I've been having some trouble with this question, I'm hoping someone could tell me if I'm on the right path.

Given the sets A, B, E, F, M and N where $\displaystyle A,B \subset M$ and $\displaystyle E,F \subset N$, and the function $\displaystyle f : M \rightarrow N$, show whether the following equalities hold:

a) $\displaystyle f^{-1}(E \cup F) = f^{-1}(E) \cup f^{-1}(F)$

b) $\displaystyle f(A \cup B) = f(A) \cup f(B)$

For part a I've done this:

Let $\displaystyle x \in f^{-1}(E \cup F)$

$\displaystyle \Leftrightarrow f(x) \in (E \cup F)$

$\displaystyle \Leftrightarrow f(x) \in E \vee f(x) \in F$

$\displaystyle \Leftrightarrow x \in f^{-1}(E) \vee x \in f^{-1}(F)$

$\displaystyle \Leftrightarrow x \in ( f^{-1}(E) \cup f^{-1}(F) )$

If I'm understanding this right A,B and E,F are proper subsets of M and N respectively, so the function f could map values between M and N that aren't in the union of A,B or E,F. But in part b) for instance f is being applied to a subset of M (A u B) and so any values outside this subset but still part of set M are irrelevant for the proof, right?

So is the proof for a) complete?