Singleton (mathematics)

In mathematics, a singleton, also known as a unit set,[1] is a set with exactly one element. For example, the set {0} is a singleton.

The term is also used for a 1-tuple (a sequence with one element).

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,[1] thus 1 and {1} are not the same thing, and the empty set is distinct from the set containing only the empty set. A set such as {{1, 2, 3}} 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 its cardinality is 1. In the standard set-theoretic construction of the natural numbers, the number 1 is defined as the singleton {0}.

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 {A, A}, which is the same as the singleton {A} (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 category theory

Structures built on singletons often serve as terminal objects or zero objects of various categories:

Definition by indicator functions

Let S be a class defined by an indicator function

b: X \to \{0, 1\}.

Then S is called a singleton if and only if there is some yX such that for all xX,

b(x) = (x = y) \,.

Traditionally, this definition was introduced by Whitehead and Russell[2] along with the definition of the natural number 1, as

1 \ \overset{\underset{\mathrm{def}}{}}{=} \ \hat{\alpha}\{(\exists x) . \alpha = \iota \jmath x\}, where \iota \jmath x \ \overset{\underset{\mathrm{def}}{}}{=} \ \hat{y}(y = x).

See also

References

  1. 1 2 Stoll, Robert (1961). Sets, Logic and Axiomatic Theories. W. H. Freeman and Company. pp. 5–6.
  2. Whitehead, Alfred North; Bertrand Russell (1910). Principia Mathematica. p. 37.
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