View source for Disjoint union

{{Short description|In mathematics, operation on sets}}
{{about|the operation on sets|the computer science meaning of the term|Tagged union|the operation on graphs|disjoint union of graphs}}
{{distinguish|Disjunctive union}}
{{inline|date=January 2022}}
{{Infobox mathematical statement
| name = Disjoint union
| image = PolygonsSetDisjointUnion.svg
| caption =
| type = [[Set (mathematics)#Basic operations|Set operation]]
| field = [[Set (mathematics)|Set theory]]
| symbolic statement = <math display=block>\bigsqcup_{i \in I} A_i = \bigcup_{i \in I} \left\{(x, i) : x \in A_i\right\}</math> 
}}

In [[mathematics]], the '''disjoint union''' (or '''discriminated union''') <math>A \sqcup B</math> of the sets {{math|''A''}} and {{math|''B''}} is the set formed from the elements of {{math|''A''}} and {{math|''B''}} labelled (indexed) with the name of the set from which they come. So, an element belonging to both {{math|''A''}} and {{math|''B''}} appears twice in the disjoint union, with two different labels. 

A disjoint union of an [[indexed family]] of sets <math>(A_i : i\in I)</math> is a set <math>A,</math> often denoted by <math display=inline>\bigsqcup_{i \in I} A_i,</math> with an [[injective function|injection]] of each <math>A_i</math> into <math>A,</math> such that the [[image (mathematics)|images]] of these injections form a [[Partition (set theory)|partition]] of <math>A</math> (that is, each element of <math>A</math> belongs to exactly one of these images). A disjoint union of a family of [[pairwise disjoint sets]] is their [[Union (set theory)|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 <math display=inline>\coprod_{i\in I} A_i</math> is often used.

The disjoint union of two sets <math>A</math> and <math>B</math> is written with [[infix notation]] as <math>A \sqcup B</math>.  Some authors use the alternative notation <math>A \uplus B</math> or <math>A \operatorname{{\cup}\!\!\!{\cdot}\,} B</math> (along with the corresponding <math display=inline>\biguplus_{i\in I} A_i</math> or <math display=inline>\operatorname{{\bigcup}\!\!\!{\cdot}\,}_{i\in I} A_i</math>). 

A standard way for building the disjoint union is to define <math>A</math> as the set of [[ordered pair]]s <math>(x, i)</math> such that <math>x \in A_i,</math> and the injection <math>A_i \to A</math> as <math>x \mapsto (x, i).</math>

== Example ==

Consider the sets <math>A_0 = \{5, 6, 7\}</math> and <math>A_1 = \{5, 6\}.</math> It is possible to index the set elements according to set origin by forming the associated sets
<math>\begin{align}
A^*_0 & = \{(5, 0), (6, 0), (7, 0)\} \\
A^*_1 & = \{(5, 1), (6, 1)\}, \\
\end{align}
</math>

where the second element in each pair matches the subscript of the origin set (for example, the <math>0</math> in <math>(5, 0)</math> matches the subscript in <math>A_0,</math> etc.). The disjoint union <math>A_0 \sqcup A_1</math> can then be calculated as follows:
<math display="block">A_0 \sqcup A_1 = A^*_0 \cup A^*_1 = \{(5, 0), (6, 0), (7, 0), (5, 1), (6, 1)\}.</math>

== Set theory definition ==

Formally, let <math>\left(A_i : i \in I\right)</math> be an [[indexed family]] of sets indexed by <math>I.</math> The '''disjoint union''' of this family is the set
<math display=block>\bigsqcup_{i \in I} A_i = \bigcup_{i \in I} \left\{(x, i) : x \in A_i\right\}.</math> 
The elements of the disjoint union are [[ordered pairs]] <math>(x, i).</math> Here <math>i</math> serves as an auxiliary index that indicates which <math>A_i</math> the element <math>x</math> came from.

Each of the sets <math>A_i</math> is canonically isomorphic to the set
<math display=block>A_i^* = \left\{(x,i) : x \in A_i\right\}.</math>
Through this isomorphism, one may consider that <math>A_i</math> is canonically embedded in the disjoint union. 
For <math>i \neq j,</math> the sets <math>A_i^*</math> and <math>A_j^*</math> are disjoint even if the sets <math>A_i</math> and <math>A_j</math> are not. 

In the extreme case where each of the <math>A_i</math> is equal to some fixed set <math>A</math> for each <math>i \in I,</math> the disjoint union is the [[Cartesian product]] of <math>A</math> and <math>I</math>:
<math display=block>\bigsqcup_{i \in I} A_i = A \times I.</math>

Occasionally, the notation
<math display=block>\sum_{i \in I} A_i</math>
is used for the disjoint union of a family of sets, or the notation <math>A + B</math> 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 [[Summation|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 [[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.

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 <math>A_i^*</math> is referred to as a {{em|copy}} of <math>A_i</math> and the notation <math>\underset{A \in C}{\,\,\bigcup\nolimits^{*}\!} A</math> is sometimes used.

== Category theory point of view ==

In [[category theory]] the disjoint union is defined as a [[coproduct]] 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 <math>\coprod</math> is frequently used, instead of <math>\bigsqcup,</math> to denote ''coproduct''.

== See also ==

* {{annotated link|Coproduct}}
* {{annotated link|Direct limit}}
* {{annotated link|Disjoint union (topology)}}
* {{annotated link|Disjoint union of graphs}}
* {{annotated link|Intersection (set theory)}}
* {{annotated link|List of set identities and relations}}
* {{annotated link|Partition of a set}}
* {{annotated link|Sum type}}
* {{annotated link|Symmetric difference}}
* {{annotated link|Tagged union}}
* {{annotated link|Union (computer science)}}

== References ==

{{reflist}}
{{reflist|group=note}}

* {{Lang Algebra | edition=3r2004 | page=60}}
* {{MathWorld |title=Disjoint Union |urlname=DisjointUnion}}

{{Set theory}}

[[Category:Basic concepts in set theory]]
[[Category:Operations on sets]]