In
mathematics, a singleton, also known as a unit set
or one-point set, is a
set with
exactly one element. For example, the set
is a singleton whose single element is
.
Properties
Within the framework of
Zermelo–Fraenkel set theory
In set theory, Zermelo–Fraenkel set theory, named after mathematicians Ernst Zermelo and Abraham Fraenkel, is an axiomatic system that was proposed in the early twentieth century in order to formulate a theory of sets free of paradoxes such ...
, the
axiom of regularity
In mathematics, the axiom of regularity (also known as the axiom of foundation) is an axiom of Zermelo–Fraenkel set theory that states that every non-empty set ''A'' contains an element that is disjoint from ''A''. In first-order logic, the ...
guarantees that no set is an element of itself. This implies that a singleton is necessarily distinct from the element it contains,
thus 1 and are not the same thing, and the
empty set is distinct from the set containing only the empty set. A set such as
is a singleton as it contains a single element (which itself is a set, however, not a singleton).
A set is a singleton
if and only if
In logic and related fields such as mathematics and philosophy, "if and only if" (shortened as "iff") is a biconditional logical connective between statements, where either both statements are true or both are false.
The connective is b ...
its
cardinality is . In
von Neumann's set-theoretic construction of the natural numbers, the number 1 is ''defined'' as the singleton
In
axiomatic set theory, the existence of singletons is a consequence of the
axiom of pairing
In axiomatic set theory and the branches of logic, mathematics, and computer science that use it, the axiom of pairing is one of the axioms of Zermelo–Fraenkel set theory. It was introduced by as a special case of his axiom of elementary se ...
: for any set ''A'', the axiom applied to ''A'' and ''A'' asserts the existence of
which is the same as the singleton
(since it contains ''A'', and no other set, as an element).
If ''A'' is any set and ''S'' is any singleton, then there exists precisely one
function
Function or functionality may refer to:
Computing
* Function key, a type of key on computer keyboards
* Function model, a structured representation of processes in a system
* Function object or functor or functionoid, a concept of object-oriente ...
from ''A'' to ''S'', the function sending every element of ''A'' to the single element of ''S''. Thus every singleton is a
terminal object
In category theory, a branch of mathematics, an initial object of a category is an object in such that for every object in , there exists precisely one morphism .
The dual notion is that of a terminal object (also called terminal element): ...
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 ...
.
A singleton has the property that every function from it to any arbitrary set is injective. The only non-singleton set with this property is the
empty set.
Every singleton set is an
ultra prefilter. If
is a set and
then the upward of
in
which is the set
is a
principal ultrafilter
In the mathematical field of order theory, an ultrafilter on a given partially ordered set (or "poset") P is a certain subset of P, namely a maximal filter on P; that is, a proper filter on P that cannot be enlarged to a bigger proper filter o ...
on
Moreover, every principal ultrafilter on
is necessarily of this form. The
ultrafilter lemma
In the mathematical field of set theory, an ultrafilter is a ''maximal proper filter'': it is a filter U on a given non-empty set X which is a certain type of non-empty family of subsets of X, that is not equal to the power set \wp(X) of X (suc ...
implies that non-
principal ultrafilters exist on every
infinite set
In set theory, an infinite set is a set that is not a finite set. Infinite sets may be countable or uncountable.
Properties
The set of natural numbers (whose existence is postulated by the axiom of infinity) is infinite. It is the only s ...
(these are called ).
Every
net
Net or net may refer to:
Mathematics and physics
* Net (mathematics), a filter-like topological generalization of a sequence
* Net, a linear system of divisors of dimension 2
* Net (polyhedron), an arrangement of polygons that can be folded up ...
valued in a singleton subset
of is an
ultranet in
The
Bell number
In combinatorial mathematics, the Bell numbers count the possible partitions of a set. These numbers have been studied by mathematicians since the 19th century, and their roots go back to medieval Japan. In an example of Stigler's law of eponymy ...
integer sequence counts the number of
partitions of a set
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 ...
(), if singletons are excluded then the numbers are smaller ().
In category theory
Structures built on singletons often serve as
terminal object
In category theory, a branch of mathematics, an initial object of a category is an object in such that for every object in , there exists precisely one morphism .
The dual notion is that of a terminal object (also called terminal element): ...
s or
zero object
In category theory, a branch of mathematics, an initial object of a category is an object in such that for every object in , there exists precisely one morphism .
The dual notion is that of a terminal object (also called terminal element): ...
s of various
categories
Category, plural categories, may refer to:
Philosophy and general uses
*Categorization, categories in cognitive science, information science and generally
*Category of being
* ''Categories'' (Aristotle)
*Category (Kant)
* Categories (Peirce)
* ...
:
* The statement above shows that the singleton sets are precisely the terminal objects in the category
Set of
sets. No other sets are terminal.
* Any singleton admits a unique
topological space
In mathematics, a topological space is, roughly speaking, a geometrical space in which closeness is defined but cannot necessarily be measured by a numeric distance. More specifically, a topological space is a set whose elements are called po ...
structure (both subsets are open). These singleton topological spaces are terminal objects in the category of topological spaces and
continuous functions. No other spaces are terminal in that category.
* Any singleton admits a unique
group
A group is a number of persons or things that are located, gathered, or classed together.
Groups of people
* Cultural group, a group whose members share the same cultural identity
* Ethnic group, a group whose members share the same ethnic ide ...
structure (the unique element serving as
identity element
In mathematics, an identity element, or neutral element, of a binary operation operating on a set is an element of the set that leaves unchanged every element of the set when the operation is applied. This concept is used in algebraic structures su ...
). These singleton groups are
zero object
In category theory, a branch of mathematics, an initial object of a category is an object in such that for every object in , there exists precisely one morphism .
The dual notion is that of a terminal object (also called terminal element): ...
s in the category of groups and
group homomorphism
In mathematics, given two groups, (''G'', ∗) and (''H'', ·), a group homomorphism from (''G'', ∗) to (''H'', ·) is a function ''h'' : ''G'' → ''H'' such that for all ''u'' and ''v'' in ''G'' it holds that
: h(u*v) = h(u) \cdot h(v)
w ...
s. No other groups are terminal in that category.
Definition by indicator functions
Let be a
class defined by an
indicator function
Then is called a ''singleton'' if and only if there is some
such that for all
Definition in ''Principia Mathematica''
The following definition was introduced by
Whitehead and
Russell
:
‘
Df.
The symbol
‘
denotes the singleton
and
denotes the class of objects identical with
aka
.
This occurs as a definition in the introduction, which, in places, simplifies the argument in the main text, where it occurs as proposition 51.01 (p.357 ibid.).
The proposition is subsequently used to define the
cardinal number
In mathematics, cardinal numbers, or cardinals for short, are a generalization of the natural numbers used to measure the cardinality (size) of sets. The cardinality of a finite set is a natural number: the number of elements in the set. T ...
1 as
:
‘
Df.
That is, 1 is the class of singletons. This is definition 52.01 (p.363 ibid.)
See also
*
*
References
*
{{Set theory
Basic concepts in set theory
1 (number)