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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.

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

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

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

Each of the sets is canonically isomorphic to the set

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

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.

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*.

- Coproduct – Category-theoretic construction
- Direct limit – Special case of colimit in category theory
- Disjoint union (topology)
- Disjoint union of graphs
- Partition of a set – Mathematical ways to group elements of a set
- Sum type
- Tagged union – Data structure used to hold a value that could take on several different, but fixed, types
- Union (computer science)

- 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*.