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In
mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, a disjoint union (or discriminated union) of a
family of sets In 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. More generally, a collection of any sets whatsoever is called a family of sets, set fami ...
(A_i : i\in I) is a set A, often denoted by \bigsqcup_ A_i, with an injection of each A_i into A, such that the images of these injections form a partition of A (that is, each element of A belongs to exactly one of these images). A disjoint union of a family of pairwise disjoint sets is their union. In
category theory Category theory is a general theory of mathematical structures and their relations that was introduced by Samuel Eilenberg and Saunders Mac Lane in the middle of the 20th century in their foundational work on algebraic topology. Nowadays, ca ...
, the disjoint union is the coproduct of the
category of sets In the mathematical field of category theory, the category of sets, denoted as Set, is the category whose objects are sets. The arrows or morphisms between sets ''A'' and ''B'' are the total functions from ''A'' to ''B'', and the composition o ...
, and thus defined
up to Two mathematical objects ''a'' and ''b'' are called equal up to an equivalence relation ''R'' * if ''a'' and ''b'' are related by ''R'', that is, * if ''aRb'' holds, that is, * if the equivalence classes of ''a'' and ''b'' with respect to ''R'' ...
a
bijection In mathematics, a bijection, also known as a bijective function, one-to-one correspondence, or invertible function, is a function between the elements of two sets, where each element of one set is paired with exactly one element of the other ...
. In this context, the notation \coprod_ A_i is often used. The disjoint union of two sets A and B is written with
infix An infix is an affix inserted inside a word stem (an existing word or the core of a family of words). It contrasts with '' adfix,'' a rare term for an affix attached to the outside of a stem, such as a prefix or suffix. When marking text for i ...
notation as A \sqcup B. Some authors use the alternative notation A \uplus B or A \operatorname B (along with the corresponding \biguplus_ A_i or \operatorname_ A_i). A standard way for building the disjoint union is to define A as the set of
ordered pair In mathematics, an ordered pair (''a'', ''b'') is a pair of objects. The order in which the objects appear in the pair is significant: the ordered pair (''a'', ''b'') is different from the ordered pair (''b'', ''a'') unless ''a'' = ''b''. (In con ...
s (x, i) such that x \in A_i, and the injection A_i \to A as x \mapsto (x, i).


Example

Consider the sets A_0 = \ and A_1 = \. It is possible to index the set elements according to set origin by forming the associated sets \begin A^*_0 & = \ \\ A^*_1 & = \, \\ \end where the second element in each pair matches the subscript of the origin set (for example, the 0 in (5, 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 = \.


Set theory definition

Formally, let \left\ be a
family of sets In 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. More generally, a collection of any sets whatsoever is called a family of sets, set fami ...
indexed by I. The disjoint union of this family is the set \bigsqcup_ A_i = \bigcup_ \left\. 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 isomorphic to the set A_i^* = \left\. Through this isomorphism, one may consider that A_i is canonically embedded in the disjoint union. For i \neq 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 is equal to some fixed set A for each i \in I, the disjoint union is the
Cartesian product In mathematics, specifically set theory, the Cartesian product of two sets ''A'' and ''B'', denoted ''A''×''B'', is the set of all ordered pairs where ''a'' is in ''A'' and ''b'' is in ''B''. In terms of set-builder notation, that is : A\t ...
of A and I: \bigsqcup_ A_i = A \times I. Occasionally, the notation \sum_ A_i is used 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 In mathematics, the cardinality of a set is a measure of the number of elements of the set. For example, the set A = \ contains 3 elements, and therefore A has a cardinality of 3. Beginning in the late 19th century, this concept was generalized ...
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 In mathematics, specifically set theory, the Cartesian product of two sets ''A'' and ''B'', denoted ''A''×''B'', is the set of all ordered pairs where ''a'' is in ''A'' and ''b'' is in ''B''. In terms of set-builder notation, that is : A\t ...
of a family of sets. In the language of
category theory Category theory is a general theory of mathematical structures and their relations that was introduced by Samuel Eilenberg and Saunders Mac Lane in the middle of the 20th century in their foundational work on algebraic topology. Nowadays, ca ...
, the disjoint union is the coproduct in the
category of sets In the mathematical field of category theory, the category of sets, denoted as Set, is the category whose objects are sets. The arrows or morphisms between sets ''A'' and ''B'' are the total functions from ''A'' to ''B'', and the composition o ...
. It therefore satisfies the associated
universal property In mathematics, more specifically in category theory, a universal property is a property that characterizes up to an isomorphism the result of some constructions. Thus, universal properties can be used for defining some objects independently fr ...
. This also means that the disjoint union is the categorical dual of the
Cartesian product In mathematics, specifically set theory, the Cartesian product of two sets ''A'' and ''B'', denoted ''A''×''B'', is the set of all ordered pairs where ''a'' is in ''A'' and ''b'' is in ''B''. In terms of set-builder notation, that is : A\t ...
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 of A_i and the notation \underset A is sometimes used.


Category theory point of view

In
category theory Category theory is a general theory of mathematical structures and their relations that was introduced by Samuel Eilenberg and Saunders Mac Lane in the middle of the 20th century in their foundational work on algebraic topology. Nowadays, ca ...
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''.


See also

* * * * * * * * * * *


References

* * {{Set theory Basic concepts in set theory Operations on sets