# Set Union is Self-Distributive/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:

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

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

## Proof

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

Thus by definition of subset:

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

$\Box$

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

Thus by definition of subset:

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

$\Box$

By definition of set equality:

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

$\blacksquare$