In
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
Injection or injected may refer to:
Science and technology
* Injective function, a mathematical function mapping distinct arguments to distinct values
* Injection (medicine), insertion of liquid into the body with a syringe
* Injection, in broadca ...
of each
into
such that the
images
An image is a visual representation of something. It can be two-dimensional, three-dimensional, or somehow otherwise feed into the visual system to convey information. An image can be an artifact, such as a photograph or other two-dimensiona ...
of these injections form a
partition
Partition may refer to:
Computing Hardware
* Disk partitioning, the division of a hard disk drive
* Memory partition, a subdivision of a computer's memory, usually for use by a single job
Software
* Partition (database), the division of a ...
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
Union commonly refers to:
* Trade union, an organization of workers
* Union (set theory), in mathematics, a fundamental operation on sets
Union may also refer to:
Arts and entertainment
Music
* Union (band), an American rock group
** ''Un ...
.
In
category theory, the disjoint union is the
coproduct
In category theory, the coproduct, or categorical sum, is a construction which includes as examples the disjoint union of sets and of topological spaces, the free product of groups, and the direct sum of modules and vector spaces. The coproduc ...
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 a
bijection. In this context, the notation
is often used.
The disjoint union of two sets
and
is written with
infix 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 pairs
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
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 ...
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 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 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 category theory, the coproduct, or categorical sum, is a construction which includes as examples the disjoint union of sets and of topological spaces, the free product of groups, and the direct sum of modules and vector spaces. The coproduc ...
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
In category theory, a branch of mathematics, duality is a correspondence between the properties of a category ''C'' and the dual properties of the opposite category ''C''op. Given a statement regarding the category ''C'', by interchanging the sou ...
of the
Cartesian product construction. See
coproduct
In category theory, the coproduct, or categorical sum, is a construction which includes as examples the disjoint union of sets and of topological spaces, the free product of groups, and the direct sum of modules and vector spaces. The coproduc ...
for more details.
For many purposes, the particular choice of auxiliary index is unimportant, and in a simplifying
abuse of notation
In mathematics, abuse of notation occurs when an author uses a mathematical notation in a way that is not entirely formally correct, but which might help simplify the exposition or suggest the correct intuition (while possibly minimizing errors a ...
, 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 the disjoint union is defined as a
coproduct
In category theory, the coproduct, or categorical sum, is a construction which includes as examples the disjoint union of sets and of topological spaces, the free product of groups, and the direct sum of modules and vector spaces. The coproduc ...
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