Union

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This editable Main Article is under development and subject to a disclaimer.

[edit intro]

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.

## Contents

* 1 Properties
* 2 General unions
* 2.1 Finite unions
* 2.2 Infinite unions
* 3 See also
* 4 References

## Properties[edit]

The union operation is:

* associative \- (A ∪ B) ∪ C = A ∪ (B ∪ C)
* commutative \- A ∪ B = B ∪ A.

## General unions[edit]

### Finite unions[edit]

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

⋃i=1nXi=X1∪(X2∪(X3∪(⋯Xn)⋯))).

### Infinite unions[edit]

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

⋃λ∈ΛXλ={x:∃λ∈Λ,x∈Xλ}.

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:

⋃X={x:∃Y∈X,x∈Y}.

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

A∪B=⋃{A,B}.

## See also[edit]

* Disjoint union
* Intersection

## References[edit]

* Paul Halmos (1960). Naive set theory. Van Nostrand Reinhold. Section 4.
* Keith J. Devlin (1979). Fundamentals of Contemporary Set Theory. Springer-Verlag, 5,10. ISBN 0-387-90441-7.