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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+ | 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=== | ||
+ | * 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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+ | ==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}} | {{stub}} | ||
+ | =="Operations" with Infinity== | ||
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+ | 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> |
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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 22:19, 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.