Union Distributes over Union/Families of Sets

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Theorem

Let $I$ be an indexing set.

Let $\family {A_\alpha}_{\alpha \mathop \in I}$ and $\family {B_\alpha}_{\alpha \mathop \in I}$ be indexed families of subsets of a set $S$.


Then:

$\displaystyle \map {\bigcup_{\alpha \mathop \in I} } {A_\alpha \cup B_\alpha} = \paren {\bigcup_{\alpha \mathop \in I} A_\alpha} \cup \paren {\bigcup_{\alpha \mathop \in I} B_\alpha}$

where $\displaystyle \bigcup_{\alpha \mathop \in I} A_\alpha$ denotes the union of $\family {A_\alpha}_{\alpha \mathop \in I}$.


Proof

\(\displaystyle x\) \(\in\) \(\displaystyle \map {\bigcup_{\alpha \mathop \in I} } {A_\alpha \cup B_\alpha}\)
\(\displaystyle \leadsto \ \ \) \(\displaystyle \exists \beta \in I: \ \ \) \(\displaystyle x\) \(\in\) \(\displaystyle A_\beta \cup B_\beta\) Definition of Union of Family
\(\displaystyle \leadsto \ \ \) \(\displaystyle x\) \(\in\) \(\displaystyle A_\beta\) Definition of Set Union
\(\, \displaystyle \lor \, \) \(\displaystyle x\) \(\in\) \(\displaystyle B_\beta\)
\(\displaystyle \leadsto \ \ \) \(\displaystyle x\) \(\in\) \(\displaystyle \bigcup_{\alpha \mathop \in I} A_\alpha\) Set is Subset of Union
\(\, \displaystyle \lor \, \) \(\displaystyle x\) \(\in\) \(\displaystyle \bigcup_{\alpha \mathop \in I} B_\alpha\) Set is Subset of Union
\(\displaystyle \leadsto \ \ \) \(\displaystyle x\) \(\in\) \(\displaystyle \paren {\bigcup_{\alpha \mathop \in I} A_\alpha} \cup \paren {\bigcup_{\alpha \mathop \in I} B_\alpha}\) Definition of Set Union

Thus by definition of subset:

$\displaystyle \map {\bigcup_{\alpha \mathop \in I} } {A_\alpha \cup B_\alpha} \subseteq \paren {\bigcup_{\alpha \mathop \in I} A_\alpha} \cup \paren {\bigcup_{\alpha \mathop \in I} B_\alpha}$

$\Box$


\(\displaystyle x\) \(\in\) \(\displaystyle \paren {\bigcup_{\alpha \mathop \in I} A_\alpha} \cup \paren {\bigcup_{\alpha \mathop \in I} B_\alpha}\)
\(\displaystyle \leadsto \ \ \) \(\displaystyle x\) \(\in\) \(\displaystyle \bigcup_{\alpha \mathop \in I} A_\alpha\) Definition of Set Union
\(\, \displaystyle \lor \, \) \(\displaystyle x\) \(\in\) \(\displaystyle \bigcup_{\alpha \mathop \in I} B_\alpha\)
\(\displaystyle \leadsto \ \ \) \(\displaystyle \exists \beta \in I: \ \ \) \(\displaystyle x\) \(\in\) \(\displaystyle A_\beta\) Definition of Union of Family
\(\displaystyle \exists \beta \in I: \ \ \) \(\, \displaystyle \lor \, \) \(\displaystyle x\) \(\in\) \(\displaystyle B_\beta\)
\(\displaystyle \leadsto \ \ \) \(\displaystyle \exists \beta \in I: \ \ \) \(\displaystyle x\) \(\in\) \(\displaystyle A_\beta \cup B_\beta\) Definition of Union of Family
\(\displaystyle \leadsto \ \ \) \(\displaystyle x\) \(\in\) \(\displaystyle \map {\bigcup_{\alpha \mathop \in I} } {A_\alpha \cup B_\alpha}\)

Thus by definition of subset:

$\displaystyle \paren {\bigcup_{\alpha \mathop \in I} A_\alpha} \cup \paren {\bigcup_{\alpha \mathop \in I} B_\alpha} \subseteq \map {\bigcup_{\alpha \mathop \in I} } {A_\alpha \cup B_\alpha}$

$\Box$


By definition of set equality:

$\displaystyle \map {\bigcup_{\alpha \mathop \in I} } {A_\alpha \cup B_\alpha} = \paren {\bigcup_{\alpha \mathop \in I} A_\alpha} \cup \paren {\bigcup_{\alpha \mathop \in I} B_\alpha}$

$\blacksquare$


Sources