In mathematics, the disjoint union (or discriminated union) <math>A \sqcup B</math> of the sets and is the set formed from the elements of and labelled (indexed) with the name of the set from which they come. So, an element belonging to both and 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 injection of each <math>A_i</math> into <math>A,</math> such that the images of these injections form a 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.
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