Almost all

In mathematics, the term "almost all" means "all but a negligible quantity". More precisely, if is a set, "almost all elements of " means "all elements of but those in a negligible subset of ". The meaning of "negligible" depends on the mathematical context; for instance, it can mean finite, countable, or null.

In contrast, "almost no" means "a negligible quantity"; that is, "almost no elements of " means "a negligible quantity of elements of ".

Meanings in different areas of mathematics

Prevalent meaning
Further information: Cofinite set

Throughout mathematics, "almost all" is sometimes used to mean "all (elements of an infinite set) except for finitely many".[1][2] This use occurs in philosophy as well.[3] Similarly, "almost all" can mean "all (elements of an uncountable set) except for countably many".[sec 1]

Examples:

Meaning in measure theory
Further information: Almost everywhere
The Cantor function as a function that has zero derivative almost everywhere

When speaking about the reals, sometimes "almost all" can mean "all reals except for a null set".[6][7][sec 2] Similarly, if S is some set of reals, "almost all numbers in S" can mean "all numbers in S except for those in a null set".[8] The real line can be thought of as a one-dimensional Euclidean space. In the more general case of an n-dimensional space (where n is a positive integer), these definitions can be generalised to "all points except for those in a null set"[sec 3] or "all points in S except for those in a null set" (this time, S is a set of points in the space).[9] Even more generally, "almost all" is sometimes used in the sense of "almost everywhere" in measure theory,[10][11][sec 4] or in the closely related sense of "almost surely" in probability theory.[11][sec 5]

Examples:

Meaning in number theory
Further information: Asymptotically almost surely

In number theory, "almost all positive integers" can mean "the positive integers in a set whose natural density is 1". That is, if A is a set of positive integers, and if the proportion of positive integers in A below n (out of all positive integers below n) tends to 1 as n tends to infinity, then almost all positive integers are in A.[16][17][sec 7]

More generally, let S be an infinite set of positive integers, such as the set of even positive numbers or the set of primes, if A is a subset of S, and if the proportion of elements of S below n that are in A (out of all elements of S below n) tends to 1 as n tends to infinity, then it can be said that almost all elements of S are in A.

Examples:

  • The natural density of cofinite sets of positive integers is 1, so each of them contains almost all positive integers.
  • Almost all positive integers are composite.[sec 7][proof 1]
  • Almost all even positive numbers can be expressed as the sum of two primes.[4]: 489 
  • Almost all primes are isolated. Moreover, for every positive integer g, almost all primes have prime gaps of more than g both to their left and to their right; that is, there is no other prime between p − g and p + g.[18]

Meaning in graph theory

In graph theory, if A is a set of (finite labelled) graphs, it can be said to contain almost all graphs, if the proportion of graphs with n vertices that are in A tends to 1 as n tends to infinity.[19] However, it is sometimes easier to work with probabilities,[20] so the definition is reformulated as follows. The proportion of graphs with n vertices that are in A equals the probability that a random graph with n vertices (chosen with the uniform distribution) is in A, and choosing a graph in this way has the same outcome as generating a graph by flipping a coin for each pair of vertices to decide whether to connect them.[21] Therefore, equivalently to the preceding definition, the set A contains almost all graphs if the probability that a coin-flip–generated graph with n vertices is in A tends to 1 as n tends to infinity.[20][22] Sometimes, the latter definition is modified so that the graph is chosen randomly in some other way, where not all graphs with n vertices have the same probability,[21] and those modified definitions are not always equivalent to the main one.

