Here is the full question:
Let $f : X → Y$ be a function from one set $X$ to another set $Y$, let $S$ be a subset of $X$, and let $U$ be a subset of $Y$. What, in general, can one say about $f^{−1}(f(S))$ and $S$? What about $f(f^{−1}(U))$ and $U$?
My reasoning:
-$f^{−1}(f(S))=S$ if and only if $f(S)$ is a one-to-one function since no two different inputs will map to one output, so we will always get our original input when we reverse the process.
-$f(f^{−1}(U))=U$ if and only if $f(U)$ is a bijective function, this way we can guarantee and there is an input to "go back to" when we reverse the process AND we will go back to the correct input (since it's unique).
Why I posted this: I feel like I am missing something in the question, I am new to analysis and always pretty unsure about my reasoning especially that there is no solutions available for this, I want to make sure about building on the correct blocks of understanding. If someone can help me point out my mistakes if there is any or suggest improvements about the way of writing proofs (I guess I use too much English) I would be grateful.
Since $S$ and $U$ are fixed, your statements are actually false. That is, the following are true:
However, since $S$ and $U$ are fixed in the problem statement, we could have $f^{-1}(f(S)) = S$ without $f$ being injective, and we could have $f(f^{-1}(U))=U$ without $f$ being surjective.
Instead, this type of question, usually, is asking you to verify which of the following are necessarily true:
As a final note, for a fixed $S$ and $U$ we still have the direction of
If you want to additionally prove those, I'm sure your instructor wouldn't mind, as the question is certainly open-ended.