Difference between revisions of "Infinite"

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(Operations with Infinity)
 
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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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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.
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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>.
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==Applications to Infinity with Sums==
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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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{{stub}}
  
  
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=="Operations" with Infinity==
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Some '''bad''' rules involving operations with infinity are as follows:
  
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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* <math>1/{\infty} = 0</math>
  
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.
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* <math>{\infty} + x  = {\pm}{\infty}</math>
  
===Equivalent formulations===
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* <math>{\infty}\cdot{x} = {\infty}</math>
* 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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None of these are true because <math>\infty</math> is not a real number which you can write equations involving.

Latest revision as of 23:19, 16 August 2013

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 $n$.

Applications to Infinity with Sums

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

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

This article is a stub. Help us out by expanding it.


"Operations" with Infinity

Some bad rules involving operations with infinity are as follows:

  • $1/{\infty} = 0$
  • ${\infty} + x  = {\pm}{\infty}$
  • ${\infty}\cdot{x} = {\infty}$

None of these are true because $\infty$ is not a real number which you can write equations involving.

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