Disjoint union

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 ${\displaystyle x\mapsto (x,i).}$