Sets:Countable - Revision history

Khanh at 20:03, 6 November 2021

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Lila at 19:16, 13 October 2021

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Lila at 19:15, 13 October 2021

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Lila: /* Introduction */

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Introduction

Lila at 19:08, 13 October 2021

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Lila: /* Formal overview */

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Formal overview

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Lila: /* History */

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History

Lila: /* Definition */

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Definition

Lila: /* Definition */

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Definition

Lila: /* Definition */

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Definition

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 This is the key definition that determines whether a total order is also a well order.
-==Resources==
+== Licensing ==
-* [https://en.wikipedia.org/wiki/Countable_set Countable set], Wikipedia
+Content obtained and/or adapted from:

← Older revision		Revision as of 19:16, 13 October 2021
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	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''.		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.		This is the key definition that determines whether a total order is also a well order.
		+
		+	==Resources==
		+	* [https://en.wikipedia.org/wiki/Countable_set Countable set], Wikipedia

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 This form of triangular mapping recursively generalizes to ''n''-tuples of natural numbers, i.e., (''a<sub>1</sub>'', ''a<sub>2</sub>'', ''a<sub>3</sub>'', ..., ''a<sub>n</sub>'') where ''a<sub>i</sub>'' 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.<ref>{{Harvard citation no brackets|Halmos|1960|page=92}}</ref>{{efn|'''Proof:''' Observe that {{math|'''N''' × '''N'''}} is countable as a consequence of the definition because the function {{math|''f'' : '''N''' × '''N''' → '''N'''}} given by  {{math|1=''f''(''m'', ''n'') = 2<sup>''m''</sup>3<sup>''n''</sup>}} is injective.<ref>{{Harvard citation no brackets|Avelsgaard|1990|page=182}}</ref>  It then follows that the Cartesian product of any two countable sets is countable, because if {{mvar|A}} and {{mvar|B}} are two countable sets there are surjections {{math|''f'' : '''N''' → ''A''}} and {{math|''g'' : '''N''' → ''B''}}. So
+'''Theorem:''' The Cartesian product of finitely many countable sets is countable.
 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.{{efn|'''Proof:''' The integers {{math|'''Z'''}} are countable because the function {{math|''f'' : '''Z''' → '''N'''}} given by {{math|1=''f''(''n'') = 2<sup>''n''</sup>}} if {{mvar|n}} is non-negative and {{math|1=''f''(''n'') = 3<sup>− ''n''</sup>}} if {{mvar|n}} is negative, is an injective function. The rational numbers {{math|'''Q'''}} are countable because the function {{math|''g'' : '''Z''' × '''N''' → '''Q'''}} given by {{math|1=''g''(''m'', ''n'') = ''m''/(''n'' + 1)}} is a surjection from the countable set {{math|'''Z''' × '''N'''}} to the rationals {{math|'''Q'''}}.}}
+'''Theorem:''' '''Z''' (the set of all integers) and '''Q''' (the set of all rational numbers) are countable.
-In a similar manner, the set of algebraic numbers is countable.<ref>{{Harvard citation no brackets|Kamke|1950|pages=3–4}}</ref>{{efn|1='''Proof:''' Per definition, every algebraic number (including complex numbers) is a root of a polynomial with integer coefficients. Given an algebraic number <math>\alpha</math>, let <math>a_0x^0+a_1x^1+a_2x^2+ \cdots + a_nx^n</math> be a polynomial with integer coefficients such that <math>\alpha</math> is the ''k''th 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 {{math|''f'' : '''A''' → '''Q'''}} given by <math>f(\alpha) = 2^{k-1} \cdot 3^{a_0} \cdot 5^{a_1} \cdot 7^{a_2} \cdots {p_{n+2}}^{a_n}</math>, while <math>p_n</math> is the ''n''-th prime.}}
+In a similar manner, the set of algebraic numbers is countable.
 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.<ref>{{Harvard citation no brackets|Avelsgaard|1990|page=180}}</ref><ref>{{Harvard citation no brackets|Fletcher|Patty|1988|page=187}}</ref>{{efn|1='''Proof:''' If {{math|''A<sub>i</sub>''}} is a countable set for each {{mvar|i}} in ''I''={1,...,n}, then for each {{mvar|n}} there is a surjective function {{math|''g<sub>i</sub>'' : '''N''' → ''A<sub>i</sub>''}} and hence the function
+'''Theorem:''' Any finite union of countable sets is countable.
 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.{{efn|1='''Proof''': As in the finite case, but ''I''='''N''' and we use the axiom of countable choice to pick for each {{mvar|i}} in {{math|'''N'''}} a surjection {{math|''g<sub>i</sub>''}} from the non-empty collection of surjections from {{math|'''N'''}} to {{math|''A<sub>i</sub>''}}.}}
+'''Theorem:''' (Assuming the axiom of countable choice) The union of countably many countable sets is countable.
 For example, given countable sets '''a''', '''b''', '''c''', ...
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 # If the function ''g'' : ''S'' → ''T'' is surjective and ''S'' is countable then ''T'' is countable.
-These follow from the definitions of countable set as injective / surjective functions.{{efn|'''Proof''': For (1) observe that if ''T'' is countable there is an injective function  ''h'' : ''T'' → '''N'''.  Then if ''f'' : ''S'' → ''T'' is injective the composition ''h'' <small> o </small> ''f'' : ''S'' → '''N''' is injective, so ''S'' is countable.
+These follow from the definitions of countable set as injective / surjective functions.
 '''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:

