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Difference between revisions of "Infinite"

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* 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>.
 
* 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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A sum works the same way. Certain sums equate to infinity, such as
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<math>\sum_{i = 3}^{\infty}{(2i - 1)}</math>
  
 
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Revision as of 13:16, 17 June 2011

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 $n$.

A sum works the same way. Certain sums equate to infinity, such as

$\sum_{i = 3}^{\infty}{(2i - 1)}$

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