Cardinality of important sets - Revision history

Khanh at 22:18, 11 November 2021

2021-11-11T22:18:14Z

Khanh at 22:17, 11 November 2021

2021-11-11T22:17:51Z

Lila: /* Resources */

2021-10-14T20:36:45Z

Resources

Lila: /* Cardinality of \R */

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Cardinality of \R

Lila: /* Aleph-nought */

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Aleph-nought

Lila: /* Cardinality of \Z */

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Cardinality of \Z

Lila: /* Cardinality of \R */

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Cardinality of \R

Lila: /* Cardinality of \Z */

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Cardinality of \Z

Lila: /* Cardinality of \Z */

2021-10-14T20:20:33Z

Cardinality of \Z

Lila: Lila moved page Cardinality:𝐍 to Cardinality of important sets: Too specific

2021-10-14T20:14:53Z

Lila moved page Cardinality:𝐍 to Cardinality of important sets: Too specific

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 ==Licensing==
 * [https://en.wikipedia.org/wiki/Natural_number Natural number, Wikipedia] under a CC BY-SA license
 * [https://proofwiki.org/wiki/Rational_Numbers_are_Countably_Infinite/Proof_1 Rational numbers are countably infinite, ProofWiki.org] under a CC BY-SA license

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 The smallest infinite cardinal number is <math>\aleph_0</math> (aleph-null). The second smallest is <math>\aleph_1</math> (aleph-one). The continuum hypothesis, which asserts that there are no sets whose cardinality is strictly between <math>\aleph_0</math> and <math>\mathfrak c</math>, means that <math>\mathfrak c = \aleph_1</math>. The truth or falsity of this hypothesis is undecidable and cannot be proven within the widely used Zermelo–Fraenkel set theory with axiom of choice (ZFC).
-==Resources==
+==Licensing==
-* [https://en.wikipedia.org/wiki/Natural_number Natural number], Wikipedia
+* [https://en.wikipedia.org/wiki/Natural_number Natural number, Wikipedia] under a CC BY-SA license
-* [https://proofwiki.org/wiki/Rational_Numbers_are_Countably_Infinite/Proof_1 Rational numbers are countably infinite], ProofWiki.org
+* [https://proofwiki.org/wiki/Rational_Numbers_are_Countably_Infinite/Proof_1 Rational numbers are countably infinite, ProofWiki.org] under a CC BY-SA license

← Older revision		Revision as of 20:36, 14 October 2021
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	==Resources==		==Resources==
	* [https://en.wikipedia.org/wiki/Natural_number Natural number], Wikipedia		* [https://en.wikipedia.org/wiki/Natural_number Natural number], Wikipedia
		+	* [https://proofwiki.org/wiki/Rational_Numbers_are_Countably_Infinite/Proof_1 Rational numbers are countably infinite], ProofWiki.org

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 ==Cardinality of <math>\R</math>==
-The '''cardinality of the continuum''' is the cardinality or "size" of the set of real numbers <math>\mathbb R</math>, sometimes called the continuum. It is an infinite cardinal number and is denoted by <math>\mathfrak c</math> (lowercase fraktur "c") or <math>|\mathbb R|</math>.
+The '''cardinality of the continuum''' is the cardinality or "size" of the set of real numbers <math>\mathbb R</math>, sometimes called the continuum. It is an infinite cardinal number and is denoted by <math>\mathfrak c</math> (lowercase fraktur "c") or <math>|\mathbb R|</math>. The irrational numbers (<math> \R\setminus \Q </math>) is also of this cardinality.
 The real numbers <math>\mathbb R</math> are more numerous than the natural numbers <math>\mathbb N</math>. Moreover, <math>\mathbb R</math> has the same number of elements as the power set of <math>\mathbb N.</math> Symbolically, if the cardinality of <math>\mathbb N</math> is denoted as <math>\aleph_0</math>, the cardinality of the continuum is

← Older revision		Revision as of 20:33, 14 October 2021
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	If the axiom of countable choice (a weaker version of the axiom of choice) holds, then <math>\,\aleph_0\,</math> is smaller than any other infinite cardinal.		If the axiom of countable choice (a weaker version of the axiom of choice) holds, then <math>\,\aleph_0\,</math> is smaller than any other infinite cardinal.
−