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 ==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.<ref>{{Cite web|date=2021-05-09|title=What Are Sets and Roster Form?|url=https://www.expii.com/t/what-are-sets-and-roster-form-4300|url-status=live|website=expii}}</ref> 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.
+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. 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 × 10<sup>32</sup>, infinite sets have more than ''n'' elements.) For example, the set of natural numbers, denotable by {0, 1, 2, 3, 4, 5, ...},{{efn|Since there is an obvious bijection between {{math|'''N'''}} and {{math|1='''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.}} 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.
+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 × 10<sup>32</sup>, infinite sets have more than ''n'' elements.) For example, the set of natural numbers, denotable by {0, 1, 2, 3, 4, 5, ...}, 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.
 [[File:Aplicación 2 inyectiva sobreyectiva02.svg|thumb|x100px|Bijective mapping from integer to even numbers]]

@@ Line 22: / Line 22: @@
 ==Introduction==
-A ''[[Set (mathematics)|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.<ref>{{Cite web|date=2021-05-09|title=What Are Sets and Roster Form?|url=https://www.expii.com/t/what-are-sets-and-roster-form-4300|url-status=live|website=expii}}</ref> 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 [[integer]]s 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.
+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.<ref>{{Cite web|date=2021-05-09|title=What Are Sets and Roster Form?|url=https://www.expii.com/t/what-are-sets-and-roster-form-4300|url-status=live|website=expii}}</ref> 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 × 10<sup>32</sup>, infinite sets have more than ''n'' elements.) For example, the set of natural numbers, denotable by {0, 1, 2, 3, 4, 5, ...},{{efn|Since there is an obvious [[bijection]] between {{math|'''N'''}} and {{math|1='''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.}} 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.
+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 × 10<sup>32</sup>, infinite sets have more than ''n'' elements.) For example, the set of natural numbers, denotable by {0, 1, 2, 3, 4, 5, ...},{{efn|Since there is an obvious bijection between {{math|'''N'''}} and {{math|1='''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.}} 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.
 [[File:Aplicación 2 inyectiva sobreyectiva02.svg|thumb|x100px|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:<math display="block">\ldots \, -\! 2\! \rightarrow \! - \! 4, \, -\! 1\! \rightarrow \! - \! 2, \, 0\! \rightarrow \! 0, \, 1\! \rightarrow \! 2, \, 2\! \rightarrow \! 4 \, \ldots</math>or, more generally, <math>n \rightarrow 2n</math> (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 (mathematics)|function]] that maps between two sets such that each element of each set corresponds to a single element in the other set.
+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:<math display="block">\ldots \, -\! 2\! \rightarrow \! - \! 4, \, -\! 1\! \rightarrow \! - \! 2, \, 0\! \rightarrow \! 0, \, 1\! \rightarrow \! 2, \, 2\! \rightarrow \! 4 \, \ldots</math>or, more generally, <math>n \rightarrow 2n</math> (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.
+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==
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 The set of real numbers is uncountable, and so is the set of all infinite sequences of natural numbers.
 ==Total orders==
-Countable sets can be [[total order|totally ordered]] in various ways, for example:
+Countable sets can be totally ordered in various ways, for example:
-*[[Well-order]]s (see also [[ordinal number]]):
+*Well-orders (see also ordinal number):
 **The usual order of natural numbers (0, 1, 2, 3, 4, 5, ...)
 **The integers in the order (0, 1, 2, 3, ...; −1, −2, −3, ...)

