# Definition:Set Union/Family of Sets/Universal Set

## Definition

Let $\mathbb U$ be a universal set.

Let $I$ be an indexing set.

Let $\left \langle {S_i} \right \rangle_{i \mathop \in I}$ be an indexed family of subsets of $\mathbb U$.

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

$\displaystyle \bigcup_{i \mathop \in I} S_i := \left\{{x \in \mathbb U: \exists i \in I: x \in S_i}\right\}$

## Also denoted as

The set $\displaystyle \bigcup_{i \mathop \in I} S_i$ can also be seen denoted as:

$\displaystyle \bigcup_I S_i$

or, if the indexing set is clear from context:

$\displaystyle \bigcup_i S_i$

However, on this website it is recommended that the full form is used.