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 .
This proves that .
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.
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