# Intersection Distributes over Intersection

## Theorem

Set intersection is distributive over itself:

$\forall A, B, C: \paren {A \cap B} \cap \paren {A \cap C} = A \cap B \cap C = \paren {A \cap C} \cap \paren {B \cap C}$

where $A, B, C$ are sets.

### Sets of Sets

Let $A$ and $B$ denote sets of sets.

Then:

$\displaystyle \bigcap \left({A \cap B}\right) = \left({\bigcap A}\right) \cap \left({\bigcap B}\right)$

where $\displaystyle \bigcap A$ denotes the intersection of $A$.

### Families of Sets

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 $\family {A_\alpha}$.

### General Result

Let $\left\langle{\mathbb S_i}\right\rangle_{i \in I}$ be an $I$-indexed family of sets of sets.

Then:

$\displaystyle \bigcap_{i \mathop \in I} \bigcap \mathbb S_i = \bigcap \bigcap_{i \mathop \in I} \mathbb S_i$

## Proof

We have:

Intersection is Associative
Intersection is Commutative
Intersection is Idempotent

The result follows from Associative Commutative Idempotent Operation is Distributive over Itself.

$\blacksquare$