# Union

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In set theory, union (denoted as ∪) is a set operation between two sets that forms a set containing the elements of both sets.

Formally, union A ∪ B means that if a ∈ A ∪ B, then a ∈ A ∨ a ∈ B, where ∨ - is logical or. We see this connection between ∪ and ∨ symbols.

## Properties

The union operation is:

## General unions

### Finite unions

The union of any finite number of sets may be defined inductively, as

${\displaystyle \bigcup _{i=1}^{n}X_{i}=X_{1}\cup (X_{2}\cup (X_{3}\cup (\cdots X_{n})\cdots ))).\,}$

### Infinite unions

The union of a general family of sets Xλ as λ ranges over a general index set Λ may be written in similar notation as

${\displaystyle \bigcup _{\lambda \in \Lambda }X_{\lambda }=\{x:\exists \lambda \in \Lambda ,~x\in X_{\lambda }\}.\,}$

We may drop the indexing notation and define the union of a set to be the set of elements of the elements of that set:

${\displaystyle \bigcup X=\{x:\exists Y\in X,~x\in Y\}.\,}$

In this notation the union of two sets A and B may be expressed as

${\displaystyle A\cup B=\bigcup \{A,B\}.\,}$