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

A disjoint union of an indexed family with an injection such that the images of these injections form a partition belongs to exactly one of these images). A disjoint union of a family of pairwise disjoint sets is their union.

In category theory, the disjoint union is the coproduct of the category of sets, and thus defined up to a bijection is often used.

is written with infix notation).

as the set of ordered pairs

Example

Set theory definition

be an indexed family The disjoint union The elements of the disjoint union are ordered pairs came from.

are not.

the disjoint union is the Cartesian product

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.

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 is referred to as a is sometimes used.

Category theory point of view

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.

to denote coproduct.

See also

References