Difference between revisions of "Cardinal number"
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| − | In [[mathematics]], '''cardinal numbers''', or '''cardinals''' for short, are a generalization of the [[Natural number|natural numbers]] used to measure the cardinality (size) of sets. | + | In [[mathematics]], '''cardinal numbers''', or '''cardinals''' for short, are a generalization of the [[Natural number|natural numbers]] used to measure the cardinality (size) of [[Set (mathematics)|sets]]. |
== Description == | == Description == | ||
| − | The cardinality of a finite set is a natural number: the number of elements in the set | + | The cardinality of a [[finite set]] is a [[natural number]]: the number of elements in the set. |
| − | Cardinality is defined in terms of bijective functions. Two sets have the same cardinality if, and only if, there is a one-to-one correspondence (bijection) between the elements of the two sets. | + | The [[Transfinite number|transfinite]] cardinal numbers describe the sizes of [[Infinite set|infinite sets]]. |
| + | |||
| + | Cardinality is defined in terms of bijective functions. Two sets have the same cardinality if, and only if, there is a one-to-one correspondence ([[bijection]]) between the elements of the two sets. | ||
In the case of finite sets, this agrees with the intuitive notion of size. | In the case of finite sets, this agrees with the intuitive notion of size. | ||
| − | In the case of infinite sets, the behavior is more complex. A fundamental theorem due to Georg Cantor shows that it is possible for infinite sets to have different cardinalities, and in particular the cardinality of the set of real numbers is greater than the cardinality of the set of natural numbers. | + | In the case of infinite sets, the behavior is more complex. A fundamental theorem due to [[Georg Cantor]] shows that it is possible for infinite sets to have different cardinalities, and in particular the cardinality of the set of [[Real number|real numbers]] is greater than the cardinality of the set of [[Natural number|natural numbers]]. |
It is also possible for a proper subset of an infinite set to have the same cardinality as the original set, something that cannot happen with proper subsets of finite sets. | It is also possible for a proper subset of an infinite set to have the same cardinality as the original set, something that cannot happen with proper subsets of finite sets. | ||
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* [[Beth number]] | * [[Beth number]] | ||
* [[Cantor's paradox]] - also known as "the paradox of the greatest cardinal" | * [[Cantor's paradox]] - also known as "the paradox of the greatest cardinal" | ||
| + | * [[Cardinality]] | ||
* [[Counting]] | * [[Counting]] | ||
* [[Names of numbers in English]] | * [[Names of numbers in English]] | ||
Latest revision as of 09:28, 22 September 2016
In mathematics, cardinal numbers, or cardinals for short, are a generalization of the natural numbers used to measure the cardinality (size) of sets.
Description
The cardinality of a finite set is a natural number: the number of elements in the set.
The transfinite cardinal numbers describe the sizes of infinite sets.
Cardinality is defined in terms of bijective functions. Two sets have the same cardinality if, and only if, there is a one-to-one correspondence (bijection) between the elements of the two sets.
In the case of finite sets, this agrees with the intuitive notion of size.
In the case of infinite sets, the behavior is more complex. A fundamental theorem due to Georg Cantor shows that it is possible for infinite sets to have different cardinalities, and in particular the cardinality of the set of real numbers is greater than the cardinality of the set of natural numbers.
It is also possible for a proper subset of an infinite set to have the same cardinality as the original set, something that cannot happen with proper subsets of finite sets.
See also
- Aleph number
- Beth number
- Cantor's paradox - also known as "the paradox of the greatest cardinal"
- Cardinality
- Counting
- Names of numbers in English
- Large cardinal
- Inclusion-exclusion principle
- Nominal number
- Ordinal number
- Regular cardinal
External links
- Cardinal number @ Wikipedia.org
