Disjoint union

In set theory, the disjoint union (or discriminated union) of a family of sets is a modified union operation that indexes the elements according to which set they originated in. Or slightly different from this, the disjoint union of a family of subsets is the usual union of the subsets which are pairwise disjointdisjoint sets means they have no element in common.

Note that these two concepts are different but strongly related. Moreover, it seems that they are essentially identical to each other in category theory. That is, both are realizations of the coproduct of category of sets.

Example

Consider the sets $A_{0}=\{1,2,3\}$  and $A_{1}=\{1,2\}$ . We can index the set elements according to set origin by forming the associated sets

{\begin{aligned}A_{0}^{*}&=\{(1,0),(2,0),(3,0)\}\\A_{1}^{*}&=\{(1,1),(2,1)\},\end{aligned}}

where the second element in each pair matches the subscript of the origin set (e.g., the $0$  in $(1,0)$  matches the subscript in $A_{0}$ , etc.). The disjoint union $A_{0}\sqcup A_{1}$  can then be calculated as follows:

$A_{0}\sqcup A_{1}=A_{0}^{*}\cup A_{1}^{*}=\{(1,0),(2,0),(3,0),(1,1),(2,1)\}.$

Set theory definition

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

$\bigsqcup _{i\in I}A_{i}=\bigcup _{i\in I}\{(x,i):x\in A_{i}\}.$

The elements of the disjoint union are ordered pairs (x, i). Here i serves as an auxiliary index that indicates which Ai the element x came from.

Each of the sets Ai is canonically isomorphic to the set

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

Through this isomorphism, one may consider that Ai is canonically embedded in the disjoint union. For ij, the sets Ai* and Aj* are disjoint even if the sets Ai and Aj are not.

In the extreme case where each of the Ai is equal to some fixed set A for each iI, the disjoint union is the Cartesian product of A and I:

$\bigsqcup _{i\in I}A_{i}=A\times I.$

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.

Disjoint unions are also sometimes written $\biguplus _{i\in I}A_{i}$  or $\ \cdot \!\!\!\!\!\bigcup _{i\in I}A_{i}$ .

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 ${\underset {A\in C}{\,\,\bigcup \nolimits ^{*}\!}}A$  is sometimes used.

Category theory point of view

In category theory the disjoint union is defined as a coproduct in the category of sets.

As such, the disjoint union is defined up to an isomorphism, and the above definition is just one realization of the coproduct, among others. When the sets are pairwise disjoint, the usual union is another realization of the coproduct. This justifies the second definition in the lead.

This categorical aspect of the disjoint union explains why $\coprod$  is frequently used, instead of $\bigsqcup$ , to denote coproduct.