Difference between revisions of "Countable"
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A [[set]] <math>S</math> is said to be '''countable''' if there is an [[injection]] <math>f:S\to\mathbb{Z}</math>. Informally, a set is countable if it has at most as many elements as does the set of [[integer]]s. The countable sets can be divided between those which are [[finite]] and those which are countably [[infinite]]. | A [[set]] <math>S</math> is said to be '''countable''' if there is an [[injection]] <math>f:S\to\mathbb{Z}</math>. Informally, a set is countable if it has at most as many elements as does the set of [[integer]]s. The countable sets can be divided between those which are [[finite]] and those which are countably [[infinite]]. | ||
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Countably infinite sets include the [[integer]]s, the [[positive integer]]s and the [[rational number]]s. | Countably infinite sets include the [[integer]]s, the [[positive integer]]s and the [[rational number]]s. | ||
| − | + | [[Uncountable]] sets include the [[real number]]s and the [[complex number]]s. | |
Revision as of 10:45, 17 July 2006
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A set 
is said to be countable if there is an injection 
. Informally, a set is countable if it has at most as many elements as does the set of integers. The countable sets can be divided between those which are finite and those which are countably infinite.
The name "countable" arises because the countably infinite sets are exactly those which can be put into bijection with the natural numbers, i.e. those whose elements can be "counted."
Countably infinite sets include the integers, the positive integers and the rational numbers.
Uncountable sets include the real numbers and the complex numbers.
