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| | Ed is what individuals call me and my wife doesn't like it at all. I've usually loved living in Kentucky but now I'm contemplating other choices. To climb is something I really enjoy doing. Distributing manufacturing is how he makes a living.<br><br>Also visit my weblog :: free psychic readings ([http://www.christmaslightshowvideos.com/profile.php?u=MiNrr click here! www.christmaslightshowvideos.com]) |
| In [[mathematics]], the phrase '''"almost all"''' has a number of specialised uses.
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| "Almost all" is sometimes used synonymously with "all but [except] [[finite set|finite]]ly many" (formally, a [[cofinite]] set) or "all but a [[countable set]]" (formally, a [[cocountable]] set); see [[almost]].
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| A simple example is that almost all prime numbers are odd, which is based on the fact that all but one prime number is odd. (The exception is the number 2, which is prime but not odd.)
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| When speaking about the [[real number|reals]], sometimes it means "all reals but a set of [[Lebesgue measure]] zero" (formally, [[almost everywhere]]). In this sense almost all reals are not a member of the [[Cantor set]] even though the Cantor set is [[Uncountable set|uncountable]].
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| In [[number theory]], if ''P''(''n'') is a property of positive [[integer]]s, and if ''p''(''N'') denotes the number of [[positive integers]] ''n'' less than ''N'' for which ''P''(''n'') holds, and if
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| :''p''(''N'')/''N'' → 1 as ''N'' → ∞ | |
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| (see [[Limit of a sequence|limit]]), then we say that "''P''(''n'') holds for almost all positive integers ''n''" (formally, [[asymptotically almost surely]]) and write
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| :<math>(\forall^\infty n) P(n).</math> | |
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| For example, the [[prime number theorem]] states that the number of [[prime numbers]] less than or equal to ''N'' is asymptotically equal to ''N''/ln ''N''. Therefore the proportion of prime integers is roughly 1/ln ''N'', which tends to 0. Thus, ''almost all'' positive integers are [[composite number|composite]] (not prime), however there are still an infinite number of primes.
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| Occasionally, "almost all" is used in the sense of "[[almost everywhere]]" in [[measure theory]], or in the closely related sense of "[[almost surely]]" in [[probability theory]].
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| ==See also==
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| *[[Generic property]]
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| *[[Sufficiently large]]
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| ==References==
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| *{{MathWorld|title=Almost All|urlname=AlmostAll}}
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| {{DEFAULTSORT:Almost All}}
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| [[Category:Mathematical terminology]]
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| [[Category:Mathematical notation]]
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Ed is what individuals call me and my wife doesn't like it at all. I've usually loved living in Kentucky but now I'm contemplating other choices. To climb is something I really enjoy doing. Distributing manufacturing is how he makes a living.
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