Disjoint union

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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 .

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 Xλ as λ ranges over a general index set Λ may be defined as