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]].
  
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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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===  
 
===Equivalent formulations===  

Revision as of 16:46, 20 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, 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 $\{1, 2, \ldots, n\}$ for a positive integer $n$.
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