# Set Theory proofs

1. Sep 7, 2004

### Ed Quanta

Let f:X->Y be a function

1) Given any subset B of Y, prove that f(f^-1(B)) is a subset of B

2) Prove that f(f^-1(B))=B for all subsets B of Y if and only if f is surjective

Help anybody?

2. Sep 7, 2004

### Hurkyl

Staff Emeritus
(If this is homework, you should post in the HW help section... lemme know and I'll move it)

Sometimes, problems become more clear just by restating it.

Note that your goal is to prove:

If x is in f(f^-1(B)) then x is in B.

So what is the criterion for x to be in f(f^-1(B))?

Ask this question a few times, and I think it solves itself.

I think the theorem and proof of (1) will provide some insight. Also, you might consider what happens if either of these conditions fails.

In the end, I again think it will almost solve itself if you dig into more detail.

3. Sep 7, 2004

### Ed Quanta

It's not homework, just some problems in my topology book that I have been thinking about. My problem with 1) which I should have stated earlier is that I don't see why f(f^-1(B)) is a subset of B, and not simply equal to it.

4. Sep 7, 2004

### Hurkyl

Staff Emeritus
Well, I think your second question gives a strong clue as to how to find an example where f(f^1(B)) != B.

It doesn't have to be complicated; try something very simple, like a function whose domain has only 1 or 2 elements, and is not surjective.