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

### 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 ∈ Λ X λ = { x : ∃ λ ∈ Λ , x ∈ X λ } . {\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:

⋃ X = { x : ∃ Y ∈ X , x ∈ Y } . {\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

A ∪ B = ⋃ { A , B } . {\displaystyle A\cup B=\bigcup \\{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.