Countable set

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In mathematics, a countable set is a set with the same cardinality (number of elements) as some subset of the set of natural numbers. A countable set is either a finite set or a countably infinite set. Whether finite or infinite, the elements of a countable set can always be counted one at a time and—although the counting may never finish—every element of the set is associated with a unique natural number.

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Georg Cantor introduced the concept of countable sets, contrasting sets that are countable with those that are uncountable. Today, countable sets form the foundation of a branch of mathematics called discrete mathematics.

A note on terminology

Although the terms "countable" and "countably infinite" as defined here are quite common, the terminology is not universal. [1] Another common style uses countable to mean what is here called countably infinite, and at most countable to mean what is here called countable. [2] [3] To avoid ambiguity, one may limit oneself to the terms "at most countable" and "countably infinite", although with respect to concision this is the worst of both worlds.[ citation needed ] The reader is advised to check the definition in use when encountering the term "countable" in the literature.

The terms enumerable [4] and denumerable [5] [6] may also be used, e.g. referring to countable and countably infinite respectively, [7] but as definitions vary the reader is once again advised to check the definition in use. [8]

Definition

The most concise definition is in terms of cardinality. A set S is countable if its cardinality |S| is less than or equal to (aleph-null), the cardinality of the set of natural numbers N. A set S is countably infinite if |S| = . A set is uncountable if it is not countable, i.e. its cardinality is greater than ; the reader is referred to Uncountable set for further discussion. [9]

Equivalently, a set S is countable iff: [5]

Similarly, a set S is countably infinite iff:

History

In 1874, in his first set theory article, Cantor proved that the set of real numbers is uncountable, thus showing that not all infinite sets are countable. [17] In 1878, he used one-to-one correspondences to define and compare cardinalities. [18] In 1883, he extended the natural numbers with his infinite ordinals, and used sets of ordinals to produce an infinity of sets having different infinite cardinalities. [19]

Introduction

A set is a collection of elements, and may be described in many ways. One way is simply to list all of its elements; for example, the set consisting of the integers 3, 4, and 5 may be denoted {3, 4, 5}, called roster form. [20] This is only effective for small sets, however; for larger sets, this would be time-consuming and error-prone. Instead of listing every single element, sometimes an ellipsis ("...") is used to represent many elements between the starting element and the end element in a set, if the writer believes that the reader can easily guess what ... represents; for example, {1, 2, 3, ..., 100} presumably denotes the set of integers from 1 to 100. Even in this case, however, it is still possible to list all the elements, because number of elements in the set is finite.

Some sets are infinite; these sets have more than n elements where n is any integer that can be specified. (No matter how large the specified integer n is, such as n = 9 × 1032, infinite sets have more than n elements.) For example, the set of natural numbers, denotable by {0, 1, 2, 3, 4, 5, ...}, [lower-alpha 1] has infinitely many elements, and we cannot use any natural number to give its size. Nonetheless, it turns out that infinite sets do have a well-defined notion of size (or more properly, cardinality, the technical term for the number of elements in a set), and not all infinite sets have the same cardinality.

Bijective mapping from integer to even numbers Aplicacion 2 inyectiva sobreyectiva02.svg
Bijective mapping from integer to even numbers

To understand what this means, we first examine what it does not mean. For example, there are infinitely many odd integers, infinitely many even integers, and (hence) infinitely many integers overall. However, it turns out that the number of even integers, which is the same as the number of odd integers, is also the same as the number of integers overall. This is because we can arrange things such that, for every integer, there is a distinct even integer:

or, more generally, (see picture). What we have done here is arrange the integers and the even integers into a one-to-one correspondence (or bijection ), which is a function that maps between two sets such that each element of each set corresponds to a single element in the other set.

However, not all infinite sets have the same cardinality. For example, Georg Cantor (who introduced this concept) demonstrated that the real numbers cannot be put into one-to-one correspondence with the natural numbers (non-negative integers), and therefore that the set of real numbers has a greater cardinality than the set of natural numbers.

