Difference between revisions of "Infinite"
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A [[set]] <math>S</math> is said to be '''infinite''' if there is a [[surjection]] <math>f:S\to\mathbb{Z}</math>. If this is not the case, <math>S</math> is said to be [[finite]]. | A [[set]] <math>S</math> is said to be '''infinite''' if there is a [[surjection]] <math>f:S\to\mathbb{Z}</math>. If this is not the case, <math>S</math> is said to be [[finite]]. | ||
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===Equivalent formulations=== | ===Equivalent formulations=== | ||
* A set is infinite if it can be put into [[bijection]] with one of its proper [[subset]]s. | * A set is infinite if it can be put into [[bijection]] with one of its proper [[subset]]s. | ||
| + | * A set is infinite if it is not empty and cannot be put into bijection with any set of the form <math>\{1, 2, \ldots, n\}</math> for a [[positive integer]] <math>n</math>. | ||
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Revision as of 19:16, 22 April 2008
A set 
is said to be infinite if there is a surjection 
. If this is not the case, is said to be finite.
In simplified language, a set is infinite if it doesn't end, i.e. you can always find another element that you haven't examined yet.
Equivalent formulations
- A set is infinite if it can be put into bijection with one of its proper subsets.
- A set is infinite if it is not empty and cannot be put into bijection with any set of the form
for a positive integer 
.
for a 
.This article is a stub. Help us out by expanding it.
