prove $f^{-1}(C \cap D) = f^{-1}(C) \cap f^{-1}(D)$ I need help with this proof:
$f: X\rightarrow Y$
$C,D\subseteq Y$
$f^{-1}(C \cap D) = f^{-1}(C) \cap f^{-1}(D)$
Thanks.

To solve these kinds of questions, you want to show that $$f^{-1}(C\cap D) \subseteq f^{-1}(C) \cap f^{-1}(D)$$ and $$f^{-1}(C\cap D) \supseteq f^{-1}(C) \cap f^{-1}(D).$$
I will show $\subseteq$ here; you should try the other direction.
Suppose $x \in f^{-1} (C \cap D)$. This means $f(x) \in (C\cap D)$, which further implies $f(x) \in C$ AND $f(x) \in D$. Thus, $x \in f^{-1}(C)$ AND $x \in f^{-1}(D)$.
