functions - Proving the Equality if f(U and V) = f(U) and f(V)

If $f:X \to Y$ is a function and $U$ and $V$ are subsets of $X$, then $f(U \cap V) = f(U) \cap f(V)$.

I am a little lost on this proof. I believe it to be true, but I am uncertain as to where to start. Any solutions would be appreciated. I have many similar proofs to prove and I would love a complete one to base my further proofs on.

Answer

You can't prove it, since it is false. Take a set $X$ with more than one point, take $x,x'\in X$ with $x\neq x'$ and let $f$ be a constant function. Then $$f\bigl(\{x\}\cap\{x’\}\bigr)=f(\emptyset)=\emptyset\neq f\bigl(\{x\}\bigr)\cap f\bigl(\{x'\}\bigr).$$

Actually, the assertion that you want to prove is equivalent to the assertion that $f$ is injective.