# 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, | + | 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 [[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>. | ||

+ | |||

+ | ==Applications to Infinity with Sums== | ||

+ | |||

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

+ | |||

+ | <math>\sum_{i = 3}^{\infty}{(2i - 1)}</math> | ||

{{stub}} | {{stub}} | ||

+ | |||

+ | |||

+ | =="Operations" with Infinity== | ||

+ | |||

+ | Some '''bad''' rules involving operations with infinity are as follows: | ||

+ | |||

+ | * <math>1/{\infty} = 0</math> | ||

+ | |||

+ | * <math>{\infty} + x = {\pm}{\infty}</math> | ||

+ | |||

+ | * <math>{\infty}\cdot{x} = {\infty}</math> | ||

+ | |||

+ | 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 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 .

## Applications to Infinity with Sums

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

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

## "Operations" with Infinity

Some **bad** rules involving operations with infinity are as follows:

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