1
$\begingroup$

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.


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.

  • 2
    If $y\in f(A\cap B)$, what does that mean? What does it mean if $x\in f^{-1}(C\cap D)$? If $x\in f^{-1}(C)\cap f^{-1}(D)$? These should follow fairly smoothly just from definitions of image, preimage, and intersection. Start by writing those out, even if it seems obvious.2012-05-14

2 Answers 2