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In mathematics, a countable set is a set with the same cardinality (number of elements) as some subset of the set of natural numbers. A countable set is either a finite set or a ''countably infinite'' set. Whether finite or infinite, the elements of a countable set can always be counted one at a time and, although the counting may never finish, every element of the set is associated with a natural number. Some authors use countable set to mean ''countably infinite'' alone. To avoid this ambiguity, the term ''at most countable'' may be used when finite sets are included and ''countably infinite'', ''enumerable'', or ''denumerable'' otherwise. Georg Cantor introduced the term ''countable set'', contrasting sets that are countable with those that are ''uncountable'' (i.e., ''nonenumerable'' or ''nondenumerable''). Today, countable sets form the foundation of a branch of mathematics called ''discrete mathematics''. ==Definition== A set is ''countable'' if there exists an injective function from to the natural numbers }.〔Since there is an obvious bijection between and }, it makes no difference whether one considers 0 a natural number or not. In any case, this article follows ISO 31-11 and the standard convention in mathematical logic, which takes 0 as a natural number.〕 If such an can be found that is also surjective (and therefore bijective), then is called ''countably infinite.'' In other words, a set is ''countably infinite'' if it has one-to-one correspondence with the natural number set, . As noted above, this terminology is not universal.: Some authors use countable to mean what is here called ''countably infinite,'' and do not include finite sets. Alternative (equivalent) formulations of the definition in terms of a bijective function or a surjective function can also be given. See below. 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「countable set」の詳細全文を読む スポンサード リンク
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