Countable set

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"Countable" redirects here. Countable set_sentence_0

For the linguistic concept, see Count noun. Countable set_sentence_1

For the political media company, see Countable (app). Countable set_sentence_2

Not to be confused with (recursively) enumerable sets. Countable set_sentence_3

In mathematics, a countable set is a set with the same cardinality (number of elements) as some subset of the set of natural numbers. Countable set_sentence_4

A countable set is either a finite set or a countably infinite set. Countable set_sentence_5

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. Countable set_sentence_6

Some authors use countable set to mean countably infinite alone. Countable set_sentence_7

To avoid this ambiguity, the term at most countable may be used when finite sets are included and countably infinite, enumerable, or denumerable otherwise. Countable set_sentence_8

Georg Cantor introduced the term countable set, contrasting sets that are countable with those that are uncountable (i.e., nonenumerable or nondenumerable). Countable set_sentence_9

Today, countable sets form the foundation of a branch of mathematics called discrete mathematics. Countable set_sentence_10

Definition Countable set_section_0

A set S is countable if there exists an injective function f from S to the natural numbers N = {0, 1, 2, 3, ...}. Countable set_sentence_11

If such an f can be found that is also surjective (and therefore bijective), then S is called countably infinite. Countable set_sentence_12

This terminology is not universal. Countable set_sentence_13

Some authors use countable to mean what is here called countably infinite, and do not include finite sets. Countable set_sentence_14

Alternative (equivalent) formulations of the definition in terms of a bijective function or a surjective function can also be given. Countable set_sentence_15

See § Formal overview without details below. Countable set_sentence_16

History Countable set_section_1

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. Countable set_sentence_17

In 1878, he used one-to-one correspondences to define and compare cardinalities. Countable set_sentence_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. Countable set_sentence_19

Introduction Countable set_section_2

A set is a collection of elements, and may be described in many ways. Countable set_sentence_20

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}. Countable set_sentence_21

This is only effective for small sets, however; for larger sets, this would be time-consuming and error-prone. Countable set_sentence_22

Instead of listing every single element, sometimes an ellipsis ("...") is used, if the writer believes that the reader can easily guess what is missing; for example, {1, 2, 3, ..., 100} presumably denotes the set of integers from 1 to 100. Countable set_sentence_23

Even in this case, however, it is still possible to list all the elements, because the set is finite. Countable set_sentence_24

Some sets are infinite; these sets have more than n elements for any integer n. For example, the set of natural numbers, denotable by {0, 1, 2, 3, 4, 5, ...}, has infinitely many elements, and we cannot use any normal number to give its size. Countable set_sentence_25

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. Countable set_sentence_26

To understand what this means, we first examine what it does not mean. Countable set_sentence_27

For example, there are infinitely many odd integers, infinitely many even integers, and (hence) infinitely many integers overall. Countable set_sentence_28

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. Countable set_sentence_29

This is because we can arrange things such that for every integer, there is a distinct even integer: ... −2→−4, −1→−2, 0→0, 1→2, 2→4, ...; or, more generally, n→2n (see picture). Countable set_sentence_30

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. Countable set_sentence_31

However, not all infinite sets have the same cardinality. Countable set_sentence_32

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. Countable set_sentence_33

A set is countable if: (1) it is finite, or (2) it has the same cardinality (size) as the set of natural numbers (i.e., denumerable). Countable set_sentence_34

Equivalently, a set is countable if it has the same cardinality as some subset of the set of natural numbers. Countable set_sentence_35

Otherwise, it is uncountable. Countable set_sentence_36

Formal overview without details Countable set_section_3

By definition, a set S is countable if there exists an injective function f : S → N from S to the natural numbers N = {0, 1, 2, 3, ...}. Countable set_sentence_37

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. Countable set_sentence_38

This view is not tenable, however, under the natural definition of size. Countable set_sentence_39

To elaborate this, we need the concept of a bijection. Countable set_sentence_40

Although a "bijection" seems 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. Countable set_sentence_41

This is where the concept of a bijection comes in: define the correspondence Countable set_sentence_42

Countable set_description_list_0

  • a ↔ 1, b ↔ 2, c ↔ 3Countable set_item_0_0

Since every element of {a, b, c} is paired with precisely one element of {1, 2, 3}, and vice versa, this defines a bijection. Countable set_sentence_43

