Finite set
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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
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 is a 3-set – a finite set with three elements – and is a 2-subset of it.
Basic properties
Any
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 2n distinct subsets. That is, the power set P(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
The
Necessary and sufficient conditions for finiteness
In Zermelo–Fraenkel set theory without the axiom of choice (ZF), the following conditions are all equivalent:[1]
- 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. (See § Set-theoretic definitions of finiteness for a different formulation of Kuratowski finiteness.)
- (Paul Stäckel) S can be given a total ordering which is well-ordered 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 P(P(S)) into itself is powerset of the powerset of S is Dedekind-finite (see below).[2]
- Every surjective function from P(P(S)) onto itself is one-to-one.
- (maximal elementwith 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[4][citation needed]), 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.
- Every surjective function from S onto itself is one-to-one.
- S is empty or every maximal element.
Foundational issues
Even for the majority of mathematicians that embrace infinite sets, in certain important contexts, the formal distinction between the finite and the infinite can remain a delicate matter. The difficulty stems from
More generally, informal notions like set, and particularly finite set, may receive interpretations across a range of formal systems varying in their axiomatics and logical apparatus. The best known axiomatic set theories include Zermelo-Fraenkel set theory (ZF), Zermelo-Fraenkel set theory with the Axiom of Choice (ZFC), Von Neumann–Bernays–Gödel set theory (NBG), Non-well-founded set theory, Bertrand Russell's Type theory and all the theories of their various models. One may also choose among classical first-order logic, various higher-order logics and intuitionistic logic.
A
Set-theoretic definitions of finiteness
In contexts where the notion of natural number sits logically prior to any notion of set, one can define a set S as finite if S admits a bijection to some set of natural numbers of the form . Mathematicians more typically choose to ground notions of number in
Various properties that single out the finite sets among all sets in the theory ZFC turn out logically inequivalent in weaker systems such as ZF or intuitionistic set theories. Two definitions feature prominently in the literature, one due to Richard Dedekind, the other to Kazimierz Kuratowski. (Kuratowski's is the definition used above.)
A set S is called
Kuratowski finiteness is defined as follows. Given any set S, the binary operation of union endows the
Readers unfamiliar with semilattices and other notions of abstract algebra may prefer an entirely elementary formulation. Kuratowski finite means S lies in the set K(S), constructed as follows. Write M for the set of all subsets X of P(S) such that:
- X contains the empty set;
- For every set T in P(S), if X contains T then X also contains the union of T with any singleton.
Then K(S) may be defined as the intersection of M.
In ZF, Kuratowski finite implies Dedekind finite, but not vice versa. In the parlance of a popular pedagogical formulation, when the axiom of choice fails badly, one may have an infinite family of socks with no way to choose one sock from more than finitely many of the pairs. That would make the set of such socks Dedekind finite: there can be no infinite sequence of socks, because such a sequence would allow a choice of one sock for infinitely many pairs by choosing the first sock in the sequence. However, Kuratowski finiteness would fail for the same set of socks.
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.[6] (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.)
- I-finite. Every non-empty set of subsets of S has a ⊆-maximal element. (This is equivalent to requiring the existence of a ⊆-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.[7])
- II-finite. Every non-empty ⊆-monotone set of subsets of S has a ⊆-maximal element.
- III-finite. The power set P(S) is Dedekind finite.
- IV-finite. S is Dedekind finite.
- V-finite. ∣S∣ = 0 or 2 ⋅ ∣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.[8]
Most of these finiteness definitions and their names are attributed to Tarski 1954 by Howard & Rubin 1998, p. 278. However, definitions I, II, III, IV and V were presented in Tarski 1924, pp. 49, 93, 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
- Ordinal number
- Peano arithmetic
Notes
- ^ "Art of Problem Solving". artofproblemsolving.com. Retrieved 2022-09-07.
- ^ The equivalence of the standard numerical definition of finite sets to the Dedekind-finiteness of the power set of the power set was shown in 1912 by Whitehead & Russell 2009, p. 288. This Whitehead/Russell theorem is described in more modern language by Tarski 1924, pp. 73–74.
- ^ Tarski 1924, pp. 48–58, demonstrated that his definition (which is also known as I-finite) is equivalent to Kuratowski's set-theoretical definition, which he then noted is equivalent to the standard numerical definition via the proof by Kuratowski 1920, pp. 130–131.
- ISBN 9780080461083.
- ^ The original paper by Kuratowski 1920 defined a set S to be finite when
- P(S)∖{∅} = ⋂{X ∈ P(P(S)∖{∅}); (∀x∈S, {x}∈X) and (∀A,B∈X, A∪B∈X)}.
- all elements of X are non-empty subsets of S,
- the set {x} is an element of X for all x in S,
- X is closed under pairwise unions.
- ^ This list of 8 finiteness concepts is presented with this numbering scheme by both Howard & Rubin 1998, pp. 278–280, and Lévy 1958, pp. 2–3, although the details of the presentation of the definitions differ in some respects which do not affect the meanings of the concepts.
- ^ de la Cruz, Dzhafarov & Hall (2006, p. 8)
- ^ Lévy 1958 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.
References
- Apostol, Tom M. (1974), Mathematical Analysis (2nd ed.), Menlo Park: LCCN 72011473
- Cohn, Paul Moritz, F.R.S. (1981), Universal Algebra, Dordrecht: )
- ISBN 978-1-108-05038-8
- ISBN 0-486-21010-3
- de la Cruz, Omar; Dzhafarov, Damir D.; Hall, Eric J. (2006), "Definitions of finiteness based on order properties" (PDF), Fundamenta Mathematicae, 189 (2): 155–172, MR 2214576
- ISBN 3-540-30989-6
- Howard, Paul; ISBN 9780821809778.
- (PDF) from the original on 2011-05-15
- Labarre, Anthony E. Jr. (1968), Intermediate Mathematical Analysis, New York: LCCN 68019130
- (PDF) from the original on 2003-07-05.
- Rudin, Walter (1976), Principles Of Mathematical Analysis (3rd ed.), New York: ISBN 0-07-054235-X
- ISBN 0-486-61630-4
- (PDF) from the original on 2011-05-15.
- MR 0060555.
- ISBN 978-1-60386-183-0.