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Disjoint union

Type	Set operation
Field	Set theory
Symbolic statement	$${\displaystyle \underset{i\in I}{⨆}{A}_i=\bigcup _{i\in I}\left\{(x,i):x\in A_i\right\}}$$

Inmathematics, thedisjoint union (ordiscriminated union)${\displaystyle A\bigsqcup B}$ of the setsA andB is the set formed from the elements ofA andB labelled (indexed) with the name of the set from which they come. So, an element belonging to bothA andB appears twice in the disjoint union, with two different labels.

A disjoint union of anindexed family of sets${\displaystyle (A_i:i\in I)}$ is a set${\displaystyle A,}$ often denoted by$\underset{i\in I}{⨆}{A}_i,$ with aninjection of each${\displaystyle A_i}$ into${\displaystyle A,}$ such that theimages of these injections form apartition of${\displaystyle A}$ (that is, each element of${\displaystyle A}$ belongs to exactly one of these images). A disjoint union of a family ofpairwise disjoint sets is theirunion.

Incategory theory, the disjoint union is thecoproduct of thecategory of sets, and thus definedup to abijection. In this context, the notation$\coprod _{i\in I}{A}_i$ is often used.

The disjoint union of two sets${\displaystyle A}$ and${\displaystyle B}$ is written withinfix notation as${\displaystyle A\bigsqcup B}$. Some authors use the alternative notation${\displaystyle A\uplus B}$ or${\displaystyle A\cup \cdot B}$ (along with the corresponding$\underset{i\in I}{⨄}{A}_i$ or${\bigcup \cdot }_{i\in I}{A}_i$).

A standard way for building the disjoint union is to define${\displaystyle A}$ as the set ofordered pairs${\displaystyle (x,i)}$ such that${\displaystyle x\in A_i,}$ and the injection${\displaystyle A_i\to A}$ as${\displaystyle x\mapsto (x,i).}$

Consider the sets${\displaystyle A_0=\{5,6,7\}}$ and${\displaystyle A_1=\{5,6\}.}$ 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${\displaystyle 0}$ in${\displaystyle (5,0)}$ matches the subscript in${\displaystyle A_0,}$ etc.). The disjoint union${\displaystyle A_0\bigsqcup A_1}$ can then be calculated as follows:$${\displaystyle A_0\bigsqcup A_1=A_0^{\ast }\cup A_1^{\ast }=\{(5,0),(6,0),(7,0),(5,1),(6,1)\}.}$$

Formally, let${\displaystyle \left(A_i:i\in I\right)}$ be anindexed family of sets indexed by${\displaystyle I.}$ Thedisjoint union of this family is the set$${\displaystyle \underset{i\in I}{⨆}{A}_i=\bigcup _{i\in I}\left\{(x,i):x\in A_i\right\}.}$$ The elements of the disjoint union areordered pairs${\displaystyle (x,i).}$ Here${\displaystyle i}$ serves as an auxiliary index that indicates which${\displaystyle A_i}$ the element${\displaystyle x}$ came from.

Each of the sets${\displaystyle A_i}$ is canonically isomorphic to the set$${\displaystyle A_i^{\ast }=\left\{(x,i):x\in A_i\right\}.}$$Through this isomorphism, one may consider that${\displaystyle A_i}$ is canonically embedded in the disjoint union. For${\displaystyle i\ne j,}$ the sets${\displaystyle A_i^{\ast }}$ and${\displaystyle A_j^{\ast }}$ are disjoint even if the sets${\displaystyle A_i}$ and${\displaystyle A_j}$ are not.

In the extreme case where each of the${\displaystyle A_i}$ is equal to some fixed set${\displaystyle A}$ for each${\displaystyle i\in I,}$ the disjoint union is theCartesian product of${\displaystyle A}$ and${\displaystyle I}$:$${\displaystyle \underset{i\in I}{⨆}{A}_i=A\times I.}$$

Occasionally, the notation$${\displaystyle \sum _{i\in I}{A}_i}$$is used for the disjoint union of a family of sets, or the notation${\displaystyle A+B}$ for the disjoint union of two sets. This notation is meant to be suggestive of the fact that thecardinality of the disjoint union is thesum of the cardinalities of the terms in the family. Compare this to the notation for theCartesian product of a family of sets.

In the language ofcategory theory, the disjoint union is thecoproduct in thecategory of sets. It therefore satisfies the associateduniversal property. This also means that the disjoint union is thecategorical dual of theCartesian product construction. SeeCoproduct for more details.

For many purposes, the particular choice of auxiliary index is unimportant, and in a simplifyingabuse of notation, the indexed family can be treated simply as a collection of sets. In this case${\displaystyle A_i^{\ast }}$ is referred to as acopy of${\displaystyle A_i}$ and the notation${\displaystyle \underset{A\in C}{{\bigcup }^{\ast }}A}$ is sometimes used.

Incategory theory the disjoint union is defined as acoproduct 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${\displaystyle \coprod }$ is frequently used, instead of${\displaystyle ⨆,}$ to denotecoproduct.

• Coproduct – Category-theoretic construction
• Direct limit – Special case of colimit in category theory
• Disjoint union (topology) – Mathematical term
• Disjoint union of graphs – Binary operation combining the vertex and edge sets of two graphs
• Intersection (set theory) – Set of elements common to all of some sets
• List of set identities and relations – Equalities for combinations of sets
• Partition of a set – Mathematical ways to group elements of a set
• Sum type – Data structure used to hold a value that could take on several different, but fixed, typesPages displaying short descriptions of redirect targets
• Symmetric difference – Elements in exactly one of two sets
• Tagged union – Data structure used to hold a value that could take on several different, but fixed, types
• Union (computer science) – Data type that allows for values that are one of multiple different data typesPages displaying short descriptions of redirect targets

• Lang, Serge (2004),Algebra,Graduate Texts in Mathematics, vol. 211 (Corrected fourth printing, revised third ed.), New York: Springer-Verlag, p. 60,ISBN 978-0-387-95385-4
• Weisstein, Eric W."Disjoint Union".MathWorld.

• Basic concepts in set theory
• Operations on sets

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