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In
mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...
, particularly
set theory Set theory is the branch of mathematical logic that studies Set (mathematics), sets, which can be informally described as collections of objects. Although objects of any kind can be collected into a set, set theory – as a branch of mathema ...
, 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 In mathematics, the natural numbers are the numbers 0, 1, 2, 3, and so on, possibly excluding 0. Some start counting with 0, defining the natural numbers as the non-negative integers , while others start with 1, defining them as the positive in ...
(possibly zero) and is called the ''
cardinality The thumb is the first digit of the hand, next to the index finger. When a person is standing in the medical anatomical position (where the palm is facing to the front), the thumb is the outermost digit. The Medical Latin English noun for thum ...
(or the
cardinal number In mathematics, a cardinal number, or cardinal for short, is what is commonly called the number of elements of a set. In the case of a finite set, its cardinal number, or cardinality is therefore a natural number. For dealing with the cas ...
)'' 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 Combinatorics is an area of mathematics primarily concerned with counting, both as a means and as an end to obtaining results, and certain properties of finite structures. It is closely related to many other areas of mathematics and has many ...
, 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 In mathematics, a bijection, bijective function, or one-to-one correspondence is a function between two sets such that each element of the second set (the codomain) is the image of exactly one element of the first set (the domain). Equival ...
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 nonempty set is finite, its elements may be written in a
sequence In mathematics, a sequence is an enumerated collection of objects in which repetitions are allowed and order matters. Like a set, it contains members (also called ''elements'', or ''terms''). The number of elements (possibly infinite) is cal ...
:
If ''n''≥2, then there are multiple such sequences. In
combinatorics Combinatorics is an area of mathematics primarily concerned with counting, both as a means and as an end to obtaining results, and certain properties of finite structures. It is closely related to many other areas of mathematics and has many ...
, a finite set with n elements is sometimes called an ''n-set'' and a
subset In mathematics, a Set (mathematics), set ''A'' is a subset of a set ''B'' if all Element (mathematics), elements of ''A'' are also elements of ''B''; ''B'' is then a superset of ''A''. It is possible for ''A'' and ''B'' to be equal; if they a ...
with k elements is called a ''k-subset''. For example, the set \ is a 3-set – a finite set with three elements – and \ 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 In mathematics, a bijection, bijective function, or one-to-one correspondence is a function between two sets such that each element of the second set (the codomain) is the image of exactly one element of the first set (the domain). Equival ...
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 Set theory is the branch of mathematical logic that studies Set (mathematics), sets, which can be informally described as collections of objects. Although objects of any kind can be collected into a set, set theory – as a branch of mathema ...
, 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^. 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 In mathematics, a Set (mathematics), set is countable if either it is finite set, finite or it can be made in one to one correspondence with the set of natural numbers. Equivalently, a set is ''countable'' if there exists an injective function fro ...
, 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: # S is a finite set. That is, S can be placed into a one-to-one correspondence with the set of those natural numbers less than some specific natural number. # ( Kazimierz Kuratowski) S has all properties which can be proved by mathematical induction beginning with the empty set and adding one new element at a time. # ( Paul Stäckel) S can be given a total ordering which is
well-order In mathematics, a well-order (or well-ordering or well-order relation) on a set is a total ordering on with the property that every non-empty subset of has a least element in this ordering. The set together with the ordering is then calle ...
ed both forwards and backwards. That is, every non-empty subset of S has both a least and a greatest element in the subset. # Every one-to-one function from \wp\bigl(\wp(S)\bigr) into itself is onto. That is, the powerset of the powerset of S is Dedekind-finite (see below). # Every surjective function from \wp\bigl(\wp(S)\bigr) onto itself is one-to-one. # (
Alfred Tarski Alfred Tarski (; ; born Alfred Teitelbaum;School of Mathematics and Statistics, University of St Andrews ''School of Mathematics and Statistics, University of St Andrews''. January 14, 1901 – October 26, 1983) was a Polish-American logician ...
) Every non-empty family of subsets of S has a minimal element with respect to inclusion. (Equivalently, every non-empty family of subsets of S has a maximal element with respect to inclusion.) # S can be well-ordered and any two well-orderings on it are order isomorphic. In other words, the well-orderings on S have exactly one order type. If the axiom of choice is also assumed (the axiom of countable choice is sufficient), then the following conditions are all equivalent: # S is a finite set. # ( Richard Dedekind) Every one-to-one function from S into itself is onto. A set with this property is called Dedekind-finite. # Every surjective function from S onto itself is one-to-one. # S is empty or every partial ordering of S contains a maximal element.


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 number In set theory, an ordinal number, or ordinal, is a generalization of ordinal numerals (first, second, th, etc.) aimed to extend enumeration to infinite sets. A finite set can be enumerated by successively labeling each element with the leas ...
s to be defined first; they are all pure "set-theoretic" definitions in terms of the equality and membership relations, not involving ω.) * I-finite. Every non-empty set of subsets of S has a \subseteq-maximal element. (This is equivalent to requiring the existence of a \subseteq-minimal element. It is also equivalent to the standard numerical concept of finiteness.) * Ia-finite. For every partition of S into two sets, at least one of the two sets is I-finite. (A set with this property which is not I-finite is called an amorphous set.) * II-finite. Every non-empty \subseteq-monotone set of subsets of S has a \subseteq-maximal element. * III-finite. The power set \wp(S) is Dedekind finite. * IV-finite. S is Dedekind finite. * V-finite. , S, =0 or 2\cdot, S, >, S, . * VI-finite. , S, =0 or , S, =1 or , S, ^2>, S, . * VII-finite. S is I-finite or not well-orderable. 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 In mathematical logic, model theory is the study of the relationship between theory (mathematical logic), formal theories (a collection of Sentence (mathematical logic), sentences in a formal language expressing statements about a Structure (mat ...
. found counter-examples to each of the reverse implications in Mostowski models. Lévy attributes most of the results to earlier papers by Mostowski and Lindenbaum. 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.


See also

* FinSet * Discrete point set *
Ordinal number In set theory, an ordinal number, or ordinal, is a generalization of ordinal numerals (first, second, th, etc.) aimed to extend enumeration to infinite sets. A finite set can be enumerated by successively labeling each element with the leas ...
* Peano arithmetic


Notes


References

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External links

* {{Set theory Basic concepts in set theory Cardinal numbers