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Suppose $F$:$X$ $\rightarrow$ $Y$ is onto. Prove that for all subsets $B \subseteq Y$, $F(F^{-1}(B)) = B$

I have no idea where to begin. Can someone please provide a hint or a guide? Thank you so very much. I am in utmost gratitude to you all. If you can, please provide any sorts of hints in the answering box so I can give you credit/check. Thank you once again

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    $\begingroup$ The two paragraphs don't seem to be connected. Are you missing something from your question? $\endgroup$ – Q the Platypus Mar 7 '16 at 5:35
  • $\begingroup$ You are right. I made a mistake. Thank you very much for correcting me. $\endgroup$ – gordon sung Mar 7 '16 at 5:41
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The notation $F(B)$ is a shorthand for $ \{y : \exists x \in B \ f(x) = y \} $ or in other words the set of all the elements of $B$ mapped through $F$.

Typically $F^{-1}$ is defined as $ F(F^{-1}(y)) = y $. So what happens to every element in a set if they are fed into $ F \circ F^-1 $ ?

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    $\begingroup$ No problem. Feel free to click on the tick mark if you think this answers your question. $\endgroup$ – Q the Platypus Mar 7 '16 at 23:27
  • $\begingroup$ good call. I have very bad ADHD so sometimes these little things fly by me. Thank you for reminding me. $\endgroup$ – gordon sung Mar 7 '16 at 23:43

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