---
tag: 05GK32H72H
kind: definition
name: Union of a family of sets
aliases: [indexed-union, arbitrary-union]
msc: ["03E20"]
defines: 0X1GSWNBN1
uses: [05Q6G4EH2G, 0JW2FHW7G3, 07KVB4JSC7]
sources:
  - url: https://en.wikipedia.org/wiki/Union_(set_theory)
    locator: Arbitrary union
---

## Definition

Let $(A_i)_{i \in I}$ be a [[family of sets|05Q6G4EH2G]].
The union of the family is the set
$$\bigcup_{i \in I} A_i = \{x \mid \exists i \in I, \, x \in A_i\}.$$
So $x$ is an element of the union when $x \in A_i$ for at least one $i \in I$.

## Remarks

For $I = \{1, 2\}$, this is the [[union|07KVB4JSC7]] $A_1 \cup A_2$.
For an empty index set, the union is the [[empty set|0ECVK59Y03]].
Example: $\bigcup_{n \ge 1} [0, n] = [0, \infty)$ in $\mathbb{R}$.

## Explanations

### L1 human

The union of a family collects every element that appears in some member of the family.
It generalizes $A \cup B$ from two sets to any number of sets.