1. ## Family of subsets

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

I not getting family of subsets. How do I start this off. Assume x belongs to the left side???

2. You are proving that the union of images is equal the image of a union.
Because you are working with images you are using existential operators.
Therefore it is best to prove each is a subset of the other.
If $\displaystyle t \in f\left( {\bigcup\limits_\Lambda {C_\lambda } } \right)$ then $\displaystyle \left( {\exists x \in \bigcup\limits_\Lambda {C_\lambda } } \right)\left[ {f(x) = t} \right]$. Then proceed to show that $\displaystyle t \in \bigcup\limits_\Lambda {f\left( {C_\lambda } \right)}$.

Then reverse the process.