# Union

From Citizendium

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

## Properties

The union operation is:

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

## General unions

### Finite unions

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

### Infinite unions

The union of a general family of sets *X*_{λ} as λ ranges over a general index set Λ may be written in similar notation as

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:

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

## See also

## References

- 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.