# Intersection Distributes over Union/Family of Sets

## Theorem

Let $I$ be an indexing set.

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

Let $B \subseteq S$.

Then:

$\ds \map {\bigcup_{\alpha \mathop \in I} } {A_\alpha \cap B} = \paren {\bigcup_{\alpha \mathop \in I} A_\alpha} \cap B$

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

### Corollary

Let $I$ and $J$ be indexing sets.

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

Then:

$\ds \bigcup_{\tuple {\alpha, \beta} \mathop \in I \times J} \paren {A_\alpha \cap B_\beta} = \paren {\bigcup_{\alpha \mathop \in I} A_\alpha} \cap \paren {\bigcup_{\beta \mathop \in J} B_\beta}$

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

## Proof

 $\ds x$ $\in$ $\ds \bigcup_{\alpha \mathop \in I} \paren {A_\alpha \cap B}$ $\ds \leadsto \ \$ $\ds \exists \alpha \in I: \,$ $\ds x$ $\in$ $\ds A_\alpha \cap B$ Definition of Union of Family $\ds \leadsto \ \$ $\ds x$ $\in$ $\ds A_\alpha$ Definition of Set Intersection $\, \ds \land \,$ $\ds x$ $\in$ $\ds B$ $\ds \leadsto \ \$ $\ds x$ $\in$ $\ds \paren {\bigcup_{\alpha \mathop \in I} A_\alpha}$ Set is Subset of Union of Family $\, \ds \land \,$ $\ds x$ $\in$ $\ds B$ $\ds \leadsto \ \$ $\ds x$ $\in$ $\ds \paren {\bigcup_{\alpha \mathop \in I} A_\alpha} \cap B$ Definition of Set Intersection

By definition of subset:

$\ds \bigcup_{\alpha \mathop \in I} \paren {A_\alpha \cap B} \subseteq \paren {\bigcup_{\alpha \mathop \in I} A_\alpha} \cap B$

$\Box$

 $\ds x$ $\in$ $\ds \paren {\bigcup_{\alpha \mathop \in I} A_\alpha} \cap B$ $\ds \leadsto \ \$ $\ds x$ $\in$ $\ds \paren {\bigcup_{\alpha \mathop \in I} A_\alpha}$ Definition of Set Intersection $\, \ds \land \,$ $\ds x$ $\in$ $\ds B$ $\ds \leadsto \ \$ $\ds \exists \alpha \in I: \,$ $\ds x$ $\in$ $\ds A_\alpha$ Definition of Union of Family $\, \ds \land \,$ $\ds x$ $\in$ $\ds B$ $\ds \leadsto \ \$ $\ds \exists \alpha \in I: \,$ $\ds x$ $\in$ $\ds A_\alpha \cap B$ Definition of Set Intersection $\ds \leadsto \ \$ $\ds x$ $\in$ $\ds \bigcup_{\alpha \mathop \in I} \paren {A_\alpha \cap B}$ Set is Subset of Union of Family

By definition of subset:

$\ds \paren {\bigcup_{\alpha \mathop \in I} A_\alpha} \cap B \subseteq \map {\bigcup_{\alpha \mathop \in I} } {A_\alpha \cap B}$

$\Box$

By definition of set equality:

$\ds\map {\bigcup_{\alpha \mathop \in I} } {A_\alpha \cap B} = \paren {\bigcup_{\alpha \mathop \in I} A_\alpha} \cap B$

$\blacksquare$