TheInfoList

In
mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and their changes (cal ...
, a singleton, also known as a unit set, is a set with exactly one element. For example, the set is a singleton containing the element ''null''. The term is also used for a 1-
tuple In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...
(a
sequence In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and t ... with one member).

# Properties

Within the framework of
Zermelo–Fraenkel set theory In set theory illustrating the intersection of two sets Set theory is the branch of mathematical logic that studies sets, which can be informally described as collections of objects. Although objects of any kind can be collected into a set ...
, 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 Empty set, non-empty Set (mathematics), set ''A'' contains an element that is Disjoint sets, disjoi ...
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 #REDIRECT Empty set #REDIRECT Empty set#REDIRECT Empty set In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry ... 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 Logic is an interdisciplinary field which studies truth and reasoning. Informal logic seeks to characterize Validity (logic), valid arguments informally, for instance by listing varieties of fallacies. Formal logic represents st ...
its
cardinality In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
is . In von Neumann's set-theoretic construction of the natural numbers, the number 1 is ''defined'' as the singleton $\.$ In
axiomatic set theory Set theory is the branch of mathematical logic that studies Set (mathematics), sets, which can be informally described as collections of objects. Although objects of any kind can be collected into a set, set theory, as a branch of mathematics, i ...
, the existence of singletons is a consequence of the
axiom of pairing In axiomatic set theory Set theory is the branch of mathematical logic that studies Set (mathematics), sets, which can be informally described as collections of objects. Although objects of any kind can be collected into a set, set theory, as ...
: 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 function key is a key on a computer A computer is a machine that can be programmed to carry out sequences of arithmetic or logical operations automatically. Modern comp ...
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 Category theory formalizes mathematical structure and its concepts in terms of a Graph labeling, labeled directed graph called a ''Category (mathematics), category'', whose nodes are called ''objects'', and whose labelled d ...
in the
category of sets In the mathematical Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities an ...
. 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 #REDIRECT Empty set #REDIRECT Empty set#REDIRECT Empty set In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry ... . Every singleton set is an ultra prefilter. If $X$ is a set and $x \in X$ then the upward of $\$ in $X,$ which is the set $\,$ is a
principal Principal may refer to: Title or rank * Principal (academia) The principal is the chief executive and the chief academic officer of a university A university ( la, universitas, 'a whole') is an educational institution, institution of higher ... ultrafilter In the mathematical field of order theory, an ultrafilter on a given partially ordered set (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 pr ...
on $X.$ Moreover, every principal ultrafilter on $X$ is necessarily of this form. The ultrafilter lemma implies that non-
principal Principal may refer to: Title or rank * Principal (academia) The principal is the chief executive and the chief academic officer of a university A university ( la, universitas, 'a whole') is an educational institution, institution of higher ... ultrafilters exist on every
infinite set In set theory illustrating the intersection (set theory), intersection of two set (mathematics), sets. Set theory is a branch of mathematical logic that studies Set (mathematics), sets, which informally are collections of objects. Although any ...
(these are called ). Every
net Net or net may refer to: Mathematics and physics * Net (mathematics) In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they ...
valued in a singleton subset $X$ of is an ultranet in $X.$ 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 File:Genji chapter symbols groupings of 5 elements.svg, The traditional Japanese symbols for the 54 chapters of the ''Tale of Genji'' are based on the 52 ways of partitioning five elements (the two red symbols represent the same partition, and th ...
(), if singletons are excluded then the numbers are smaller ().

# In category theory

Structures built on singletons often serve as
terminal object In category theory Category theory formalizes mathematical structure and its concepts in terms of a Graph labeling, labeled directed graph called a ''Category (mathematics), category'', whose nodes are called ''objects'', and whose labelled d ...
s or
zero object In category theory Category theory formalizes mathematical structure and its concepts in terms of a Graph labeling, labeled directed graph called a ''Category (mathematics), category'', whose nodes are called ''objects'', and whose labelled d ...
s of various
categories Category, plural categories, may refer to: Philosophy and general uses *Categorization Categorization is the human ability and activity of recognizing shared features or similarities between the elements of the experience of the world (such ...
: * 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 Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no gener ...
structure (both subsets are open). These singleton topological spaces are terminal objects in the category of topological spaces and
continuous function In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...
s. No other spaces are terminal in that category. * Any singleton admits a unique
group A group is a number A number is a mathematical object used to counting, count, measurement, measure, and nominal number, label. The original examples are the natural numbers 1, 2, 3, 4, and so forth. Numbers can be represented in language with ...
structure (the unique element serving as
identity element In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and th ...
). These singleton groups are
zero object In category theory Category theory formalizes mathematical structure and its concepts in terms of a Graph labeling, labeled directed graph called a ''Category (mathematics), category'', whose nodes are called ''objects'', and whose labelled d ...
s in the category of groups and
group homomorphism In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no ge ... s. No other groups are terminal in that category.

# Definition by indicator functions

Let be a
class Class or The Class may refer to: Common uses not otherwise categorized * Class (biology), a taxonomic rank * Class (knowledge representation), a collection of individuals or objects * Class (philosophy), an analytical concept used differently f ...
defined by an
indicator function In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities a ...
$b : X \to \.$ Then is called a ''singleton'' if and only if there is some $y \in X$ such that for all $x \in X,$ $b(x) = (x = y).$

# Definition in ''Principia Mathematica''

The following definition was introduced by Whitehead and Russell :$\iota$$x = \hat\left(y = x\right)$ Df. The symbol $\iota$$x$ denotes the singleton $\$ and $\hat\left(y = x\right)$ denotes the class of objects identical with $x$ 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 1 as :$1=\hat\left(\left(\exists x\right)\alpha=\iota$$x\right)$ Df. That is, 1 is the class of singletons. This is definition 52.01 (p.363 ibid.)