Let $f: A \rightarrow B$ be given. Prove:
(a) For each subset $X \subset A, X \subset f^{-1}(f(X)).$
At first I didn't understand why $X$ wouldn't just be equal to $f^{-1}(f(X))$, but I guess that is the difference between an inverse image and an inverse function, right? For instance if I had $A = \{1,2,3\}, B = \{1,2\}, X = \{1,2\}, W = \{2,3\}$, and $f(x) = 1$ then $X, W \subset A$, but $f^{-1}(f(X)) = \{1,2,3\} \neq X$ and of course $X \subset f^{-1}(f(X))$, right?
So am I understanding at least that much correctly? My problem is that I don't know what I am actually supposed to write down to prove this in general, could someone please help me with that?