Discriminated union links here. For discriminated unions in computer science, see tagged union

In set theory, a disjoint union (or discriminated union) is a union of a collection of sets whose members are pairwise disjoint.

Formally, if $C$ is a collection of sets, then

$\backslash bigcup\_\{A\; \backslash in\; C\}\; A$

is a disjoint union if and only if for all A and B in C

$A\; \backslash neq\; B\; \backslash implies\; A\; \backslash cap\; B\; =\; \backslash varnothing.$

The term disjoint union 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 the above sense. This allows one to take the disjoint union of a collection of sets that are not in fact disjoint.

Formally, let {A_i : i ∈ I} be a family of sets indexed by I. The disjoint union of this family is the set

$\backslash coprod\_\{i\backslash in\; I\}A\_i\; =\; \backslash bigcup\_\{i\backslash in\; I\}\backslash \{(x,i)\; :\; x\; \backslash in\; A\_i\backslash \}$

The elements of the disjoint union are ordered pairs (x, i). Here i serves as an auxiliary index that indicates which A_i the element x came from. Each of the sets A_i is canonically embedded in the disjoint union as the set

$A\_i^*\; =\; \backslash \{(x,i)\; :\; x\; \backslash in\; A\_i\backslash \}$

For i ≠ j, the sets A_i* and A_j* are disjoint even if the sets A_i and A_j are not.

Consider the extreme case where each of the A_i are equal to some fixed set A for each i ∈ I. In this case one can show that the disjoint union of this family is the Cartesian product of A and I:

$\backslash coprod\_\{i\backslash in\; I\}A\_i\; =\; A\; \backslash times\; I.$

One may occasionally see the notation

$\backslash sum\_\{i\backslash in\; I\}A\_i$

for the disjoint union of a family of sets, or the notation A + B for the disjoint union of two sets. This notation is meant to be suggestive of the fact that the cardinality of the disjoint union is the sum of the cardinalities of the terms in the family. Compare this to the notation for the Cartesian product of a family of sets.

In the language of category theory, the disjoint union is the coproduct in the category of sets. It therefore satisfies the associated universal property. This also means that the disjoint union is the categorical dual of the Cartesian product construction. See coproduct for more details.

See also

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• coproduct
• disjoint union (topology)
• tagged union

Set theory

disjunkte Vereinigung