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 in1 and in2 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 .
The disjoint union of any finite number of sets may be defined inductively, as
The disjoint union of a general family of sets Xλ as λ ranges over a general index set Λ may be defined as