Formal overview

By definition, a set S is countable if there exists an injective function f : SN from S to the natural numbers N = {0, 1, 2, 3, ...}. It simply means that every element in S has the correspondence to a different element in N.

It might seem natural to divide the sets into different classes: put all the sets containing one element together; all the sets containing two elements together; ...; finally, put together all infinite sets and consider them as having the same size. This view is not tenable, however, under the natural definition of size.

To elaborate this, we need the concept of a bijection. Although a "bijection" may seem a more advanced concept than a number, the usual development of mathematics in terms of set theory defines functions before numbers, as they are based on much simpler sets. This is where the concept of a bijection comes in: define the correspondence

a ↔ 1, b ↔ 2, c ↔ 3

Since every element of {a, b, c} is paired with precisely one element of {1, 2, 3}, and vice versa, this defines a bijection.

We now generalize this situation; we define that two sets are of the same size, if and only if there is a bijection between them. For all finite sets, this gives us the usual definition of "the same size".

As for the case of infinite sets, consider the sets A = {1, 2, 3, ... }, the set of positive integers, and B = {2, 4, 6, ... }, the set of even positive integers. We claim that, under our definition, these sets have the same size, and that therefore B is countably infinite. Recall that to prove this, we need to exhibit a bijection between them. This can be achieved using the assignment n ↔ 2n, so that

1 ↔ 2, 2 ↔ 4, 3 ↔ 6, 4 ↔ 8, ....

As in the earlier example, every element of A has been paired off with precisely one element of B, and vice versa. Hence they have the same size. This is an example of a set of the same size as one of its proper subsets, which is impossible for finite sets.

Likewise, the set of all ordered pairs of natural numbers (the Cartesian product of two sets of natural numbers, N × N) is countably infinite, as can be seen by following a path like the one in the picture:

The Cantor pairing function assigns one natural number to each pair of natural numbers Pairing natural.svg
The Cantor pairing function assigns one natural number to each pair of natural numbers

The resulting mapping proceeds as follows:

0 ↔ (0, 0), 1 ↔ (1, 0), 2 ↔ (0, 1), 3 ↔ (2, 0), 4 ↔ (1, 1), 5 ↔ (0, 2), 6 ↔ (3, 0), ....

This mapping covers all such ordered pairs.

This form of triangular mapping recursively generalizes to n-tuples of natural numbers, i.e., (a1, a2, a3, ..., an) where ai and n are natural numbers, by repeatedly mapping the first two elements of a n-tuple to a natural number. For example, (0, 2, 3) can be written as ((0, 2), 3). Then (0, 2) maps to 5 so ((0, 2), 3) maps to (5, 3), then (5, 3) maps to 39. Since a different 2-tuple, that is a pair such as (a, b), maps to a different natural number, a difference between two n-tuples by a single element is enough to ensure the n-tuples being mapped to different natural numbers. So, an injection from the set of n-tuples to the set of natural numbers N is proved. For the set of n-tuples made by the Cartesian product of finitely many different sets, each element in each tuple has the correspondence to a natural number, so every tuple can be written in natural numbers then the same logic is applied to prove the theorem.

Theorem: The Cartesian product of finitely many countable sets is countable. [21] [lower-alpha 2]

The set of all integers Z and the set of all rational numbers Q may intuitively seem much bigger than N. But looks can be deceiving. If a pair is treated as the numerator and denominator of a vulgar fraction (a fraction in the form of a/b where a and b ≠ 0 are integers), then for every positive fraction, we can come up with a distinct natural number corresponding to it. This representation also includes the natural numbers, since every natural number is also a fraction N/1. So we can conclude that there are exactly as many positive rational numbers as there are positive integers. This is also true for all rational numbers, as can be seen below.

