Union is Associative/Family of Sets

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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:

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

$\blacksquare$


Also see


Sources