Finite set

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

{{bi|left=1.6|$\backslash displaystyle\; f\backslash colon\; S\backslash to\; n$}}

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:

{{bi|left=1.6|$\backslash displaystyle\; x\_1,x\_2,\backslash ldots,x\_n\; \backslash quad\; (x\_i\; \backslash in\; S,\; \backslash \; 1\; \backslash le\; i\; \backslash le\; n).$}}

If n≥2, then there are multiple such sequences.

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 $\backslash \{5,6,7\backslash \}$ is a 3-set – a finite set with three elements – and $\backslash \{6,7\backslash \}$ is a 2-subset of it.

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

{{bi|left=1.6|$\backslash displaystyle\; |S\; \backslash cup\; T|\; \backslash le\; |S|\; +\; |T|.$}}

In fact, by the inclusion–exclusion principle:

{{bi|left=1.6|$\backslash displaystyle\; |S\; \backslash cup\; T|\; =\; |S|\; +\; |T|\; -\; |S\backslash cap\; T|.$}}

More generally, the union of any finite number of finite sets is finite. The Cartesian product of finite sets is also finite, with:

{{bi|left=1.6|$\backslash displaystyle\; |S\; \backslash times\; T|\; =\; |S|\backslash times|T|.$}}

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 $\backslash wp(S)$ 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, 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.

{{anchor|Tarski finite}}

In Zermelo–Fraenkel set theory without the axiom of choice (ZF), the following conditions are all equivalent:{{Cite web |title=Art of Problem Solving |url=https://artofproblemsolving.com/wiki/index.php/Zermelo-Fraenkel_Axioms |access-date=2022-09-07 |website=artofproblemsolving.com}}

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.
2. (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.
3. (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.
4. Every one-to-one function from $\backslash wp\backslash bigl(\backslash wp(S)\backslash bigr)$ into itself is onto. That is, the powerset of the powerset of $S$ is Dedekind-finite (see below).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 {{harvnb|Whitehead|Russell|2009|p=288}}. This Whitehead/Russell theorem is described in more modern language by {{harvnb|Tarski|1924|pp=73–74}}.
5. Every surjective function from $\backslash wp\backslash bigl(\backslash wp(S)\backslash bigr)$ onto itself is one-to-one.
6. (Alfred Tarski) Every non-empty family of subsets of $S$ has a minimal element with respect to inclusion.{{harvnb|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 {{harvnb|Kuratowski|1920|pp=130–131}}. (Equivalently, every non-empty family of subsets of $S$ has a maximal element with respect to inclusion.)
7. $S$ 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),{{cite book |last1=Herrlich |first1=Horst |title=Axiom of Choice |date=2006 |publisher=Springer |series=Lecture Notes in Mathematics |volume=1876 |doi=10.1007/11601562 |isbn=3-540-30989-6 |url=https://link.springer.com/book/10.1007/11601562 |access-date=18 July 2023|contribution=Proposition 4.13|page=48}} then the following conditions are all equivalent:

1. $S$ is a finite set.
2. (Richard Dedekind) Every one-to-one function from $S$ into itself is onto. A set with this property is called Dedekind-finite.
3. Every surjective function from $S$ onto itself is one-to-one.
4. $S$ is empty or every partial ordering of $S$ contains a maximal element.

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.This list of 8 finiteness concepts is presented with this numbering scheme by both {{harvnb|Howard|Rubin|1998|pp=278–280}}, and {{harvnb|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. (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 $\backslash subseteq$-maximal element. (This is equivalent to requiring the existence of a $\backslash subseteq$-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.{{harvtxt|de la Cruz|Dzhafarov|Hall|2006|p=8}})
• II-finite. Every non-empty $\backslash subseteq$-monotone set of subsets of $S$ has a $\backslash subseteq$-maximal element.
• III-finite. The power set $\backslash wp(S)$ is Dedekind finite.
• IV-finite. $S$ is Dedekind finite.
• V-finite. $|S|=0$ or $2\backslash cdot|S|>|S|$.
• 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.{{harvnb|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.

Most of these finiteness definitions and their names are attributed to {{harvnb|Tarski|1954}} by {{harvnb|Howard|Rubin|1998|p=278}}. However, definitions I, II, III, IV and V were presented in {{harvnb|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.

• FinSet
• Discrete point set
• Ordinal number
• Peano arithmetic

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Category:Basic concepts in set theory

Category:Cardinal numbers