Set Intersection is Self-Distributive

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Theorem

Set intersection is self-distributive:

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

where $A, B, C$ are sets.


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:

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

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


General Result

Let $\family {\mathbb S_i} _{i \mathop \in I}$ be an $I$-indexed family of sets of sets.

Then:

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


Proof

We have:

Intersection is Associative
Intersection is Commutative
Set Intersection is Idempotent

The result follows from Associative Commutative Idempotent Operation is Self-Distributive.

$\blacksquare$