Let $f: X \rightarrow Y$ be a function from $X$ to $Y$. Let $S$ be a subset of $X$, and $U$ be a subset of $Y$.
(iii) In general, what's the relation between $f^{-1}f^{-1}(U)$ and $U$?
Need help for 3.4.2 (iii). My solutions to part(i) and part(ii) are in general that there are no relation except in certain cases such as when f is bijective.
For part(iii), I thought it might be something similar but I tried to think of a couple of counterexamples and they all failed in some way or another so I thought that it may be the case that for (iii) both sets are always the same. I tried to proof it as below but got stuck and am not sure how to proceed. If f is injective, then clearly it works but in general im not sure how to proceed.
Let $x$ be an element of $f^{-1}(f(f^{-1}(U)))$, then there must exist $y \in f(f^{-1}(U))$, such that $f(x) = y$. For such $y$ to exist, there must also exist some $z \in f^{-1}(U)$, such that $f(z) = y$.
Best Answer
They are the same set for an arbitrary $f$.
The inclusion $f^{-1}(U)\subseteq f^{-1}(f(f^{-1}(U)))$ is a consequence of what you already have seen in (i).
Conversely, let $x\in f^{-1}(f(f^{-1}(U)))$. Thus, $f(x)\in f(f^{-1}(U))$ so there exists $z\in f^{-1}(U)$ such that $f(x)=f(z)$. Now, since $z\in f^{-1}(U)$ then $f(z)\in U$, hence $f(x)\in U$. Therefore, $x\in f^{-1}(U)$.