# Union (set theory)

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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, the union A ∪ B is defined by the following: a ∈ A ∪ B if and only if ( 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

$\bigcup_{i=1}^n X_i = X_1 \cup (X_2 \cup (X_3 \cup (\cdots X_n)\cdots))) . \,$

### 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

$\bigcup_{\lambda\in \Lambda} X_\lambda = \{ x : \exists \lambda \in \Lambda,~x \in X_\lambda \} .\,$

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:

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

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

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