Set Theory Proofs: f:X->Y Function and Subset B of Y

[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?

 

[Hurkyl]

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

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

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.

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

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.

 

[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.

 

[Hurkyl]

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.