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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 singleton, also known as a unit set or one-point set, is a
set Set, The Set, SET or SETS may refer to: Science, technology, and mathematics Mathematics *Set (mathematics), a collection of elements *Category of sets, the category whose objects and morphisms are sets and total functions, respectively Electro ...
with exactly one element. For example, the set \ is a singleton whose single element is 0.


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 a ...
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 In mathematics, the empty set is the unique set having no elements; its size or cardinality (count of elements in a set) is zero. Some axiomatic set theories ensure that the empty set exists by including an axiom of empty set, while in othe ...
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 bic ...
its
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 ...
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 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, is mostly concern ...
, 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 set ...
: 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 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 the category of sets. 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 In mathematics, the empty set is the unique set having no elements; its size or cardinality (count of elements in a set) is zero. Some axiomatic set theories ensure that the empty set exists by including an axiom of empty set, while in othe ...
. 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 ultrafilter on X. Moreover, every principal ultrafilter on X is necessarily of this form. The ultrafilter lemma implies that non- principal ultrafilters exist on every infinite set (these are called ). Every net valued in a singleton subset X of is an ultranet in X. The Bell number integer sequence counts the number of partitions of a set (), if singletons are excluded then the numbers are smaller ().


In category theory

Structures built on singletons often serve as terminal objects or zero objects of various categories: * The statement above shows that the singleton sets are precisely the terminal objects in the category
Set Set, The Set, SET or SETS may refer to: Science, technology, and mathematics Mathematics *Set (mathematics), a collection of elements *Category of sets, the category whose objects and morphisms are sets and total functions, respectively Electro ...
of
set Set, The Set, SET or SETS may refer to: Science, technology, and mathematics Mathematics *Set (mathematics), a collection of elements *Category of sets, the category whose objects and morphisms are sets and total functions, respectively Electro ...
s. 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 poin ...
structure (both subsets are open). These singleton topological spaces are terminal objects in the category of topological spaces and
continuous function In mathematics, a continuous function is a function such that a continuous variation (that is a change without jump) of the argument induces a continuous variation of the value of the function. This means that there are no abrupt changes in val ...
s. No other spaces are terminal in that category. * Any singleton admits a unique group 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 objects in the category of groups and group homomorphisms. 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 ...
defined by an
indicator function In mathematics, an indicator function or a characteristic function of a subset of a set is a function that maps elements of the subset to one, and all other elements to zero. That is, if is a subset of some set , one has \mathbf_(x)=1 if x\i ...
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 :\iotax = \hat(y = x) Df. The symbol \iotax denotes the singleton \ and \hat(y = x) 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 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. ...
1 as :1=\hat((\exists x)\alpha=\iotax) 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)