Editing Disjoint union (section)

== 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.