Countable set

In mathematics, a set is countable if it has the same cardinality (the number of elements of the set) as some subset of the set of natural numbers N = {0, 1, 2, 3, ...}. Equivalently, a set S is countable if there exists an injective function f : SN from S to N; it simply means that every element in S corresponds to a different element in N.

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 due to the infinite number of the elements to be counted — every element of the set is associated with a unique natural number.

Georg Cantor introduced the concept of countable sets, contrasting sets that are countable with those that are uncountable. Today, countable sets form the foundation of a branch of mathematics called discrete mathematics.

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