Almost

In set theory, when dealing with sets of infinite size, the term almost or nearly is used to refer to all but a finite (or a countable) amount of negligible elements in the set.[1][2]

For other uses, see Almost (disambiguation).

More specifically, given an set ${\displaystyle S}$ that is a subset of another countably infinite set ${\displaystyle L}$, ${\displaystyle S}$ is said to be almost ${\displaystyle L}$ if the set difference ${\displaystyle L\backslash S}$ is finite in size. Alternatively, if ${\displaystyle L}$ is uncountable set, then ${\displaystyle S}$ can also be said to be almost ${\displaystyle L}$ if ${\displaystyle L\backslash S}$ is countable in size.[3]

For example:

• The set ${\displaystyle S=\{n\in \mathbb {N} \,|\,n\geq k\}}$ is almost ${\displaystyle \mathbb {N} }$ for any ${\displaystyle k}$ in ${\displaystyle \mathbb {N} }$, because only finitely many natural numbers are less than ${\displaystyle k}$.
• The set of prime numbers is not almost ${\displaystyle \mathbb {N} }$, because there are infinitely many natural numbers that are not prime numbers.
• The set of transcendental numbers are almost ${\displaystyle \mathbb {R} }$, because the algebraic real numbers form a countable subset of the set of real number (the latter of which is uncountable).[4]

This use of "almost" 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.[5] 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)—as both sets are uncountable in nature.

See also

• Almost all
• Almost surely
• Approximation
• List of mathematical jargon

References

1. "The Definitive Glossary of Higher Mathematical Jargon — Almost". Math Vault. 2019-08-01. Retrieved 2019-11-16.
2. Halmos, Paul R. (1962). Algebraic Logic. New York: Chelsea Publishing Company. p. 114.
3. Schwartzman, Steven (1994). The words of mathematics : an etymological dictionary of mathematical terms used in English. Washington, DC: Mathematical Association of America. pp. 22. ISBN 0883855119. OCLC 30573178.
4. "Almost All Real Numbers are Transcendental - ProofWiki". proofwiki.org. Retrieved 2019-11-16.
5. "Theorem 36: the Cantor set is an uncountable set with zero measure". Theorem of the week. 2010-09-30. Retrieved 2019-11-16.