In
mathematicsMathematics is the science and study of quantity, structure, space, and change. Mathematicians seek out patterns, formulate new conjectures, and establish truth by rigorous deduction from appropriately chosen axioms and definitions....
, the term
disjoint union may refer to one of two different concepts:
- In set theory
The modern study of set theory was initiated by Cantor and Dedekind in the 1870s. After the discovery of paradoxes in informal set theory, numerous axiom systems were proposed in the early twentieth century, of which the Zermelo–Fraenkel axioms, with the axiom of choice, are the best-known.The...
, a disjoint union (or discriminated union) is a modified unionIn set theory, the union of a collection of sets is the set of all distinct elements in the collection. The union of a collection of sets gives a set .- Definition :A simple example:...
operation which indexes the elements according to which set they originated in;
- In probability theory
Probability theory is the branch of mathematics concerned with analysis of random phenomena. The central objects of probability theory are random variables, stochastic processes, and events: mathematical abstractions of non-deterministic events or measured quantities that may either be single...
, a disjoint union is the usual union of sets which are, however, themselves pairwise disjoint.
Formally, let {
Ai :
i ∈
I} be a
family of setsIn set theory and related branches of mathematics, a collection F of subsets of a given set S is called a family of subsets of S, or a family of sets over S...
indexed by
I.
In
mathematicsMathematics is the science and study of quantity, structure, space, and change. Mathematicians seek out patterns, formulate new conjectures, and establish truth by rigorous deduction from appropriately chosen axioms and definitions....
, the term
disjoint union may refer to one of two different concepts:
- In set theory
The modern study of set theory was initiated by Cantor and Dedekind in the 1870s. After the discovery of paradoxes in informal set theory, numerous axiom systems were proposed in the early twentieth century, of which the Zermelo–Fraenkel axioms, with the axiom of choice, are the best-known.The...
, a disjoint union (or discriminated union) is a modified unionIn set theory, the union of a collection of sets is the set of all distinct elements in the collection. The union of a collection of sets gives a set .- Definition :A simple example:...
operation which indexes the elements according to which set they originated in;
- In probability theory
Probability theory is the branch of mathematics concerned with analysis of random phenomena. The central objects of probability theory are random variables, stochastic processes, and events: mathematical abstractions of non-deterministic events or measured quantities that may either be single...
, a disjoint union is the usual union of sets which are, however, themselves pairwise disjoint.
Set theory definition
Formally, let {
Ai :
i ∈
I} be a
family of setsIn set theory and related branches of mathematics, a collection F of subsets of a given set S is called a family of subsets of S, or a family of sets over S...
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
Ai the element
x came from. Each of the sets
Ai is canonically embedded in the disjoint union as the set
For
i ≠
j, 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
i ∈
I, the disjoint union is the
Cartesian productIn mathematics, a Cartesian product is the direct product of two sets. The Cartesian product is named after René Descartes, whose formulation of analytic geometry gave rise to this concept....
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
cardinalityIn mathematics, the cardinality of a set is a measure of the "number of elements of the set". For example, the set A = {2, 4, 6} contains 3 elements, and therefore A has a cardinality of 3...
of the disjoint union is the
sumSUM can refer to:* The State University of Management* Soccer United Marketing* Society for the Establishment of Useful Manufactures* StartUp-Manager* Software User’s Manual,as from DOD-STD-2 167A, and MIL-STD-498...
of the cardinalities of the terms in the family. Compare this to the notation for the
Cartesian productIn mathematics, a Cartesian product is the direct product of two sets. The Cartesian product is named after René Descartes, whose formulation of analytic geometry gave rise to this concept....
of a family of sets.
In the language of
category theoryIn mathematics, category theory deals in an abstract way with mathematical structures and relationships between them: it abstracts from sets and functions to objects linked in diagrams by morphisms or arrows....
, the disjoint union is the
coproductIn category theory, the coproduct, or categorical sum, is the category-theoretic construction which subsumes the disjoint union of sets and of topological spaces, the free product of groups, and the direct sum of modules and vector spaces. The coproduct of a family of objects is essentially the...
in the
category of setsIn mathematics, the category of sets, denoted as Set, is the category whose objects are all sets and whose morphisms are all functions. It is the most basic and the most commonly used category in mathematics.-Properties of the category of sets:...
. It therefore satisfies the associated
universal propertyIn various branches of mathematics, a useful construction is often viewed as the “most efficient solution” to a certain problem. The definition of a universal property uses the language of category theory to make this notion precise and to study it abstractly.This article gives a general treatment...
. This also means that the disjoint union is the categorical dual of the
Cartesian productIn mathematics, a Cartesian product is the direct product of two sets. The Cartesian product is named after René Descartes, whose formulation of analytic geometry gave rise to this concept....
construction. See
coproductIn category theory, the coproduct, or categorical sum, is the category-theoretic construction which subsumes the disjoint union of sets and of topological spaces, the free product of groups, and the direct sum of modules and vector spaces. The coproduct of a family of objects is essentially the...
for more details.
For many purposes, the particular choice of auxiliary index is unimportant, and in a simplifying
abuse of notationIn mathematics, abuse of notation occurs when an author uses a mathematical notation in a way that is not formally correct but that seems likely to simplify the exposition or suggest the correct intuition . Abuse of notation should be contrasted with misuse of notation, which should be avoided...
, the indexed family can be treated simply as a collection of sets. In this case is referred to as a
copy of 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
In category theory, the coproduct, or categorical sum, is the category-theoretic construction which subsumes the disjoint union of sets and of topological spaces, the free product of groups, and the direct sum of modules and vector spaces. The coproduct of a family of objects is essentially the...
- Disjoint union (topology)
In general topology and related areas of mathematics, the disjoint union of a family of topological spaces is a space formed by equipping the disjoint union of the underlying sets with a natural topology called the disjoint union topology...
- Disjoint union of graphs
- Tagged union
In computer science, a tagged union, also called a variant, variant record, discriminated union, or disjoint union, is a data structure used to hold a value that could take on several different, but fixed types. Only one of the types can be in use at any one time, and a tag field explicitly...
- Union (computer science)
In computer science, a union is a value that may have any of several representations or formats; or a data structure that consists of a variable which may hold such a value. Some programming languages support special data types, called union types, to describe such values and variables...