Singleton (mathematics)


In mathematics, 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.

Properties

Within the framework of Zermelo–Fraenkel set theory, the axiom of regularity 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.
A set is a singleton if and only if 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: for any set A, the axiom applied to A and A asserts the existence of, which is the same as the singleton .
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.
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:
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
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.
The proposition is subsequently used to define the cardinal number 1 as
That is, 1 is the class of singletons. This is definition 52.01