Finite

In mathematics, a set is called finite if and only if there is a bijection between the set and some set of the form {1, 2, ..., n} where n is a natural number. All finite sets are countable, but not all countable sets are finite.

~ ~ ~ ~ ~ ~ ~ ~ ~ ~

~ ~ ~ ~ ~ ~ ~ ~ ~ ~

It is a theorem (assuming the axiom of choice) that a set is finite if and only if there exists no bijection between the set and any of its proper subsets. Equivalently, a set is finite if its cardinality, i.e. the number of its elements, is a natural number. For instance, the set of integers between -15 and 3 is finite, since it has 17 elements. The set of all prime numbers is not finite. Sets that are not finite are called infinite.

~ ~ ~ ~ ~ ~ ~ ~ ~ ~

In physics, finite additionally means "non-zero", for instance in a sentence like "if the distance of the two objects is finite...".

~ ~ ~ ~ ~ ~ ~ ~ ~ ~

See also: infinity

~ ~ ~ ~ ~ ~ ~ ~ ~ ~

Set: :This article is about sets in mathematics. For other senses, see set (disambiguation)....

Bijection: In mathematics, a bijection, or a bijective function, is a function f from a set X to a set Y with the property that, for every y in Y, there is exactly one x in X such that f(x) = y and no unmapped element exists in either X or Y....

Natural number: Natural number can mean either a positive integer (, , , , ...) or a non-negative integer (, , , , , ...)....