In set theory, a disjoint union or discriminated union is a set union in which each element of the resulting union is disjoint from each of the others; the intersection over a disjoint union is the empty set.

The term is also used to refer to a modified union operation which indexes the elements according to which set they originated in, ensuring that the result is a disjoint union. In computer science, this concept is important to construction of many data structures and is implemented directly by tagged unions and algebraic data types.

Formally, if C is a collection of sets, then

<math>

\mathcal{A} = \bigcup_{A \in C} A <math>

is a disjoint union if and only if

<math>

\forall A,B \in C \quad st. \ A \ne B: A \cap B = \empty <math>

As mentioned, one can take the disjoint union of sets that are not in fact disjoint by using an indexing device. For example given A₁ and A₂, which may have common elements, with union B, the disjoint union as a subset of B x {1,2} is the union of A₁ x {1} and A₂x{2}.

See also: Basic Set Theory