This article is about the operation on sets. For the computer science meaning of the term, see Tagged union. For the operation on graphs, see disjoint union of graphs.
Not to be confused with Disjunctive union.
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Disjoint union
Type
Set operation
Field
Set theory
Symbolic statement
In mathematics, the disjoint union (or discriminated 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.
A disjoint union of an indexed 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
In mathematics, the disjointunion (or discriminated union) A ⊔ B {\displaystyle A\sqcup B} of the sets A and B is the set formed from the elements of...
formal logic, two sets are said to be disjoint sets if they have no element in common. Equivalently, two disjoint sets are sets whose intersection is the...
mathematics, the disjointunion of graphs is an operation that combines two or more graphs to form a larger graph. It is analogous to the disjointunion of sets...
computer science, a tagged union, also called a variant, variant record, choice type, discriminated union, disjointunion, sum type, or coproduct, is...
σ-compact. A manifold need not be connected, but every manifold M is a disjointunion of connected manifolds. These are just the connected components of M...
characteristic of their disjointunion is the sum of their Euler characteristics, since homology is additive under disjointunion: χ ( M ⊔ N ) = χ ( M )...
or categorical sum, is a construction which includes as examples the disjointunion of sets and of topological spaces, the free product of groups, and the...
product types (i.e., tuples and records) and sum types (i.e., tagged or disjointunions, coproduct types or variant types). The values of a product type typically...
The term discriminated union may refer to: Disjointunion in set theory. Tagged union in computer science. This disambiguation page lists articles associated...
manifold. Two manifolds of the same dimension are cobordant if their disjointunion is the boundary of a compact manifold one dimension higher. The boundary...
complements and countable disjointunions π-system – Family of sets closed under intersection Ring of sets – Family closed under unions and relative complements...
{\displaystyle o(S)} and o ( T ) {\displaystyle o(T)} , the type of the disjointunion is o ( S ) ⊕ o ( T ) {\displaystyle o(S)\oplus o(T)} , while the type...
tangent vectors in M {\displaystyle M} . As a set, it is given by the disjointunion of the tangent spaces of M {\displaystyle M} . That is, T M = ⨆ x ∈...
In type theory, a union has a sum type; this corresponds to disjointunion in mathematics. Depending on the language and type, a union value may be used...
connected. disjoint 1. Two subgraphs are edge disjoint if they share no edges, and vertex disjoint if they share no vertices. 2. The disjointunion of two...
equation can be deduced from the former using that, for any X that is the disjointunion of a closed and open subscheme U and V, respectively, ζ X ( s ) = ζ...
space is a topological space that cannot be represented as the union of two or more disjoint non-empty open subsets. Connectedness is one of the principal...
c j ) {\displaystyle z^{\omega (c_{j})}} . If the target class is a disjointunion of two other classes, C = A + B {\displaystyle {\mathcal {C}}={\mathcal...
countable unions of pairwise disjoint sets: if A 1 , A 2 , A 3 , … {\displaystyle A_{1},A_{2},A_{3},\ldots } is a sequence of pairwise disjoint sets in...
\mathbb {Z} /2)} corresponding to the trivial double covering, the disjointunion of two copies of X. For any topological space X, the cap product is...
regard to disjointunions of graphs: A graph invariant is additive if, for all two graphs G and H, the value of the invariant on the disjointunion of G and...
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