Set Union is Self-Distributive

Theorem

Set union is self-distributive:

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

where $A, B, C$ are sets.

Sets of Sets

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

Then:

$\ds \bigcup \paren {A \cup B} = \paren {\bigcup A} \cup \paren {\bigcup B}$

where $\ds \bigcup A$ denotes the union 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:

$\ds \map {\bigcup_{\alpha \mathop \in I} } {A_\alpha \cup B_\alpha} = \paren {\bigcup_{\alpha \mathop \in I} A_\alpha} \cup \paren {\bigcup_{\alpha \mathop \in I} B_\alpha}$

where $\ds \bigcup_{\alpha \mathop \in I} A_\alpha$ denotes the union of $\family {A_\alpha}_{\alpha \mathop \in I}$.

General Result

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

Then:

$\ds \bigcup_{i \mathop \in I} \bigcup \mathbb S_i = \bigcup \bigcup_{i \mathop \in I} \mathbb S_i$

Proof

We have:

• Union is Associative
• Union is Commutative
• Set Union is Idempotent

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

$\blacksquare$