TITLE: Maps - question about $f(A \cup B)=f(A) \cup f(B)$ and $ f(A \cap B)=f(A) \cap f(B)$
QUESTION [6 upvotes]: I am struggling to prove this map statement on sets. 
The statement is: 
Let $f:X \rightarrow Y$ be a map. 
i) $\forall_{A,B \subset X}: f(A \cup B)=f(A) \cup f(B)$
ii) $\forall_{A,B \subset X}: f(A \cap B) \subset f(A) \cap f(B)$
iii) $f$ is injective $ \Longleftrightarrow$ $\forall_{A,B \subset X}: f(A \cap B)=f(A) \cap f(B)$
My problem is: I know how to operate on sets, I know how to operate on sets, but I don't know how and where to start the proof,  the biggest problem in mathematics, I think. 
Thanks for help!

REPLY [2 votes]: Recall that $y\in f(C)\iff \exists x\in C:f(x)=y$. Using this you should be able to show that $y\in f(A\cup B)\iff y\in f(A)\text{ or }y\in f(B)\iff y\in f(A)\cup f(B)$ and so on.