We now generalize this situation and define two sets as of the same size, if and only if there is a bijection between them. Countable set_sentence_44

For all finite sets, this gives us the usual definition of "the same size". Countable set_sentence_45

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. Countable set_sentence_46

We claim that, under our definition, these sets have the same size, and that therefore B is countably infinite. Countable set_sentence_47

Recall that to prove this, we need to exhibit a bijection between them. Countable set_sentence_48

This can be achieved using the assignment n ↔ 2n, so that Countable set_sentence_49

Countable set_description_list_1

  • 1 ↔ 2, 2 ↔ 4, 3 ↔ 6, 4 ↔ 8, ....Countable set_item_1_1

As in the earlier example, every element of A has been paired off with precisely one element of B, and vice versa. Countable set_sentence_50

Hence they have the same size. Countable set_sentence_51

This is an example of a set of the same size as one of its proper subsets, which is impossible for finite sets. Countable set_sentence_52

Likewise, the set of all ordered pairs of natural numbers is countably infinite, as can be seen by following a path like the one in the picture: Countable set_sentence_53

The resulting mapping proceeds as follows: Countable set_sentence_54

Countable set_description_list_2

  • 0 ↔ (0,0), 1 ↔ (1,0), 2 ↔ (0,1), 3 ↔ (2,0), 4 ↔ (1,1), 5 ↔ (0,2), 6 ↔ (3,0) ....Countable set_item_2_2

This mapping covers all such ordered pairs. Countable set_sentence_55

If each pair is treated as the numerator and denominator of a vulgar fraction, then for every positive fraction, we can come up with a distinct number corresponding to it. Countable set_sentence_56

This representation includes also the natural numbers, since every natural number is also a fraction N/1. Countable set_sentence_57

So we can conclude that there are exactly as many positive rational numbers as there are positive integers. Countable set_sentence_58

This is true also for all rational numbers, as can be seen below. Countable set_sentence_59

Theorem: The Cartesian product of finitely many countable sets is countable. Countable set_sentence_60

This form of triangular mapping recursively generalizes to vectors of finitely many natural numbers, by repeatedly mapping the first two elements to a natural number. Countable set_sentence_61

For example, (0,2,3) maps to (5,3), which maps to 39. Countable set_sentence_62

Sometimes more than one mapping is useful: the set to be shown to be countably infinite is mapped onto another set, then this other set is mapped onto to the natural numbers. Countable set_sentence_63

For example, the positive rational numbers can easily be mapped to (a subset of) the pairs of natural numbers because p/q maps to (p, q). Countable set_sentence_64

The following theorem concerns infinite subsets of countably infinite sets. Countable set_sentence_65

Theorem: Every subset of a countable set is countable. Countable set_sentence_66

In particular, every infinite subset of a countably infinite set is countably infinite. Countable set_sentence_67

For example, the set of prime numbers is countable, by mapping the n-th prime number to n: Countable set_sentence_68

Countable set_unordered_list_3

  • 2 maps to 1Countable set_item_3_3
  • 3 maps to 2Countable set_item_3_4
  • 5 maps to 3Countable set_item_3_5
  • 7 maps to 4Countable set_item_3_6
  • 11 maps to 5Countable set_item_3_7
  • 13 maps to 6Countable set_item_3_8
  • 17 maps to 7Countable set_item_3_9
  • 19 maps to 8Countable set_item_3_10
  • 23 maps to 9Countable set_item_3_11
  • ...Countable set_item_3_12

There are also sets which are "naturally larger than" N. For instance, Z the set of all integers or Q, the set of all rational numbers, which intuitively may seem much bigger than N. But looks can be deceiving, for we assert: Countable set_sentence_69

Theorem: Z (the set of all integers) and Q (the set of all rational numbers) are countable. Countable set_sentence_70

In a similar manner, the set of algebraic numbers is countable. Countable set_sentence_71

These facts follow easily from a result that many individuals find non-intuitive. Countable set_sentence_72

Theorem: Any finite union of countable sets is countable. Countable set_sentence_73

With the foresight of knowing that there are uncountable sets, we can wonder whether or not this last result can be pushed any further. Countable set_sentence_74

The answer is "yes" and "no", we can extend it, but we need to assume a new axiom to do so. Countable set_sentence_75

