# Intersection Distributes over Intersection/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:

$\displaystyle \map {\bigcap_{\alpha \mathop \in I} } {A_\alpha \cap B_\alpha} = \paren {\bigcap_{\alpha \mathop \in I} A_\alpha} \cap \paren {\bigcap_{\alpha \mathop \in I} B_\alpha}$

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

## Proof

 $\displaystyle x$ $\in$ $\displaystyle \map {\bigcap_{\alpha \mathop \in I} } {A_\alpha \cap B_\alpha}$ $\displaystyle \leadsto \ \$ $\displaystyle \forall \alpha \in I: \ \$ $\displaystyle x$ $\in$ $\displaystyle A_\alpha \cap B_\alpha$ Definition of Intersection of Family $\displaystyle \leadsto \ \$ $\displaystyle \forall \alpha \in I: \ \$ $\displaystyle x$ $\in$ $\displaystyle A_\alpha$ Definition of Set Intersection $\, \displaystyle \land \,$ $\displaystyle x$ $\in$ $\displaystyle B_\alpha$ $\displaystyle \leadsto \ \$ $\displaystyle x$ $\in$ $\displaystyle \bigcap_{\alpha \mathop \in I} A_\alpha$ Definition of Intersection of Family $\, \displaystyle \land \,$ $\displaystyle x$ $\in$ $\displaystyle \bigcap_{\alpha \mathop \in I} B_\alpha$ Definition of Intersection of Family $\displaystyle \leadsto \ \$ $\displaystyle x$ $\in$ $\displaystyle \paren {\bigcap_{\alpha \mathop \in I} A_\alpha} \cap \paren {\bigcap_{\alpha \mathop \in I} B_\alpha}$ Definition of Set Intersection

Thus by definition of subset:

$\displaystyle \map {\bigcap_{\alpha \mathop \in I} } {A_\alpha \cap B_\alpha} \subseteq \paren {\bigcap_{\alpha \mathop \in I} A_\alpha} \cap \paren {\bigcap_{\alpha \mathop \in I} B_\alpha}$

$\Box$

 $\displaystyle x$ $\in$ $\displaystyle \paren {\bigcap_{\alpha \mathop \in I} A_\alpha} \cap \paren {\bigcap_{\alpha \mathop \in I} B_\alpha}$ $\displaystyle \leadsto \ \$ $\displaystyle x$ $\in$ $\displaystyle \bigcap_{\alpha \mathop \in I} A_\alpha$ Definition of Set Intersection $\, \displaystyle \land \,$ $\displaystyle x$ $\in$ $\displaystyle \bigcap_{\alpha \mathop \in I} B_\alpha$ $\displaystyle \leadsto \ \$ $\displaystyle \forall \alpha \in I: \ \$ $\displaystyle x$ $\in$ $\displaystyle A_\alpha$ Definition of Intersection of Family $\, \displaystyle \land \,$ $\displaystyle x$ $\in$ $\displaystyle B_\alpha$ $\displaystyle \leadsto \ \$ $\displaystyle \forall \alpha \in I: \ \$ $\displaystyle x$ $\in$ $\displaystyle A_\alpha \cap B_\alpha$ Definition of Set Intersection $\displaystyle \leadsto \ \$ $\displaystyle x$ $\in$ $\displaystyle \map {\bigcap_{\alpha \mathop \in I} } {A_\alpha \cap B_\alpha}$ Definition of Intersection of Family

Thus by definition of subset:

$\displaystyle \paren {\bigcap_{\alpha \mathop \in I} A_\alpha} \cap \paren {\bigcap_{\alpha \mathop \in I} B_\alpha} \subseteq \map {\bigcap_{\alpha \mathop \in I} } {A_\alpha \cap B_\alpha}$

$\Box$

By definition of set equality:

$\displaystyle \map {\bigcap_{\alpha \mathop \in I} } {A_\alpha \cap B_\alpha} = \paren {\bigcap_{\alpha \mathop \in I} A_\alpha} \cap \paren {\bigcap_{\alpha \mathop \in I} B_\alpha}$

$\blacksquare$