Difference between revisions of "Countable"
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| − | A [[set]] X is '''countable''' if and only if there is a [[bijection]] from X to | + | A [[set]] X is '''countable''' if and only if there is a [[bijection]] from X to a subset set of [[Natural Numbers|natural numbers]]. Countable sets include finite sets, the set of [[integer]]s, and the set of [[rational number]]s. |
| − | There is no smallest countable set. Indeed, the set of natural numbers is in bijection with the natural numbers without 0: subtracting 1 from every number gives a bijection from the first set to the second. Repeating this process shows that for any initial segment of the the natural numbers (such as {1, 2, ..., n}), we have a bijection between the set of natural numbers and the set of natural numbers without this segment. The bijection is simply subraction by n. | + | There is no smallest infinite countable set. Indeed, the set of natural numbers is in bijection with the natural numbers without 0: subtracting 1 from every number gives a bijection from the first set to the second. Repeating this process shows that for any initial segment of the the natural numbers (such as {1, 2, ..., n}), we have a bijection between the set of natural numbers and the set of natural numbers without this segment. The bijection is simply subraction by n. |
[[Category:Set theory]] | [[Category:Set theory]] | ||
Revision as of 16:45, February 24, 2008
A set X is countable if and only if there is a bijection from X to a subset set of natural numbers. Countable sets include finite sets, the set of integers, and the set of rational numbers.
There is no smallest infinite countable set. Indeed, the set of natural numbers is in bijection with the natural numbers without 0: subtracting 1 from every number gives a bijection from the first set to the second. Repeating this process shows that for any initial segment of the the natural numbers (such as {1, 2, ..., n}), we have a bijection between the set of natural numbers and the set of natural numbers without this segment. The bijection is simply subraction by n.