The use of the term "almost all" in graph theory is not standard; the term "asymptotically almost surely" is more commonly used for this concept.[20]

Example:

Meaning in topology

In topology[24] and especially dynamical systems theory[25][26][27] (including applications in economics),[28] "almost all" of a topological space's points can mean "all of the space's points except for those in a meagre set". Some use a more limited definition, where a subset contains almost all of the space's points only if it contains some open dense set.[26][29][30]

Example:

Meaning in algebra

In abstract algebra and mathematical logic, if U is an ultrafilter on a set X, "almost all elements of X" sometimes means "the elements of some element of U".[31][32][33][34] For any partition of X into two disjoint sets, one of them will necessarily contain almost all elements of X. It is possible to think of the elements of a filter on X as containing almost all elements of X, even if it isn't an ultrafilter.[34]

Proofs
  1. The prime number theorem shows that the number of primes less than or equal to n is asymptotically equal to n/ln(n). Therefore, the proportion of primes is roughly ln(n)/n, which tends to 0 as n tends to infinity, so the proportion of composite numbers less than or equal to n tends to 1 as n tends to infinity.[17]

See also

References

Primary sources
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  2. Cahen, Paul-Jean; Chabert, Jean-Luc (7 December 2010) [First published 2000]. "Chapter 4: What's New About Integer-Valued Polynomials on a Subset?". In Hazewinkel, Michiel (ed.). Non-Noetherian Commutative Ring Theory. Mathematics and Its Applications. Vol. 520. Springer. p. 85. doi:10.1007/978-1-4757-3180-4. ISBN 978-1-4419-4835-9.
  3. Gärdenfors, Peter (22 August 2005). The Dynamics of Thought. Synthese Library. Vol. 300. Springer. pp. 190–191. ISBN 978-1-4020-3398-8.
  4. Courant, Richard; Robbins, Herbert; Stewart, Ian (18 July 1996). What is Mathematics? An Elementary Approach to Ideas and Methods (2nd ed.). Oxford University Press. ISBN 978-0-19-510519-3.
  5. Movshovitz-hadar, Nitsa; Shriki, Atara (2018-10-08). Logic In Wonderland: An Introduction To Logic Through Reading Alice's Adventures In Wonderland - Teacher's Guidebook. World Scientific. p. 38. ISBN 978-981-320-864-3. This can also be expressed in the statement: 'Almost all prime numbers are odd.'
  6. Korevaar, Jacob (1 January 1968). Mathematical Methods: Linear Algebra / Normed Spaces / Distributions / Integration. Vol. 1. New York: Academic Press. pp. 359–360. ISBN 978-1-4832-2813-6.
  7. Natanson, Isidor P. (June 1961). Theory of Functions of a Real Variable. Vol. 1. Translated by Boron, Leo F. (revised ed.). New York: Frederick Ungar Publishing. p. 90. ISBN 978-0-8044-7020-9.
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  9. Helmberg, Gilbert (December 1969). Introduction to Spectral Theory in Hilbert Space. North-Holland Series in Applied Mathematics and Mechanics. Vol. 6 (1st ed.). Amsterdam: North-Holland Publishing Company. p. 320. ISBN 978-0-7204-2356-3.
  10. Vestrup, Eric M. (18 September 2003). The Theory of Measures and Integration. Wiley Series in Probability and Statistics. United States: Wiley-Interscience. p. 182. ISBN 978-0-471-24977-1.
  11. Billingsley, Patrick (1 May 1995). Probability and Measure (PDF). Wiley Series in Probability and Statistics (3rd ed.). United States: Wiley-Interscience. p. 60. ISBN 978-0-471-00710-4. Archived from the original (PDF) on 23 May 2018.
  12. Niven, Ivan (1 June 1956). Irrational Numbers. Carus Mathematical Monographs. Vol. 11. Rahway: Mathematical Association of America. pp. 2–5. ISBN 978-0-88385-011-4.
  13. Baker, Alan (1984). A concise introduction to the theory of numbers. Cambridge University Press. p. 53. ISBN 978-0-521-24383-4.
