Finite set

In mathematics, a finite set is a set that has a finite number of elements. For example,

is a finite set with five elements. The number of elements of a finite set is a natural number (a non-negative integer) and is called the cardinality of the set. A set that is not finite is called infinite. For example, the set of all positive integers is infinite:

Finite sets are particularly important in combinatorics, the mathematical study of counting. Many arguments involving finite sets rely on the pigeonhole principle, which states that there cannot exist an injective function from a larger finite set to a smaller finite set.

Definition and terminology

Formally, a set S is called finite if there exists a bijection

for some natural number n. The number n is the set's cardinality, denoted as |S|. The empty set {} or Ø is considered finite, with cardinality zero.

If a set is finite, its elements may be written — in many ways — in a sequence:

In combinatorics, a finite set with n elements is sometimes called an n-set and a subset with k elements is called a k-subset. For example, the set {5,6,7} is a 3-set – a finite set with three elements – and {6,7} is a 2-subset of it.

Hyperfinite set

In non-standard analysis, a branch of mathematics, a hyperfinite set or *-finite set is a type of internal set. An internal set H of internal cardinality g ∈ *N (the hypernaturals) is hyperfinite if and only if there exists an internal bijection between G = {1,2,3,...,g} and H. Hyperfinite sets share the properties of finite sets: A hyperfinite set has minimal and maximal elements, and a hyperfinite union of a hyperfinite collection of hyperfinite sets may be derived. The sum of the elements of any hyperfinite subset of *R always exists, leading to the possibility of well-defined integration.

Hyperfinite sets can be used to approximate other sets. If a hyperfinite set approximates an interval, it is called a near interval with respect to that interval. Consider a hyperfinite set with a hypernatural n. K is a near interval for [a,b] if k₁ = a and k_n = b, and if the difference between successive elements of K is infinitesimal. Phrased otherwise, the requirement is that for every r ∈ [a,b] there is a k_i ∈ K such that k_i ≈ r. This, for example, allows for an approximation to the unit circle, considered as the set for θ in the interval [0,2π].