le 80$ . This is a contradiction because the assumption is $x+y>80$ .

Since the assumption that $x\le 40$ and $y\le 40$ leads to a contradiction of a known fact, the assumption must be false. Therefore, the conclusion that $x\le 40$ and $y\le 40$ must be true. Thus, at least one of the games had to cost more than $\$40$ .