functions - Verifying a proposition on image and preimage: $f(Acap B)subseteq f(A)cap f(B)$ and $f^{-1}(Ccap D)=f^{-1}(C)cap f^{-1}(D)$

I need help in verifying the following please:

Let $f:X\to Y$ be a function, and let $A,B\subseteq X$, and let $C,D\subseteq Y$. Then

• $f(A\cap B)\subseteq f(A)\cap f(B)$


• $f^{-1}(C\cap D)=f^{-1}(C)\cap f^{-1}(D)$

I am not quite certain how to get started, and would appreciate any help. Thanks.

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Edit:

Please excuse my ignorance, but I think that part of the trouble I am having is notational, since I only have this proposition from one book and the definitions of image and preimage from another. Here is what I have for the definitions:

Let $f: X \to Y$ be a function. Then the image of $A$ under $f$ is

$$f(A) := \{f(a) \in Y)(a \in A)\}$$
and the preimage of $C$ under $f$ is
$$f^{-1}(C) := \{(x \in X)(f(x) \in C)\}.$$

So are the above definitions `missing' something? Thanks again.