Finite Set

A finite set is a mathematical set whose elements can be counted and whose total number is a natural number. For example, the set {1, 2, 3} has three elements, and is therefore finite. The empty set, denoted ∅, is also finite, and is the set whose number of elements is zero.

Formally, a set A is finite if there exists a natural number n and a one-to-one correspondence (bijection) between A and the set {1, 2, …, n}. The number n is called the cardinality of the set, and is written |A| or #A. A set that does not satisfy this condition is called an infinite set or unbounded set.

Basic Properties

The union of two finite sets is a finite set, as are their intersection and difference. Every subset of a finite set is finite, and its cardinality does not exceed that of the larger set. The power set of a finite set with n elements has 2 raised to the power n elements, which is also finite.

Distinction from Infinite Sets

An important property that distinguishes finite sets is that a finite set cannot be put in one-to-one correspondence with any of its proper subsets; this is the Dedekind method used to define finiteness. Infinite sets, such as the set of natural numbers, have a proper subset whose cardinality equals that of the whole set.

Principles of Counting

Finite sets form the foundation of the branch of mathematics known as counting or combinatorics. The addition principle and the inclusion–exclusion principle calculate the number of elements in overlapping sets. The pigeonhole principle states that if more items are distributed among a finite set that is smaller, at least one place must receive two items.

Applications

The concept of finite sets applies throughout ordinary mathematics, mathematical science, and logic. In computer science, stored data is always a finite set, which affects search and sorting algorithms. Finite sets are also used in defining algebraic structures such as finite groups.