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 $\ds \bigcap_{\alpha \mathop \in I} A_\alpha$ 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:
The result follows from Associative Commutative Idempotent Operation is Self-Distributive.
$\blacksquare$