Let $f: S \rightarrow T.$ If $B \subseteq T$ and $f^{-1}(B) = \{s \in S| f(s) \in B \}.$
Give short answers to these questions, provide, where appropriate, simple examples to support your answers.
1- If $B \neq \emptyset,$ can it be that $f^{-1}(B) = \emptyset $?
2- Can it happen that $f^{-1}(B) = S $?
3- Is it true that $f(f^{-1}(B)) = B $?
4- Is it true that $f^{-1}(f(A)) = A?$
** My trials:**
1-No, for example if $B = \{ 1\}$ then we are sure by the definition of $f^{-1}(B)$ we are sure that there exists $s \in S$ such that $f(s) = 1 \in B$, hence $f^{-1}(B) \neq \emptyset .$
2- yes, but I am unable to build an example, my difficulty is in choosing the function definition, could anyone help me in this?
3- & 4- I do not know how to think about them, I am caring about teaching me to think about the question not about the final answer, could anyone help me in these 2 questions please?
Your answer to 1. is incorrect. Just consider a function which is not surjective, and let $B$ be a set of elements that is disjoint from the range of $f$.
Regarding 2, the answer is yes, just take $B=T$.
3 and 4 are false in general. To get a counterexample of 3, just choose any $B$ that contains an element not belonging to the image of $f$. For 4, you can get a counterexample by considering the function $f\colon \{1,2\}\to \{1\}$ and $A=\{1\}$.