In mathematics, particularly set theory, a finite set is a set that has a finite number of elements. Informally, a finite set is a set which one could in principle count and finish counting. For example, is a finite set with five elements. The number of elements of a finite set is a natural number (possibly zero) and is called the cardinality (or the cardinal number) of the set. A set that is not a finite set is called an infinite set. 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 (natural numbers are defined as sets in Zermelo-Fraenkel set theory). The number n is the set's cardinality, denoted as |S|. 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.

Basic properties

Any proper subset of a finite set S is finite and has fewer elements than S itself. As a consequence, there cannot exist a bijection between a finite set S and a proper subset of S. Any set with this property is called Dedekind-finite. Using the standard ZFC axioms for set theory, every Dedekind-finite set is also finite, but this implication cannot be proved in ZF (Zermelo–Fraenkel axioms without the axiom of choice) alone. The axiom of countable choice, a weak version of the axiom of choice, is sufficient to prove this equivalence. Any injective function between two finite sets of the same cardinality is also a surjective function (a surjection). Similarly, any surjection between two finite sets of the same cardinality is also an injection. The union of two finite sets is finite, with In fact, by the inclusion–exclusion principle: More generally, the union of any finite number of finite sets is finite. The Cartesian product of finite sets is also finite, with: Similarly, the Cartesian product of finitely many finite sets is finite. A finite set with n elements has 2^n distinct subsets. That is, the power set \wp(S) of a finite set S is finite, with cardinality 2^{|S|}. Any subset of a finite set is finite. The set of values of a function when applied to elements of a finite set is finite. All finite sets are countable, but not all countable sets are finite. (Some authors, however, use "countable" to mean "countably infinite", so do not consider finite sets to be countable.) The free semilattice over a finite set is the set of its non-empty subsets, with the join operation being given by set union.

Necessary and sufficient conditions for finiteness

In Zermelo–Fraenkel set theory without the axiom of choice (ZF), the following conditions are all equivalent: If the axiom of choice is also assumed (the axiom of countable choice is sufficient), then the following conditions are all equivalent:

Other concepts of finiteness

In ZF set theory without the axiom of choice, the following concepts of finiteness for a set S are distinct. They are arranged in strictly decreasing order of strength, i.e. if a set S meets a criterion in the list then it meets all of the following criteria. In the absence of the axiom of choice the reverse implications are all unprovable, but if the axiom of choice is assumed then all of these concepts are equivalent. (Note that none of these definitions need the set of finite ordinal numbers to be defined first; they are all pure "set-theoretic" definitions in terms of the equality and membership relations, not involving ω.) The forward implications (from strong to weak) are theorems within ZF. Counter-examples to the reverse implications (from weak to strong) in ZF with urelements are found using model theory. Most of these finiteness definitions and their names are attributed to by. However, definitions I, II, III, IV and V were presented in, together with proofs (or references to proofs) for the forward implications. At that time, model theory was not sufficiently advanced to find the counter-examples. Each of the properties I-finite thru IV-finite is a notion of smallness in the sense that any subset of a set with such a property will also have the property. This is not true for V-finite thru VII-finite because they may have countably infinite subsets.