# Union Distributes over Intersection/Family of Sets

## Theorem

Let $I$ be an indexing set.

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

Let $B \subseteq S$.

Then:

$\displaystyle \map {\bigcap_{\alpha \mathop \in I} } {A_\alpha \cup B} = \paren {\bigcap_{\alpha \mathop \in I} A_\alpha} \cup B$

where $\displaystyle \bigcap_{\alpha \mathop \in I} A_\alpha$ denotes the intersection 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:

$\displaystyle \bigcap_{\tuple{\alpha, \beta} \mathop \in I \times J} \paren {A_\alpha \cup B_\beta} = \paren {\bigcap_{\alpha \mathop \in I} A_\alpha} \cup \paren {\bigcap_{\beta \mathop \in J} B_\beta}$

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

## Proof

 $\displaystyle x$ $\in$ $\displaystyle \map {\bigcap_{\alpha \mathop \in I} } {A_\alpha \cup B}$ $\displaystyle \leadsto \ \$ $\displaystyle \forall \alpha \in I: \ \$ $\displaystyle x$ $\in$ $\displaystyle A_\alpha \cup B$ Intersection is Subset $\displaystyle \leadsto \ \$ $\displaystyle \forall \alpha \in I: \ \$ $\displaystyle x$ $\in$ $\displaystyle A_\alpha$ Definition of Set Union $\, \displaystyle \lor \,$ $\displaystyle x$ $\in$ $\displaystyle B$ $\displaystyle \leadsto \ \$ $\displaystyle x$ $\in$ $\displaystyle \paren {\bigcap_{\alpha \mathop \in I} A_\alpha}$ Definition of Intersection of Family $\, \displaystyle \lor \,$ $\displaystyle x$ $\in$ $\displaystyle B$ $\displaystyle \leadsto \ \$ $\displaystyle x$ $\in$ $\displaystyle \paren {\bigcap_{\alpha \mathop \in I} A_\alpha} \cup B$ Definition of Set Union

By definition of subset:

$\displaystyle \map {\bigcap_{\alpha \mathop \in I} } {A_\alpha \cup B} \subseteq \paren {\bigcap_{\alpha \mathop \in I} A_\alpha} \cup B$

$\Box$

 $\displaystyle x$ $\in$ $\displaystyle \paren {\bigcap_{\alpha \mathop \in I} A_\alpha} \cup B$ $\displaystyle \leadsto \ \$ $\displaystyle x$ $\in$ $\displaystyle \paren {\bigcap_{\alpha \mathop \in I} A_\alpha}$ Definition of Set Union $\, \displaystyle \lor \,$ $\displaystyle x$ $\in$ $\displaystyle B$ $\displaystyle \leadsto \ \$ $\displaystyle \forall \alpha \in I: \ \$ $\displaystyle x$ $\in$ $\displaystyle A_\alpha$ Intersection is Subset $\, \displaystyle \lor \,$ $\displaystyle x$ $\in$ $\displaystyle B$ $\displaystyle \leadsto \ \$ $\displaystyle \forall \alpha \in I: \ \$ $\displaystyle x$ $\in$ $\displaystyle A_\alpha \cup B$ Definition of Set Union $\displaystyle \leadsto \ \$ $\displaystyle x$ $\in$ $\displaystyle \map {\bigcap_{\alpha \mathop \in I} } {A_\alpha \cup B}$ Definition of Intersection of Family

By definition of subset:

$\displaystyle \paren {\bigcap_{\alpha \mathop \in I} A_\alpha} \cup B \subseteq \map {\bigcap_{\alpha \mathop \in I} } {A_\alpha \cup B}$

$\Box$

By definition of set equality:

$\displaystyle \map {\bigcap_{\alpha \mathop \in I} } {A_\alpha \cup B} = \paren {\bigcap_{\alpha \mathop \in I} A_\alpha} \cup B$

$\blacksquare$