In set theory, a disjoint union (or discriminated union) is a union of a collection of sets whose members are pairwise disjoint. Set theory is the mathematical theory of sets, which represent collections of abstract objects. ... In set theory and other branches of mathematics, the union of a collection of sets is the set that contains everything that belongs to any of the sets, but nothing else. ... The notion of a set is one of the most important and fundamental concepts in modern mathematics. ... In mathematics, two sets are said to be disjoint if they have no element in common. ...

Formally, if C is a collection of sets, then

is a disjoint union if and only if for all A and B in C In mathematics, philosophy, logic and technical fields that depend on them, iff is used as an abbreviation for if and only if. It is often, not always, written italicized: iff. ...

The term disjoint union is also used to refer to a modified union operation which indexes the elements according to which set they originated in, ensuring that the result is a disjoint union in the above sense. This allows one to take the disjoint union of a collection of sets that are not in fact disjoint.

Formally, let {A_i : i ∈ I} be a family of sets indexed by I. The disjoint union of this family is the set In mathematics, it is a common practice to index or label a collection of objects by some set I called an index set. ...

The elements of the disjoint union are ordered pairs (x, i). Here i serves as an auxilary index that indicates which A_i the element x came from. Note that each of the sets A_i can be canonically embedded in the disjoint union as the set An ordered pair is a collection of two objects such that one can be distinguished as the first element and the other as the second element. ...

Observe that for i ≠ j, the sets A_i* and A_j* are disjoint even if the sets A_i and A_j are not.

Consider the extreme case where each of the A_i are equal to some fixed set A for each i ∈ I. In this case one can show that the disjoint union of this family is the Cartesian product of A and I: In mathematics, the Cartesian product (or direct product) X Y of two sets X and Y is the set of all possible ordered pairs whose first component is a member of X and whose second component is a member of Y. This concept is named after Ren Descartes. ...

One may occasionally see the notation

for the disjoint union of a family of sets, or the notation A + B 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. The cardinality of a set is a property that describes the size of the set by describing it using a cardinal number. ... Addition is one of the basic operations of arithmetic. ... In mathematics, the Cartesian product (or direct product) X Y of two sets X and Y is the set of all possible ordered pairs whose first component is a member of X and whose second component is a member of Y. This concept is named after Ren Descartes. ...

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. Category theory is a mathematical theory that deals in an abstract way with mathematical structures and relationships between them. ... In category theory, the coproduct, or categorical sum, is the dual notion to the categorical product. ... In mathematics, the category of sets is the category whose objects are all sets and whose morphisms are all functions. ... In category theory, abstract algebra and other fields of mathematics, frequently constructions are defined or characterised by an abstract property which requires the existence of a unique morphism under certain conditions. ... In category theory, an abstract branch of mathematics, the dual of a category C is the category formed by reversing all the morphisms of C. That is, we take Cop to be the category with objects that are those of C, but with the morphisms from X to Y in... In mathematics, the Cartesian product (or direct product) X Y of two sets X and Y is the set of all possible ordered pairs whose first component is a member of X and whose second component is a member of Y. This concept is named after Ren Descartes. ... In category theory, the coproduct, or categorical sum, is the dual notion to the categorical product. ...

See also

• coproduct
• disjoint union (topology)
• tagged union