Theorem: (Assuming the axiom of countable choice) The union of countably many countable sets is countable. Countable set_sentence_76

For example, given countable sets a, b, c, ... Countable set_sentence_77

Using a variant of the triangular enumeration we saw above: Countable set_sentence_78

Countable set_unordered_list_4

  • a0 maps to 0Countable set_item_4_13
  • a1 maps to 1Countable set_item_4_14
  • b0 maps to 2Countable set_item_4_15
  • a2 maps to 3Countable set_item_4_16
  • b1 maps to 4Countable set_item_4_17
  • c0 maps to 5Countable set_item_4_18
  • a3 maps to 6Countable set_item_4_19
  • b2 maps to 7Countable set_item_4_20
  • c1 maps to 8Countable set_item_4_21
  • d0 maps to 9Countable set_item_4_22
  • a4 maps to 10Countable set_item_4_23
  • ...Countable set_item_4_24

This only works if the sets a, b, c, ... are disjoint. Countable set_sentence_79

If not, then the union is even smaller and is therefore also countable by a previous theorem. Countable set_sentence_80

We need the axiom of countable choice to index all the sets a, b, c, ... simultaneously. Countable set_sentence_81

Theorem: The set of all finite-length sequences of natural numbers is countable. Countable set_sentence_82

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). Countable set_sentence_83

So we are talking about a countable union of countable sets, which is countable by the previous theorem. Countable set_sentence_84

Theorem: The set of all finite subsets of the natural numbers is countable. Countable set_sentence_85

The elements of any finite subset can be ordered into a finite sequence. Countable set_sentence_86

There are only countably many finite sequences, so also there are only countably many finite subsets. Countable set_sentence_87

The following theorem gives equivalent formulations in terms of a bijective function or a surjective function. Countable set_sentence_88

A proof of this result can be found in Lang's text. Countable set_sentence_89

(Basic) Theorem: Let S be a set. Countable set_sentence_90

The following statements are equivalent: Countable set_sentence_91

Countable set_ordered_list_5

  1. S is countable, i.e. there exists an injective function f : S → N.Countable set_item_5_25
  2. Either S is empty or there exists a surjective function g : N → S.Countable set_item_5_26
  3. Either S is finite or there exists a bijection h : N → S.Countable set_item_5_27

Corollary: Let S and T be sets. Countable set_sentence_92

Countable set_ordered_list_6

  1. If the function f : S → T is injective and T is countable then S is countable.Countable set_item_6_28
  2. If the function g : S → T is surjective and S is countable then T is countable.Countable set_item_6_29

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). Countable set_sentence_93

A proof is given in the article Cantor's theorem. Countable set_sentence_94

As an immediate consequence of this and the Basic Theorem above we have: Countable set_sentence_95

Proposition: The set P(N) is not countable; i.e. it is uncountable. Countable set_sentence_96

For an elaboration of this result see Cantor's diagonal argument. Countable set_sentence_97

The set of real numbers is uncountable (see Cantor's first uncountability proof), and so is the set of all infinite sequences of natural numbers. Countable set_sentence_98

Some technical details Countable set_section_4

The proofs of the statements in the above section rely upon the existence of functions with certain properties. Countable set_sentence_99

This section presents functions commonly used in this role, but not the verifications that these functions have the required properties. Countable set_sentence_100

The Basic Theorem and its Corollary are often used to simplify proofs. Countable set_sentence_101

Observe that N in that theorem can be replaced with any countably infinite set. Countable set_sentence_102

Proposition: Any finite set is countable. Countable set_sentence_103

Proof: Let S be such a set. Countable set_sentence_104

Two cases are to be considered: Either S is empty or it isn't. Countable set_sentence_105

1.) Countable set_sentence_106

The empty set is even itself a subset of the natural numbers, so it is countable. Countable set_sentence_107

2.) Countable set_sentence_108

If S is nonempty and finite, then by definition of finiteness there is a bijection between S and the set {1, 2, ..., n} for some positive natural number n. This function is an injection from S into N. Countable set_sentence_109

Proposition: Any subset of a countable set is countable. Countable set_sentence_110

Proof: The restriction of an injective function to a subset of its domain is still injective. Countable set_sentence_111

Proposition: If S is a countable set then S ∪ {x} is countable. Countable set_sentence_112

