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In mathematics, a singleton, also known as a unit set, is a set with exactly one element. For example, the set 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,〔 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 A set is a singleton if and only if its cardinality is . 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 , 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. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Singleton (mathematics)」の詳細全文を読む スポンサード リンク
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