Difference between revisions of "Countable set"
(Comment: In some texts, this definition is called a "countably infinite" set) |
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| − | A set | + | {{TEX|done}}{{MSC|03E20}} |
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| + | A set [[Equipotent sets|equipotent]] to the set of natural numbers and hence of the same [[cardinality]]. For example, the set of integers, the set of [[rational number]]s or the set of [[algebraic number]]s. | ||
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| + | An ''uncountable set'' is one which is not countable: for example, the set of [[real number]]s is uncountable, by [[Cantor theorem|Cantor's theorem]]. | ||
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| + | The [[Union of sets|union]] and [[Cartesian product]] of two countable sets is again countable; the union of a countable family of countable sets is also countable. | ||
====Comments==== | ====Comments==== | ||
| − | In some texts, this definition is that a "countably infinite" set, and a "countable" set refers to one which is finite or countably infinite: that is, a set of the same cardinality as some subset of the natural numbers. | + | In some texts, this definition is that of a "countably infinite" or "denumerable" set, and a "countable" set refers to one which is finite or countably infinite: that is, a set of the same cardinality as some subset of the natural numbers. |
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| + | ====References==== | ||
| + | * P. R. Halmos, "Naive Set Theory", Springer (1960) {{ISBN|0-387-90092-6}} | ||
Latest revision as of 08:50, 26 November 2023
2020 Mathematics Subject Classification: Primary: 03E20 [MSN][ZBL]
A set equipotent to the set of natural numbers and hence of the same cardinality. For example, the set of integers, the set of rational numbers or the set of algebraic numbers.
An uncountable set is one which is not countable: for example, the set of real numbers is uncountable, by Cantor's theorem.
The union and Cartesian product of two countable sets is again countable; the union of a countable family of countable sets is also countable.
Comments
In some texts, this definition is that of a "countably infinite" or "denumerable" set, and a "countable" set refers to one which is finite or countably infinite: that is, a set of the same cardinality as some subset of the natural numbers.
References
- P. R. Halmos, "Naive Set Theory", Springer (1960) ISBN 0-387-90092-6
Countable set. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Countable_set&oldid=33594