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 | + | {{stub}} |
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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 [[element]]s as does the set of [[integer]]s. 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 number]]s, i.e. those whose elements can be "counted." | The name "countable" arises because the countably infinite sets are exactly those which can be put into [[bijection]] with the [[natural number]]s, i.e. those whose elements can be "counted." | ||
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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. | |
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| + | ===Properties=== | ||
| + | * Any finite set is countable. | ||
Latest revision as of 20:35, 26 September 2008
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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.
Properties
- Any finite set is countable.
