Question

let f be a function X to Y prove that "f" is one to one function if and only if f(A∩B)=f(A)∩f(B) for all subsets "A" and "B" of X. WHen S is a set,we define f(S)={f(x)|x∈S}

Answer 1

Can you say how far you've gotten?
Some things that might help generally:
1) To prove "M if and only if K" you have to prove M->K and K->M
2) Try using your simplest proof techniques first (direct proof, then proof by contrapositive, contradiction, etc.) many different techniques may work it's just a matter of finding the easiest one.
3) Use the last expression to expand the earlier one. That might make the proof easier. You could also try making up actual sets and both a one-to-one function and a non-one-to-one function to see how the proof might work (remember to switch back into using variables before writing your actual proof, otherwise it will only prove the case for an instance, lacking rigor and generality).

Answer 2

TY lhm. that helped me solve the problem

Answer 3

uhhh well TY lhm can solve the peoblem