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]].
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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]].
  
 
In simplified language, if a set is infinite, that means that it doesn't end, i.e. you can always find another element that you haven't examined yet.
 
In simplified language, if a set is infinite, that means that it doesn't end, i.e. you can always find another element that you haven't examined yet.
  
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===Equivalent formulations===
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* A set is infinite if it can be put into [[bijection]] with one of its proper [[subset]]s. 
  
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* 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>.

Revision as of 12:20, 17 July 2006

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A set $S$ is said to be infinite if there is a surjection $f:S\to\mathbb{Z}$. If this is not the case, $S$ is said to be finite.

In simplified language, if a set is infinite, that means that 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 $\{1, 2, \ldots, n\}$ for a positive integer $n$.
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