Theorem:Z (the set of all integers) and Q (the set of all rational numbers) are countable. [lower-alpha 3]

In a similar manner, the set of algebraic numbers is countable. [23] [lower-alpha 4]

Sometimes more than one mapping is useful: a set A to be shown as countable is one-to-one mapped (injection) to another set B, then A is proved as countable if B is one-to-one mapped to the set of natural numbers. For example, the set of positive rational numbers can easily be one-to-one mapped to the set of natural number pairs (2-tuples) because p/q maps to (p, q). Since the set of natural number pairs is one-to-one mapped (actually one-to-one correspondence or bijection) to the set of natural numbers as shown above, the positive rational number set is proved as countable.

Theorem: Any finite union of countable sets is countable. [24] [25] [lower-alpha 5]

With the foresight of knowing that there are uncountable sets, we can wonder whether or not this last result can be pushed any further. The answer is "yes" and "no", we can extend it, but we need to assume a new axiom to do so.

Theorem: (Assuming the axiom of countable choice) The union of countably many countable sets is countable. [lower-alpha 6]

For example, given countable sets a, b, c, ...

Enumeration for countable number of countable sets Countablepath.svg
Enumeration for countable number of countable sets

Using a variant of the triangular enumeration we saw above:

This only works if the sets a, b, c, ... are disjoint. If not, then the union is even smaller and is therefore also countable by a previous theorem.

We need the axiom of countable choice to index all the sets a, b, c, ... simultaneously.

Theorem: The set of all finite-length sequences of natural numbers is countable.

This set is the union of the length-1 sequences, the length-2 sequences, the length-3 sequences, each of which is a countable set (finite Cartesian product). So we are talking about a countable union of countable sets, which is countable by the previous theorem.

Theorem: The set of all finite subsets of the natural numbers is countable.

The elements of any finite subset can be ordered into a finite sequence. There are only countably many finite sequences, so also there are only countably many finite subsets.

Theorem: Let S and T be sets.

  1. If the function f : ST is injective and T is countable then S is countable.
  2. If the function g : ST is surjective and S is countable then T is countable.

These follow from the definitions of countable set as injective / surjective functions. [lower-alpha 7]

Cantor's theorem asserts that if A is a set and P(A) is its power set, i.e. the set of all subsets of A, then there is no surjective function from A to P(A). A proof is given in the article Cantor's theorem. As an immediate consequence of this and the Basic Theorem above we have:

Proposition: The set P(N) is not countable; i.e. it is uncountable.

For an elaboration of this result see Cantor's diagonal argument.

The set of real numbers is uncountable, [lower-alpha 8] and so is the set of all infinite sequences of natural numbers.

Minimal model of set theory is countable

If there is a set that is a standard model (see inner model) of ZFC set theory, then there is a minimal standard model (see Constructible universe). The Löwenheim–Skolem theorem can be used to show that this minimal model is countable. The fact that the notion of "uncountability" makes sense even in this model, and in particular that this model M contains elements that are:

was seen as paradoxical in the early days of set theory, see Skolem's paradox for more.

The minimal standard model includes all the algebraic numbers and all effectively computable transcendental numbers, as well as many other kinds of numbers.

Total orders

Countable sets can be totally ordered in various ways, for example:

In both examples of well orders here, any subset has a least element; and in both examples of non-well orders, some subsets do not have a least element. This is the key definition that determines whether a total order is also a well order.

