disjoint union

Disjoint union of two sets

Let A and B be sets. Then their disjoint unionMathworldPlanetmathPlanetmath is the union


where is an object chosen so that A and B×{} are disjoint. Normally B×{} is identified with B in the obvious way. The element is almost never mentioned; it serves only as a “tag” to make the two sets disjoint. If A and B are already disjoint, then AB is isomorphic to AB; this is the most common situation in practice.

Disjoint union of many sets

If we have a collectionMathworldPlanetmath of sets {Ai} indexed by some set I, then the disjoint union


Observe that we have a natural isomorphism AiAi×{i}, and that the images of any pair of these isomorphismsMathworldPlanetmathPlanetmathPlanetmathPlanetmath have empty intersectionMathworldPlanetmathPlanetmath. This is also often called being pairwise disjoint and is a much stronger condition than that the intersection of all the images is empty. As before, if the Ai are already pairwise disjoint, then



If one is working in some categoryMathworldPlanetmath, the term “disjoint union” often means “coproductMathworldPlanetmath”.

For example, as sets, is two copies of the real line. As topological spacesMathworldPlanetmath, is again two copies of the real line with a topology whose open sets are pairs of real open sets, one for each copy of . This is the coproduct in the category of topological spaces.

Of course, there are many categories where this usage is unnatural. For example, in the category of pointed sets, the coproduct is the disjoint union with the distinguished points identified. In the category of abelian groups, the coproduct is the direct sum.

Another closely related usage should be mentioned. Occasionally an author will write “…and AB is a disjoint union…”. What this means is that AB is isomorphic to AB, which is to say that A and B are already disjoint.

Title disjoint union
Canonical name DisjointUnion
Date of creation 2013-03-22 14:13:24
Last modified on 2013-03-22 14:13:24
Owner yark (2760)
Last modified by yark (2760)
Numerical id 9
Author yark (2760)
Entry type Definition
Classification msc 03E99
Related topic CardinalityOfDisjointUnionOfFiniteSets