Editing Disjoint union (section)

{{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 = [[File:PolygonsSetDisjointUnion.svg|220px|class=skin-invert-image]]
| 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>