Definition:Set Union/Set of Sets

Definition
Let $\mathbb S$ be a set of sets.

The union of $\mathbb S$ is:
 * $\displaystyle \bigcup \mathbb S := \set {x: \exists X \in \mathbb S: x \in X}$

That is, the set of all elements of all elements of $\mathbb S$.

Remark: The existence of $\displaystyle \bigcup \mathbb S$ is an independent axiom.

Thus the general union of two sets can be defined as:
 * $\displaystyle \bigcup \set {S, T} = S \cup T$

Also denoted as
Some sources denote $\displaystyle \bigcup \mathbb S$ as $\displaystyle \bigcup_{S \mathop \in \mathbb S} S$.

Also see

 * Union of Doubleton for a proof that $\displaystyle \bigcup \set {S, T} = S \cup T$


 * Definition:Intersection of Set of Sets