Finite

Finite describes objects with bounded size or extent. In mathematics, a set is finite if it has a finite number of elements; equivalently there exists a bijection with {1,...,n} for some natural number n. An object is finite if it has finitely many components or terms, such as a finite sequence.

Subsets of finite sets are finite, and the union of finitely many finite sets is finite; the Cartesian product of finite sets is finite. A set is finite exactly when its cardinality is a nonnegative integer. The contrast is with infinite sets, which have infinitely many elements. For example, {1,2,3} is finite; the set of natural numbers N is infinite.

In computer science and algebra, finite objects appear as finite automata (machines with a finite number of states), finite fields (fields with finitely many elements), and finite groups (groups with finite order). Finite structures are central in combinatorics, probability with finite sample spaces, and algorithm design, where finiteness guarantees termination and computability.