# Proving Surjections Problem

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

• The two paragraphs don't seem to be connected. Are you missing something from your question? – Q the Platypus Mar 7 '16 at 5:35
• You are right. I made a mistake. Thank you very much for correcting me. – gordon sung Mar 7 '16 at 5:41

## 1 Answer

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 $F \circ F^-1$ ?

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