	==Cardinality of <math>\N</math>==		==Cardinality of <math>\N</math>==

← Older revision		Revision as of 20:33, 14 October 2021
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	If the domain is restricted to <math>\mathbb{Z}</math> then each and every member of <math>\mathbb{Z}</math> has one and only one corresponding member of <math>\mathbb{N}</math> and by the definition of cardinal equality the two sets have equal cardinality.		If the domain is restricted to <math>\mathbb{Z}</math> then each and every member of <math>\mathbb{Z}</math> has one and only one corresponding member of <math>\mathbb{N}</math> and by the definition of cardinal equality the two sets have equal cardinality.
		+
		+	==Cardinality of <math>\Q</math>==
		+	The set of all rational numbers <math>\Q</math> is countably infinite and has the same cardinality as the set of all natural numbers {{math\|ℵ_0}}.
		+
		+	The rational numbers are arranged thus:
		+
		+	:<math>\dfrac 0 1, \dfrac 1 1, \dfrac {-1} 1, \dfrac 1 2, \dfrac {-1} 2, \dfrac 2 1, \dfrac {-2} 1, \dfrac 1 3, \dfrac 2 3, \dfrac {-1} 3, \dfrac {-2} 3, \dfrac 3 1, \dfrac 3 2, \dfrac {-3} 1, \dfrac {-3} 2, \dfrac 1 4, \dfrac 3 4, \dfrac {-1} 4, \dfrac {-3} 4, \dfrac 4 1, \dfrac 4 3, \dfrac {-4} 1, \dfrac {-4} 3 \ldots</math>
		+
		+	It is clear that every rational number will appear somewhere in this list.
		+
		+	Thus it is possible to set up a bijection between each rational number and its position in the list, which is an element of <math>\N</math>. Note that this is NOT true for the irrational numbers, which have the same cardinality as the real numbers (see below).

	==Cardinality of <math>\R</math>==		==Cardinality of <math>\R</math>==

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 The real numbers <math>\mathbb R</math> are more numerous than the natural numbers <math>\mathbb N</math>. Moreover, <math>\mathbb R</math> has the same number of elements as the power set of <math>\mathbb N.</math> Symbolically, if the cardinality of <math>\mathbb N</math> is denoted as <math>\aleph_0</math>, the cardinality of the continuum is
-{{block indent|<math>\mathfrak c = 2^{\aleph_0} > \aleph_0 \,. </math>}}
+<math>\mathfrak c = 2^{\aleph_0} > \aleph_0 \,. </math>
 This was proven by Georg Cantor in his uncountability proof of 1874, part of his groundbreaking study of different infinities. The inequality was later stated more simply in his diagonal argument in 1891. Cantor defined cardinality in terms of bijective functions: two sets have the same cardinality if, and only if, there exists a bijective function between them.
 Between any two real numbers ''a''&nbsp;<&nbsp;''b'', no matter how close they are to each other, there are always infinitely many other real numbers, and Cantor showed that they are as many as those contained in the whole set of real numbers. In other words, the open interval (''a'',''b'') is equinumerous with <math>\mathbb R.</math> This is also true for several other infinite sets, such as any ''n''-dimensional Euclidean space <math>\mathbb R^n</math> (see space filling curve). That is,
-{{block indent|<math>|(a,b)| = |\mathbb R| = |\mathbb R^n|.</math>}}
+<math>|(a,b)| = |\mathbb R| = |\mathbb R^n|.</math>
 The smallest infinite cardinal number is <math>\aleph_0</math> (aleph-null). The second smallest is <math>\aleph_1</math> (aleph-one). The continuum hypothesis, which asserts that there are no sets whose cardinality is strictly between <math>\aleph_0</math> and <math>\mathfrak c</math>, means that <math>\mathfrak c = \aleph_1</math>. The truth or falsity of this hypothesis is undecidable and cannot be proven within the widely used Zermelo–Fraenkel set theory with axiom of choice (ZFC).

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 ==Cardinality of <math>\Z</math>==
-The [[cardinality]] of the set of integers is equal to {{math|ℵ{{sub|0}}}} (aleph-null). This is readily demonstrated by the construction of a bijection, that is, a function that is injective and surjective from <math>\mathbb{Z}</math> to <math>\mathbb{N}</math>.
+The [[cardinality]] of the set of integers is equal to {{math|ℵ_0}} (aleph-null). This is readily demonstrated by the construction of a bijection, that is, a function that is injective and surjective from <math>\mathbb{Z}</math> to <math>\mathbb{N}</math>.
 If <math>\mathbb{N}_0 \equiv \{0, 1, 2, ...\}</math> then consider the function:
 :<math>f(x) = \begin{cases} 2|x|,  & \mbox{if } x \leq 0\\ 2x-1, & \mbox{if }  x > 0. \end{cases} </math>
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 If the domain is restricted to <math>\mathbb{Z}</math> then each and every member of <math>\mathbb{Z}</math> has one and only one corresponding member of <math>\mathbb{N}</math> and by the definition of cardinal equality the two sets have equal cardinality.
 ==Resources==
 * [https://en.wikipedia.org/wiki/Natural_number Natural number], Wikipedia

← Older revision	Revision as of 20:14, 14 October 2021
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