  14. Granville, Andrew; Rudnick, Zeev (7 January 2007). Equidistribution in Number Theory, An Introduction. Nato Science Series II. Vol. 237. Springer. p. 11. ISBN 978-1-4020-5404-4.
  15. Burk, Frank (3 November 1997). Lebesgue Measure and Integration: An Introduction. A Wiley-Interscience Series of Texts, Monographs, and Tracts. United States: Wiley-Interscience. p. 260. ISBN 978-0-471-17978-8.
  16. Hardy, G. H. (1940). Ramanujan: Twelve Lectures on Subjects Suggested by His Life and Work. Cambridge University Press. p. 50.
  17. Hardy, G. H.; Wright, E. M. (December 1960). An Introduction to the Theory of Numbers (4th ed.). Oxford University Press. pp. 8–9. ISBN 978-0-19-853310-8.
  18. Prachar, Karl (1957). Primzahlverteilung. Grundlehren der mathematischen Wissenschaften (in German). Vol. 91. Berlin: Springer. p. 164. Cited in Grosswald, Emil (1 January 1984). Topics from the Theory of Numbers (2nd ed.). Boston: Birkhäuser. p. 30. ISBN 978-0-8176-3044-7.
  19. Babai, László (25 December 1995). "Automorphism Groups, Isomorphism, Reconstruction". In Graham, Ronald; Grötschel, Martin; Lovász, László (eds.). Handbook of Combinatorics. Vol. 2. Netherlands: North-Holland Publishing Company. p. 1462. ISBN 978-0-444-82351-9.
  20. Spencer, Joel (9 August 2001). The Strange Logic of Random Graphs. Algorithms and Combinatorics. Vol. 22. Springer. pp. 3–4. ISBN 978-3-540-41654-8.
  21. Bollobás, Béla (8 October 2001). Random Graphs. Cambridge Studies in Advanced Mathematics. Vol. 73 (2nd ed.). Cambridge University Press. pp. 34–36. ISBN 978-0-521-79722-1.
  22. Grädel, Eric; Kolaitis, Phokion G.; Libkin, Leonid; Marx, Maarten; Spencer, Joel; Vardi, Moshe Y.; Venema, Yde; Weinstein, Scott (11 June 2007). Finite Model Theory and Its Applications. Texts in Theoretical Computer Science (An EATCS Series). Springer. p. 298. ISBN 978-3-540-00428-8.
  23. Buckley, Fred; Harary, Frank (21 January 1990). Distance in Graphs. Addison-Wesley. p. 109. ISBN 978-0-201-09591-3.
  24. Oxtoby, John C. (1980). Measure and Category. Graduate Texts in Mathematics. Vol. 2 (2nd ed.). United States: Springer. pp. 59, 68. ISBN 978-0-387-90508-2. While Oxtoby does not explicitly define the term there, Babai has borrowed it from Measure and Category in his chapter "Automorphism Groups, Isomorphism, Reconstruction" of Graham, Grötschel and Lovász's Handbook of Combinatorics (vol. 2), and Broer and Takens note in their book Dynamical Systems and Chaos that Measure and Category compares this meaning of "almost all" to the measure theoretic one in the real line (though Oxtoby's book discusses meagre sets in general topological spaces as well).
  25. Baratchart, Laurent (1987). "Recent and New Results in Rational L2 Approximation". In Curtain, Ruth F. (ed.). Modelling, Robustness and Sensitivity Reduction in Control Systems. NATO ASI Series F. Vol. 34. Springer. p. 123. doi:10.1007/978-3-642-87516-8. ISBN 978-3-642-87516-8.
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  27. Sharkovsky, A. N.; Kolyada, S. F.; Sivak, A. G.; Fedorenko, V. V. (30 April 1997). Dynamics of One-Dimensional Maps. Mathematics and Its Applications. Vol. 407. Springer. p. 33. doi:10.1007/978-94-015-8897-3. ISBN 978-94-015-8897-3.
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  29. Albertini, Francesca; Sontag, Eduardo D. (1 September 1991). "Transitivity and Forward Accessibility of Discrete-Time Nonlinear Systems". In Bonnard, Bernard; Bride, Bernard; Gauthier, Jean-Paul; Kupka, Ivan (eds.). Analysis of Controlled Dynamical Systems. Progress in Systems and Control Theory. Vol. 8. Birkhäuser. p. 29. doi:10.1007/978-1-4612-3214-8. ISBN 978-1-4612-3214-8.
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