Math 310-01/02

11-17-2017

1. Define by

Prove that .

First, suppose , where . I need to show that . That is, I must show .

Suppose on the contrary that . Then

This contradiction shows that , so .

Conversely, suppose , so .

First, . For if , then

This contradiction shows , so .

Now

This proves that .

Hence, .

2. Define by

Prove that f is injective, but f is not surjective.

If , then , so . Since for all x and y, this is a contradiction. Therefore, f is not surjective.

Suppose . Then

Taking logs on both sides of , I get . Adding this equation to , I get , so . This allows me to cancel a and c from to get . Therefore, , and f is injective.

*Without work, all life goes rotten. But when work is soulless,
life stifles and dies.* - *Albert Camus*

Copyright 2017 by Bruce Ikenaga