# Union Distributes over Union/Families of Sets

## 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:

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

$\blacksquare$