# Finite set

set that has a finite number of elements

In mathematics, a finite set is a set that is not infinite. A finite set has a certain number of elements. The elements of the set can be numbered like {1, 2, ..., n} and n must either be a natural number or zero. An infinite set is a set with an unlimited number of elements.

Another definition is to say a set is finite if its cardinality (the number of its elements) is a natural number. A set with n elements is called an n-set.

A definition that is harder to understand, but which is often used by mathematicians, is to say that a set is finite if there is no strict subset that can be put in 1-to-1 correspondence with the set itself. (A strict subset of a set is one that is not equal to the set itself). For example, the set of even numbers can be put in one-to-one correspondence with the set of natural numbers (2n corresponds to n), but the set of even numbers is also a strict subset of the natural numbers (there are many numbers which are not in the set of even numbers, but which are natural numbers, for example the number 1). This means that the set of natural numbers is infinite.