Proof: If x ∈ S there is nothing to be shown. Countable set_sentence_113

Otherwise let f: S → N be an injection. Countable set_sentence_114

Define g: S ∪ {x} → N by g(x) = 0 and g(y) = f(y) + 1 for all y in S. This function g is an injection. Countable set_sentence_115

Proposition: If A and B are countable sets then A ∪ B is countable. Countable set_sentence_116

Proof: Let f: A → N and g: B → N be injections. Countable set_sentence_117

Define a new injection h: A ∪ B → N by h(x) = 2f(x) if x is in A and h(x) = 2g(x) + 1 if x is in B but not in A. Countable set_sentence_118

Proposition: The Cartesian product of two countable sets A and B is countable. Countable set_sentence_119

Proof: Observe that N × N is countable as a consequence of the definition because the function f : N × N → N given by f(m, n) = 23 is injective. Countable set_sentence_120

It then follows from the Basic Theorem and the Corollary that the Cartesian product of any two countable sets is countable. Countable set_sentence_121

This follows because if A and B are countable there are surjections f : N → A and g : N → B. Countable set_sentence_122

So Countable set_sentence_123

Countable set_description_list_7

  • f × g : N × N → A × BCountable set_item_7_30

is a surjection from the countable set N × N to the set A × B and the Corollary implies A × B is countable. Countable set_sentence_124

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. Countable set_sentence_125

Proposition: The integers Z are countable and the rational numbers Q are countable. Countable set_sentence_126

Proof: The integers Z are countable because the function f : Z → N given by f(n) = 2 if n is non-negative and f(n) = 3 if n is negative, is an injective function. Countable set_sentence_127

The rational numbers Q are countable because the function g : Z × N → Q given by g(m, n) = m/(n + 1) is a surjection from the countable set Z × N to the rationals Q. Countable set_sentence_128

Proposition: The algebraic numbers A are countable. Countable set_sentence_129

Proposition: If An is a countable set for each n in N then the union of all An is also countable. Countable set_sentence_130

Proof: This is a consequence of the fact that for each n there is a surjective function gn : N → An and hence the function Countable set_sentence_131

given by G(n, m) = gn(m) is a surjection. Countable set_sentence_132

Since N × N is countable, the Corollary implies that the union is countable. Countable set_sentence_133

We use the axiom of countable choice in this proof to pick for each n in N a surjection gn from the non-empty collection of surjections from N to An. Countable set_sentence_134

A topological proof for the uncountability of the real numbers is described at finite intersection property. Countable set_sentence_135

Minimal model of set theory is countable Countable set_section_5

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). Countable set_sentence_136

The Löwenheim–Skolem theorem can be used to show that this minimal model is countable. Countable set_sentence_137

The fact that the notion of "uncountability" makes sense even in this model, and in particular that this model M contains elements that are: Countable set_sentence_138

Countable set_unordered_list_8

  • subsets of M, hence countable,Countable set_item_8_31
  • but uncountable from the point of view of M,Countable set_item_8_32

was seen as paradoxical in the early days of set theory, see Skolem's paradox for more. Countable set_sentence_139

The minimal standard model includes all the algebraic numbers and all effectively computable transcendental numbers, as well as many other kinds of numbers. Countable set_sentence_140

Total orders Countable set_section_6

Countable sets can be totally ordered in various ways, for example: Countable set_sentence_141

Countable set_unordered_list_9

  • Well-orders (see also ordinal number):Countable set_item_9_33
    • The usual order of natural numbers (0, 1, 2, 3, 4, 5, ...)Countable set_item_9_34
    • The integers in the order (0, 1, 2, 3, ...; −1, −2, −3, ...)Countable set_item_9_35
  • Other (not well orders):Countable set_item_9_36
    • The usual order of integers (..., −3, −2, −1, 0, 1, 2, 3, ...)Countable set_item_9_37
    • The usual order of rational numbers (Cannot be explicitly written as an ordered list!)Countable set_item_9_38

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. Countable set_sentence_142

This is the key definition that determines whether a total order is also a well order. Countable set_sentence_143

See also Countable set_section_7

Countable set_unordered_list_10


Credits to the contents of this page go to the authors of the corresponding Wikipedia page: en.wikipedia.org/wiki/Countable set.