Definition:Set Union/Set of Sets

Definition
Let $I$ be an indexing set.

Let $\left \langle {X_i} \right \rangle_{i \mathop \in I}$ be a family of subsets of a set $S$.

Then the union of $\left \langle {X_i} \right \rangle$ is defined as:


 * $\displaystyle \bigcup_{i \mathop \in I} X_i = \left\{{y: \exists i \in I: y \in X_i}\right\}$

If the indexing set is clear from context, the notation $\displaystyle \bigcup_i X_i$ can be used.

The indexing set itself can be disposed of, as follows:

If $\mathbb S$ is a set of sets, then the union of $\mathbb S$ is:
 * $\displaystyle \bigcup \mathbb S = \left\{{x: \exists X \in \mathbb S: x \in X}\right\}$

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

Thus:
 * $\displaystyle S \cup T = \bigcup \left\{{S, T}\right\}$