# Union is Associative/Family of Sets

## Theorem

Let $\family {S_i}_{i \mathop \in I}$ and $\family {I_\lambda}_{\lambda \mathop \in \Lambda}$ be indexed families of sets.

Let $\displaystyle I = \bigcup_{\lambda \mathop \in \Lambda} I_\lambda$ denote the union of $\family {I_\lambda}_{\lambda \mathop \in \Lambda}$.

Then:

$\displaystyle \bigcup_{i \mathop \in I} S_i = \bigcup_{\lambda \mathop \in \Lambda} \paren {\bigcup_{i \mathop \in I_\lambda} S_i}$

## Proof

For every $\lambda \in \Lambda$, let $\displaystyle T_\lambda = \bigcup_{i \mathop \in I_\lambda} S_i$.

Then:

 $\displaystyle x$ $\in$ $\displaystyle \bigcup_{i \mathop \in I} S_i$ $\displaystyle \leadstoandfrom \ \$ $\, \displaystyle \exists i \in I: \,$ $\displaystyle x$ $\in$ $\displaystyle S_i$ Definition of Union of Family $\displaystyle \leadstoandfrom \ \$ $\, \displaystyle \exists \lambda \in \Lambda: \exists i \in I_\lambda: \,$ $\displaystyle x$ $\in$ $\displaystyle S_i$ $\displaystyle \leadstoandfrom \ \$ $\, \displaystyle \exists \lambda \in \Lambda: \,$ $\displaystyle x$ $\in$ $\displaystyle \bigcup_{i \mathop \in I_\lambda} S_i = T_\lambda$ $\displaystyle \leadstoandfrom \ \$ $\displaystyle x$ $\in$ $\displaystyle \bigcup_{\lambda \mathop \in \Lambda} T_\lambda$

Thus:

$\displaystyle \bigcup_{i \mathop \in I} S_i = \bigcup_{\lambda \mathop \in \Lambda} T_\lambda = \bigcup_{\lambda \mathop \in \Lambda} \paren {\bigcup_{i \mathop \in I_\lambda} S_i}$

$\blacksquare$