U+2A06 N-Ary Square Union Operator

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

The disjoint union of two sets $A$ and $B$ is written with infix notation as $A\bigsqcup B$. Some authors use the alternative notation $A\uplus B$ or $A\cup \cdot B$ (along with the corresponding $\underset{i\in I}{⨄}{A}_i$ or ${\bigcup \cdot }_{i\in I}{A}_i$).

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