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

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## Definition

Let $\mathbb U$ be a universal set.

Let $I$ be an indexing set.

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

Then the **union** of $\family {S_i}$ is defined and denoted as:

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

## 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.

## Sources

- 1982: P.M. Cohn:
*Algebra Volume 1*(2nd ed.) ... (previous) ... (next): Chapter $1$: Sets and mappings: $\S 1.2$: Sets