# Functions-Set Theory Proof that $f(C \cup D) = f(C) \cup f(D)$ [duplicate]

Possible Duplicate:
Prove $f(S \cup T) = f(S) \cup f(T)$

I'm revisiting set theory and am troubled by this question.

Let $f:A \rightarrow B$, and $C \subset A$, $D \subset A$.

Prove that $f(C \cup D) = f(C) \cup f(D)$.

Any thoughts?

## marked as duplicate by Martin Sleziak, Stefan Hansen, Fabian, user642796, Old JohnJan 24 '13 at 10:06

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## 2 Answers

I'll show $\subseteq$. Let $y\in f(C\cup D)$. Then there exists an $x\in C\cup D$ such that $f(x)=y$. This means $x\in C$ or $x\in D$, hence $f(x)\in f(C)$ or $f(x)\in f(D)$. This implies $f(x)\in f(C)\cup f(D)$ and we've established $$f(C\cup D)\subseteq f(C)\cup f(D).$$ Approach the other containment in a similar manner.

if $f : A \to B$ and $E \subset A$ then by definition $\{ f(e) \in B | e \in E \}$.

So you want to prove

$$\{ f(e) \in B | e \in C \cap D \} = \{ f(e) \in B | e \in C \} \cap \{ f(e) \in B | e \in D \}$$

so

$$\{ f(e) \in B | e \in C \cap D \} = \{ f(e) \in B | e \in C \& e \in D \}$$

done