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Concept
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 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 , the disjoint union is the coproduct of the , 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
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.
In the language of , the disjoint union is the coproduct in the . It therefore satisfies the associated universal property.
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