Almost

Almost

In mathematics, especially in set theory, when dealing with sets of infinite size, the term almost or nearly is used to mean "all the elements except for finitely many".

In other words, an infinite set "S" that is a subset of another infinite set "L", is almost "L" if the subtracted set "L""S" is of finite size.

Examples:
*The set S = { n in mathbf{N} | n ge k } is almost N for any "k" in N, because only finitely many natural numbers are less than "k".
*The set of prime numbers is not almost N because there are infinitely many natural numbers that are not prime numbers.

This is conceptually similar to the "almost everywhere" concept of measure theory, but is not the same. For example, the Cantor set is uncountably infinite, but has Lebesgue measure zero. So a real number in (0, 1) is a member of the complement of the Cantor set "almost everywhere", but it is not true that the complement of the Cantor set is "almost" the real numbers in (0, 1).

ee also

*Almost all
*Almost surely


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  • almost — • During the next week Morel s temper was almost unbearable D. H. Lawrence, 1913. Almost has a special role in diluting or ‘downgrading’ adjectives and adverbs that express an extreme, as if the user wants to keep the notion at arm s length or to …   Modern English usage

  • Almost — Al most ([add]l m[=o]st), adv. [AS. ealm[ae]st, [ae]lm[ae]st, quite the most, almost all; eal (OE. al) all + m?st most.] Nearly; well nigh; all but; for the greatest part. [1913 Webster] Almost thou persuadest me to be a Christian. Acts xxvi. 28 …   The Collaborative International Dictionary of English

  • almost — I adverb approximately, close to, nearly, on the brink of, on the verge of, scarcely, within sight of II index approximate, quasi Burton s Legal Thesaurus. William C. Burton. 2006 …   Law dictionary

  • almost as — index quasi Burton s Legal Thesaurus. William C. Burton. 2006 …   Law dictionary

  • almost — (adv.) O.E. eallmæst nearly all, for the most part, lit. mostly all; see ALL (Cf. all) + MOST (Cf. most). Modern form from 15c …   Etymology dictionary

  • almost — *nearly, approximately, well nigh …   New Dictionary of Synonyms

  • almost — [adv] nearly, very nearly about, about to, all but, approximately, around, as good as, bordering on, close to, close upon, essentially, for all practical purposes, for the greatest part, in effect, in the neighborhood of, in the vicinity of, just …   New thesaurus

  • almost — ► ADVERB ▪ very nearly. ORIGIN Old English …   English terms dictionary

  • almost — [ôl′mōst΄, ôl mōst′] adv. [OE eallmæst: see ALL & MOST] very nearly but not completely; all but …   English World dictionary

  • almost — adverb very nearly but not completely: We ve almost finished. | We stayed there for almost a week. | almost every: They sold almost everything. | almost all: Almost all the children here speak two languages. | an almost impossible task | wines… …   Longman dictionary of contemporary English

  • almost — al|most [ ɔlmoust ] adverb *** nearly but not completely: Are you ready? Almost! I m just putting my shoes on. It s almost a year since she died. The baby s almost walking now. almost all: Almost all of the students here are from South America.… …   Usage of the words and phrases in modern English

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