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]]. | + | {{stub}} |
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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]]. | ||
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. | 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. | ||
+ | ===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>. |
Revision as of 11:20, 17 July 2006
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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, 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.
Equivalent formulations
- 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 .