1. ## Family of subsets

Prove: If $\displaystyle A \rightarrow B$ is a function and { $\displaystyle C_\lambda \mid \lambda \epsilon \Lambda$ } is a family of subsets of A, then $\displaystyle f ( \bigcup_{\lambda \epsilon \Lambda} C_\lambda) = \bigcup_{\lambda \epsilon \Lambda} f (C_\lambda)$

2. Originally Posted by tigergirl
Prove: If $\displaystyle A \rightarrow B$ is a function and { $\displaystyle C_\lambda \mid \lambda \epsilon \Lambda$ } is a family of subsets of A, then $\displaystyle f ( \bigcup_{\lambda \epsilon \Lambda} C_\lambda) = \bigcup_{\lambda \epsilon \Lambda} f (C_\lambda)$
You need to show some of your own effort.
1) What does it mean to say $\displaystyle s \in f\left( {\bigcup\limits_{\lambda \in \Lambda } {C_\lambda } } \right)?$
2) What does it mean to say $\displaystyle t \in \bigcup\limits_{\lambda \in \Lambda } {f(C_\lambda )}?$
3. Basically, the problem is to show that $\displaystyle f(C_1\cup C_2)=f(C_1)\cup f(C_2)$. Both inclusions are pretty obvious. I don't think the extension to infinite families $\displaystyle \Lambda$ (from $\displaystyle \Lambda=\{1,2\}$ just considered) presents any problems.