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Type | Set operation |
---|---|
Field | Set theory |
Statement | 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. |
Symbolic statement |
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). 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. In this context, the notation is often used.
The disjoint union of two sets and is written with infix notation as . Some authors use the alternative notation or (along with the corresponding or ).
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
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:
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
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.