Disjoint union

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In set theory, a disjoint union (or discriminated union) is a modified union operation which indexes the elements according to which set they originated in.

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

$\coprod_{i\in I}A_i = \bigcup_{i\in I}\{(x,i) : x \in A_i\}$

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

$A_i^* = \{(x,i) : x \in A_i\}$

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

In the extreme case where each of the A_i are equal to some fixed set A for each i ∈ I, the disjoint union is the Cartesian product of A and I:

$\coprod_{i\in I}A_i = A \times I.$

One may occasionally see the notation

$\sum_{i\in I}A_i$

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.

For many purposes, the particular choice of auxiliary index is unimportant, and in a simplifying abuse of notation, the indexed family can be treated simply as a collection of sets. In this case $A_i^*$ is referred to as a copy of $A i$ and the notation $\bigcup_{A \in C}{^*} A$ is sometimes used.

The term disjoint union (but not discriminated union) is also used to refer to a union of a collection of sets whose members are pairwise disjoint. In this case the disjoint union may be identified with the simple union, and the same notation is used. This usage is more common in computer science.

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

$\bigcup_{A \in C} A$