Proof involving surjective/onto and image/preimage of sets

[mathmajor2013]

EXERCISE: Suppose f is surjective, and B is a subset of Y. Prove that f(f^-1(B))=B.

SOLUTION: We must show that f(f^-1(B)) is a subset of B and that B is a subset of f(f^-1(B)). I have already proven that f(f^-1(B)) is a subset of B. Now I must prove that B is a subset of f(f^-1(B)) when f is surjective. Fix x is an element of B.

After this I am lost. Help please! I know that the surjectivity must come in handy at some point since B is not a subset of f(f^-1(B)) for all f.

 

[llello]

Well

$$f^{-1} : B \rightarrow Y$$

So let x be in B as you said. Now $$f^{-1}(x)$$ is an element of Y provided that there is some y in Y such that $$f^{-1}(x) = y$$. This is where you need to use surjectivity. Hope this helps.