Disjoint union

This article includes a list of references, related reading, or external links, but its sources remain unclear because it lacks inline citations. Please help improve this article by introducing more precise citations. (January 2022) (Learn how and when to remove this message)

Disjoint union

Type	Set operation
Field	Set theory
Symbolic statement	$${\displaystyle \bigsqcup _{i\in I}A_{i}=\bigcup _{i\in I}\left\{(x,i):x\in A_{i}\right\}}$$

In mathematics, the disjoint union (or discriminated union) ${\displaystyle A\sqcup B}$ 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 ${\displaystyle (A_{i}:i\in I)}$ is a set ${\displaystyle A,}$ often denoted by ${\textstyle \bigsqcup _{i\in I}A_{i},}$ with an injection of each ${\displaystyle A_{i}}$ into ${\displaystyle A,}$ such that the images of these injections form a partition of ${\displaystyle A}$ (that is, each element of ${\displaystyle A}$ 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 ${\textstyle \coprod _{i\in I}A_{i}}$ is often used.

The disjoint union of two sets ${\displaystyle A}$ and ${\displaystyle B}$ is written with infix notation as ${\displaystyle A\sqcup B}$. Some authors use the alternative notation ${\displaystyle A\uplus B}$ or ${\displaystyle A\operatorname {{\cup }\!\!\!{\cdot }\,} B}$ (along with the corresponding ${\textstyle \biguplus _{i\in I}A_{i}}$ or ${\textstyle \operatorname {{\bigcup }\!\!\!{\cdot }\,} _{i\in I}A_{i}}$).

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