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 ...
is a set
often denoted by
with an
injection of each
into
such that the
images of these injections form a
partition of
(that is, each element of
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
is often used.
The disjoint union of two sets
and
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
. Some authors use the alternative notation
or
(along with the corresponding
or
).
A standard way for building the disjoint union is to define
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
such that
and the injection
as
Example
Consider the sets
and
It is possible to index the set elements according to set origin by forming the associated sets
where the second element in each pair matches the subscript of the origin set (for example, the
in
matches the subscript in
etc.). The disjoint union
can then be calculated as follows:
Set theory definition
Formally, let
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
The disjoint union of this family is the set
The elements of the disjoint union are
ordered pairs Here
serves as an auxiliary index that indicates which
the element
came from.
Each of the sets
is canonically isomorphic to the set
Through this isomorphism, one may consider that
is canonically embedded in the disjoint union.
For
the sets
and
are disjoint even if the sets
and
are not.
In the extreme case where each of the
is equal to some fixed set
for each
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
and
:
Occasionally, the notation
is used for the disjoint union of a family of sets, or the notation
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
is referred to as a of
and the notation
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
is frequently used, instead of
to denote ''coproduct''.
See also
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References
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{{Set theory
Basic concepts in set theory
Operations on sets