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 {A_{i} : i ∈ I} be a family of sets indexed by I. The disjoint union of this family is the set
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
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:
One may occasionally see the notation
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 is referred to as a copy of A_{i} and the notation 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: A∩B = ∅. Then the union of all sets in collection C is called the disjoint union of sets:
As such, the term “disjoint union” is simply a shorthand for “union of sets which are pairwise disjoint”.
See also
 Coproduct
 Disjoint union (topology)
 Disjoint union of graphs
 Tagged union
 Union (computer science)
References
 Weisstein, Eric W., "Disjoint Union" from MathWorld.
Categories: Basic concepts in set theory
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