# Disjoint union

In mathematics, the **disjoint union** of two sets *X* and *Y* is a set which contains "copies" of each of *X* and *Y*: it is denoted or, less often, .

There are *injection maps* in_{1} and in_{2} from *X* and *Y* to the disjoint union, which are injective functions with disjoint images.

If *X* and *Y* are disjoint, then the usual union is also a disjoint union. In general, the disjoint union can be realised in a number of ways, for example as

The disjoint union has a universal property: if there is a set *Z* with maps and , then there is a map such that the compositions and .

The disjoint union is commutative, in the sense that there is a natural bijection between and ; it is associative again in the sense that there is a natural bijection between and .

## General unions

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

The disjoint union of a general family of sets *X*_{λ} as λ ranges over a general index set Λ may be defined as

## References

- Michael D. Potter (1990).
*Sets: An Introduction*. Oxford University Press, 36-37. ISBN 0-19-853399-3.