Singleton (mathematics)
In mathematics, a singleton, also known as a unit set, 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 member).
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 {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 {}.
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).
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