Intersection Distributes over Intersection

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Theorem

Set intersection is distributive over itself:

$\forall A, B, C: \left({A \cap B}\right) \cap \left({A \cap C}\right) = A \cap B \cap C = \left({A \cap C}\right) \cap \left({B \cap C}\right)$

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.

$\blacksquare$