Editing Almost

{{short description|Term in set theory}}
{{Other uses}}
{{More citations needed|date=January 2021}}

In [[set theory]], when dealing with [[set (mathematics)|sets]] of [[infinite set|infinite size]], the term '''almost''' or '''nearly''' is used to refer to all but a negligible amount of elements in the set. The notion of "negligible" depends on the context, and may mean "of measure zero" (in a [[measure space]]), "finite" (when infinite sets are involved), or "countable" (when [[uncountably infinite set]]s are involved).

For example:
*The set <math> S = \{ n \in \mathbb{N}\,|\, n \ge k \} </math> is almost <math>\mathbb{N}</math> for any <math>k</math> in '''<math>\mathbb{N}</math>''', because only finitely many [[natural number]]s are less than ''<math>k</math>''.
*The set of [[prime number]]s is not almost '''<math>\mathbb{N}</math>''', because there are infinitely many natural numbers that are not prime numbers.
*The set of [[transcendental number]]s are almost '''<math>\mathbb{R}</math>''', because the [[algebraic number|algebraic]] [[real number|real]] numbers form a [[countable]] [[subset]] of the set of real numbers (which is uncountable).<ref>{{Cite web|url=https://proofwiki.org/wiki/Almost_All_Real_Numbers_are_Transcendental|title=Almost All Real Numbers are Transcendental - ProofWiki|website=proofwiki.org|access-date=2019-11-16}}</ref>
*The [[Cantor set]] is uncountably infinite, but has [[Lebesgue measure]] zero.<ref>{{Cite web|url=https://theoremoftheweek.wordpress.com/2010/09/30/theorem-36-the-cantor-set-is-an-uncountable-set-with-zero-measure/|title=Theorem 36: the Cantor set is an uncountable set with zero measure|date=2010-09-30|website=Theorem of the week|language=en|access-date=2019-11-16}}</ref> So almost all real numbers in (0,&thinsp;1) are members of the [[complement (set theory)|complement]] of the Cantor set.

==See also==
{{Wiktionarypar|almost}}
*[[Almost periodic function]] - and Operators
*[[Almost all]]
*[[Almost surely]]
*[[Approximation]]
*[[List of mathematical jargon]]

== References ==
<references />{{Set theory}}

[[Category:Mathematical terminology]]
[[Category:Set theory]]


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