See also

Notes

  1. Since there is an obvious bijection between N and N* = {1, 2, 3, ...}, it makes no difference whether one considers 0 a natural number or not. In any case, this article follows ISO 31-11 and the standard convention in mathematical logic, which takes 0 as a natural number.
  2. Proof: Observe that N × N is countable as a consequence of the definition because the function f : N × NN given by f(m, n) = 2m3n is injective. [22] It then follows that the Cartesian product of any two countable sets is countable, because if A and B are two countable sets there are surjections f : NA and g : NB. So
    f × g : N × NA × B
    is a surjection from the countable set N × N to the set A × B and the Corollary implies A × B is countable. This result generalizes to the Cartesian product of any finite collection of countable sets and the proof follows by induction on the number of sets in the collection.
  3. Proof: The integers Z are countable because the function f : ZN given by f(n) = 2n if n is non-negative and f(n) = 3n if n is negative, is an injective function. The rational numbers Q are countable because the function g : Z × NQ given by g(m, n) = m/(n + 1) is a surjection from the countable set Z × N to the rationals Q.
  4. Proof: Per definition, every algebraic number (including complex numbers) is a root of a polynomial with integer coefficients. Given an algebraic number , let be a polynomial with integer coefficients such that is the kth root of the polynomial, where the roots are sorted by absolute value from small to big, then sorted by argument from small to big. We can define an injection (i. e. one-to-one) function f : AQ given by , while is the n-th prime.
  5. Proof: If Ai is a countable set for each i in I={1,...,n}, then for each n there is a surjective function gi : NAi and hence the function
    given by G(i, m) = gi(m) is a surjection. Since I × N is countable, the union is countable.
  6. Proof: As in the finite case, but I=N and we use the axiom of countable choice to pick for each i in N a surjection gi from the non-empty collection of surjections from N to Ai.
  7. Proof: For (1) observe that if T is countable there is an injective function h : TN. Then if f : ST is injective the composition h o f : SN is injective, so S is countable. For (2) observe that if S is countable, either S is empty or there is a surjective function h : NS. Then if g : ST is surjective, either S and T are both empty, or the composition g o h : NT is surjective. In either case T is countable.
  8. See Cantor's first uncountability proof, and also Finite intersection property#Applications for a topological proof.

Citations

  1. Manetti, Marco (19 June 2015). Topology. Springer. p. 26. ISBN   978-3-319-16958-3.
  2. Rudin 1976 , Chapter 2
  3. Tao & 2016 181
  4. Kamke 1950 , p. 2
  5. 1 2 Lang 1993 , §2 of Chapter I
  6. Apostol 1969 , p. 23, Chapter 1.14
  7. Thierry, Vialar (4 April 2017). Handbook of Mathematics. BoD - Books on Demand. p. 24. ISBN   978-2-9551990-1-5.
  8. Mukherjee, Subir Kumar (2009). First Course in Real Analysis. Academic Publishers. p. 22. ISBN   978-81-89781-90-3.
  9. Yaqub, Aladdin M. (24 October 2014). An Introduction to Metalogic. Broadview Press. ISBN   978-1-4604-0244-3.
  10. Singh, Tej Bahadur (17 May 2019). Introduction to Topology. Springer. p. 422. ISBN   978-981-13-6954-4.
  11. 1 2 Katzourakis, Nikolaos; Varvaruca, Eugen (2 January 2018). An Illustrative Introduction to Modern Analysis. CRC Press. ISBN   978-1-351-76532-9.
  12. Halmos 1960 , p. 91
  13. Weisstein, Eric W. "Countable Set". mathworld.wolfram.com. Retrieved 2020-09-06.
  14. Kamke 1950 , p. 2
  15. Dlab, Vlastimil; Williams, Kenneth S. (9 June 2020). Invitation To Algebra: A Resource Compendium For Teachers, Advanced Undergraduate Students And Graduate Students In Mathematics. World Scientific. p. 8. ISBN   978-981-12-1999-3.
  16. Tao & 2016 182
  17. Stillwell, John C. (2010), Roads to Infinity: The Mathematics of Truth and Proof, CRC Press, p. 10, ISBN   9781439865507, Cantor's discovery of uncountable sets in 1874 was one of the most unexpected events in the history of mathematics. Before 1874, infinity was not even considered a legitimate mathematical subject by most people, so the need to distinguish between countable and uncountable infinities could not have been imagined.
  18. Cantor 1878, p. 242.
  19. Ferreirós 2007, pp. 268, 272273.
  20. "What Are Sets and Roster Form?". expii. 2021-05-09.
  21. Halmos 1960 , p. 92
  22. Avelsgaard 1990, p. 182
  23. Kamke 1950 , pp. 3–4
  24. Avelsgaard 1990 , p. 180
  25. Fletcher & Patty 1988 , p. 187

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