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.
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:
Set theory definition
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.
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.
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.
Disjoint union (topology) – space formed by equipping the disjoint union of the underlying sets with a natural topology called the disjoint union topologyPages displaying wikidata descriptions as a fallback