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 * there is an injective and surjective (and therefore bijective) mapping between {{mvar|S}} and '''N'''. In other words, a set is countably infinite if it has one-to-one correspondence with the natural number set, {{math|'''N'''}}.
 * The elements of {{mvar|S}} can be arranged in an infinite sequence <math>a_0, a_1, a_2, \ldots</math>, where <math>a_i</math> is distinct from <math>a_j</math> for <math>i\neq j</math> and every element of {{mvar|S}} is listed.
 ==Introduction==

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 Equivalently, a set {{mvar|S}} is countable iff:
 * there exists an injective function from {{mvar|S}} to '''N'''.
-* {{mvar|S}} is empty or there exists a [[surjective function]] from '''N''' to {{mvar|S}}, or .
+* {{mvar|S}} is empty or there exists a surjective function from '''N''' to {{mvar|S}}, or .
-* there exists a [[bijective]] mapping between {{mvar|S}} and a subset of '''N'''.
+* there exists a bijective mapping between {{mvar|S}} and a subset of '''N'''.
 * {{mvar|S}} is either finite or countably infinite.
-Similarly, a set {{mvar|S}} is countably infinite [[iff]]:
+Similarly, a set {{mvar|S}} is countably infinite iff:
 * there is an injective and surjective (and therefore bijective) mapping between {{mvar|S}} and '''N'''. In other words, a set is countably infinite if it has one-to-one correspondence with the natural number set, {{math|'''N'''}}.
 * The elements of {{mvar|S}} can be arranged in an infinite sequence <math>a_0, a_1, a_2, \ldots</math>, where <math>a_i</math> is distinct from <math>a_j</math> for <math>i\neq j</math> and every element of {{mvar|S}} is listed.

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 ==Definition==
-The most concise definition is in terms of cardinality. A set {{mvar|S}} is ''countable'' if its cardinality |S| is less than or equal to <math>\aleph_0</math> ([[aleph-null]]), the cardinality of the set of natural numbers {{math|1='''N'''}}. A set {{mvar|S}} is ''countably infinite'' if |S| = <math>\aleph_0</math>. A set is ''uncountable'' if it is not countable, i.e. its cardinality is greater than  <math>\aleph_0</math>.
+The most concise definition is in terms of cardinality. A set {{mvar|S}} is ''countable'' if its cardinality |S| is less than or equal to <math>\aleph_0</math> (aleph-null), the cardinality of the set of natural numbers {{math|1='''N'''}}. A set {{mvar|S}} is ''countably infinite'' if |S| = <math>\aleph_0</math>. A set is ''uncountable'' if it is not countable, i.e. its cardinality is greater than  <math>\aleph_0</math>.
 Equivalently, a set {{mvar|S}} is countable iff: