Disjoint union

Summary

In mathematics, a disjoint union (or discriminated union) of a family of sets is a set often denoted by with an injection of each into such that the images of these injections form a partition of (that is, each element of belongs to exactly one of these images). The disjoint union of a family of pairwise disjoint sets is their union. In terms of category theory, the disjoint union is the coproduct of the category of sets, and thus defined up to a bijection.

The disjoint union of the sets A and B is the set formed from the elements of A and B labelled (indexed) with the name of the set from which they come. So, an element belonging to both A and B appears twice in the disjoint union, with two different labels.

A standard way for building the disjoint union is to define as the set of ordered pairs such that and the injection as

ExampleEdit

Consider the sets   and   It is possible to index the set elements according to set origin by forming the associated sets

 
where the second element in each pair matches the subscript of the origin set (for example, the   in   matches the subscript in   etc.). The disjoint union   can then be calculated as follows:
 

Set theory definitionEdit

Formally, let   be a family of sets indexed by   The disjoint union of this family is the set

 
The elements of the disjoint union are ordered pairs   Here   serves as an auxiliary index that indicates which   the element   came from.

Each of the sets   is canonically isomorphic to the set

 
Through this isomorphism, one may consider that   is canonically embedded in the disjoint union. For   the sets   and   are disjoint even if the sets   and   are not.

In the extreme case where each of the   is equal to some fixed set   for each   the disjoint union is the Cartesian product of   and  :

 

Occasionally, the notation

 
is used for the disjoint union of a family of sets, or the notation   for the disjoint union of two sets. This notation is meant to be suggestive of the fact that the cardinality of the disjoint union is the sum of the cardinalities of the terms in the family. Compare this to the notation for the Cartesian product of a family of sets.

Disjoint unions are also sometimes written   or  

In the language of category theory, the disjoint union is the coproduct in the category of sets. It therefore satisfies the associated universal property. This also means that the disjoint union is the categorical dual of the Cartesian product construction. See coproduct for more details.

For many purposes, the particular choice of auxiliary index is unimportant, and in a simplifying abuse of notation, the indexed family can be treated simply as a collection of sets. In this case   is referred to as a copy of   and the notation   is sometimes used.

Category theory point of viewEdit

In category theory the disjoint union is defined as a coproduct in the category of sets.

As such, the disjoint union is defined up to an isomorphism, and the above definition is just one realization of the coproduct, among others. When the sets are pairwise disjoint, the usual union is another realization of the coproduct. This justifies the second definition in the lead.

This categorical aspect of the disjoint union explains why   is frequently used, instead of   to denote coproduct.

See alsoEdit

ReferencesEdit

  • Lang, Serge (2004), Algebra, Graduate Texts in Mathematics, vol. 211 (Corrected fourth printing, revised third ed.), New York: Springer-Verlag, p. 60, ISBN 978-0-387-95385-4
  • Weisstein, Eric W. "Disjoint Union". MathWorld.