Disjoint union


Disjoint union

In mathematics, the term disjoint union may refer to one of two different concepts:

  • In set theory, a disjoint union (or discriminated union) is a modified union operation that indexes the elements according to which set they originated in; disjoint sets have no element in common.
  • In probability theory (or more generally in measure theory), a disjoint union is the usual union of sets that are, however, themselves pairwise disjoint.

Contents

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 embedded in the disjoint union as the set


    A_i^* = \{(x,i) : x \in A_i\}.

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 are 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.

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 Ai and the notation \bigcup_{A \in C}{^*} A is sometimes used.

Probability theory definition

Let C be a collection of pairwise disjoint sets. That is, for all sets A≠B in C, the intersection of these sets is empty: AB = ∅. Then the union of all sets in collection C is called the disjoint union of sets:


    \bigsqcup_{A \in C} A \equiv \bigcup_{A \in C} A

As such, the term “disjoint union” is simply a shorthand for “union of sets which are pairwise disjoint”.

See also

References


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