A function of the inverse function of a set contained in a set.

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I'm doing an intro course on set theory and need to prove:
Suppose that $$ f : X \rightarrow Y, A \subset X, B \subset Y $$ Prove that:
a) $ f(f^{-1}(B)) \subset B $ and give an example where equality does not hold
b) $ A \subset f^{-1}(f(A)) $ and give an example where equality does not hold.

I have an idea that this has something to do with surjections, injections and bijections since the the function is not defined explicitly as any one, but I'm lost....

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2 Answers

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Let $y \in f(f^{-1}(B))$ Then clearly, $y = f(x)$ for some $x \in f^{-1}(B)$. Now, by definition of $f^{-1}(B)$, this means that $y \in B$.

For the second part, let $a \in A$. Then let $b = f(a)$. What can you say about $f^{-1}(f\{b\})$?

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For a counterexample to (a), consider a function not surjective, eg. take $B=\mathbb R$ but $f=\sin\colon\mathbb R\to\mathbb R$. Then $f(f^{-1}(B))=f(\mathbb R)=[-1,1]$.

For a counterexample to (b), consider a non-injective function (for instance, take it to be periodic), eg. take $A=[0,2\pi]$ but $f=\sin$ (again). Then $f^{-1}(f(A))=f^{-1}([-1,1])=\mathbb R$.

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