Finite set

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In mathematics, a finite set simply is set which has finitely many elements, i.e., its cardinality (the number of its elements) is a natural number. A set which is not finite is an infinite set and has at least as many elements as there are natural numbers.

A characteristic property of finite sets (which, in set theory is used to define finite sets) is the following:

A set is finite if and only if every proper subset has fewer elements than the set.

In other words, if — from a given set with n elements (n a natural number) — one or more elements are removed, then the remaining set has less (at most n-1) elements. (Note that the empty set also has this property, even though it has no proper subset.)

On the other hand, removing an element from an infinite set does not change its size. Simple examples, such as Galileo's paradox and Hilbert's hotel, show that even infinitely many elements — if they are chosen accordingly — can be removed without changing its size.

The definition can be formalised in terms of functions. A set X is finite if and only if, for any function f from X to itself, f is surjective if and only if f is injective.