Chapter 0, Problem 1RE

State, with a reason, whether each of the following statements is true or false.

1. If $a$ and $b$ are integers with $a-b>0$ and $b-a>0$, then $a\ne b$.

2. If $a$ and $b$ are integers with $a-b\ge 0$ and $b-a\ge 0$, then $a\ne b$.

To determine

Whether the statements given below are true of false.

If $a$ and $b$ are integers with $a-b>0$ and $b-a>0$, then $a\ne b$.

If $a$ and $b$ are integers with $a-b\ge 0$ and $b-a\ge 0$, then $a\ne b$.

Explanation of Solution

$a-b>0$ and $b-a>0$ means $a>b$ and $b>a$.

We see that the hypothesis “$a$ and $b$ are integers with $a-b>0$ and $b-a>0$”is always false.

Thus, the implication is true.

$a-b\ge 0$ and $b-a\ge 0$ means $a\ge b$ and $b\ge a$.

That is, $a=b$.

Thus, the given implication is false.