Almost all
Template:Short description In mathematics, the term "almost all" means "all but a negligible quantity". More precisely, if <math>X</math> is a set, "almost all elements of <math>X</math>" means "all elements of <math>X</math> but those in a negligible subset of <math>X</math>". 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 <math>X</math>" means "a negligible quantity of elements of <math>X</math>".
Meanings in different areas of mathematics
[edit]Prevalent meaning
[edit]Template:Further Throughout mathematics, "almost all" is sometimes used to mean "all (elements of an infinite set) except for finitely many".Template:R This use occurs in philosophy as well.Template:R Similarly, "almost all" can mean "all (elements of an uncountable set) except for countably many".Template:R
Examples:
- Almost all positive integers are greater than 1012.Template:R
- Almost all prime numbers are odd (2 is the only exception).<ref>Template:Cite book</ref>
- Almost all polyhedra are irregular (as there are only nine exceptions: the five platonic solids and the four Kepler–Poinsot polyhedra).
- If P is a nonzero polynomial, then P(x) ≠ 0 for almost all x (if not all x).
Meaning in measure theory
[edit]
When speaking about the reals, sometimes "almost all" can mean "all reals except for a null set".Template:RTemplate:R 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".Template:R 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"Template:R or "all points in S except for those in a null set" (this time, S is a set of points in the space).Template:R Even more generally, "almost all" is sometimes used in the sense of "almost everywhere" in measure theory,Template:RTemplate:R or in the closely related sense of "almost surely" in probability theory.Template:RTemplate:R
Examples:
- In a measure space, such as the real line, countable sets are null. The set of rational numbers is countable, so almost all real numbers are irrational.Template:R
- Georg Cantor's first set theory article proved that the set of algebraic numbers is countable as well, so almost all reals are transcendental.Template:RTemplate:R
- Almost all reals are normal.Template:R
- The Cantor set is also null. Thus, almost all reals are not in it even though it is uncountable.Template:R
- The derivative of the Cantor function is 0 for almost all numbers in the unit interval.Template:R It follows from the previous example because the Cantor function is locally constant, and thus has derivative 0 outside the Cantor set.
Meaning in number theory
[edit]Template:Further 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.Template:RTemplate:R
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.Template:RTemplate:Refn
- Almost all even positive numbers can be expressed as the sum of two primes.Template:R
- Almost all primes are isolated. Moreover, for every positive integer Template:Mvar, almost all primes have prime gaps of more than Template:Mvar both to their left and to their right; that is, there is no other prime between Template:Math and Template:Math.Template:R
Meaning in graph theory
[edit]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.Template:R However, it is sometimes easier to work with probabilities,Template:R 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.Template:R 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.Template:R 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,Template:R 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.Template:R
Example:
- Almost all graphs are asymmetric.Template:R
- Almost all graphs have diameter 2.Template:R
Meaning in topology
[edit]In topologyTemplate:R and especially dynamical systems theoryTemplate:R (including applications in economics),Template:R "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.Template:R
Example:
- Given an irreducible algebraic variety, the properties that hold for almost all points in the variety are exactly the generic properties.Template:R This is due to the fact that in an irreducible algebraic variety equipped with the Zariski topology, all nonempty open sets are dense.
Meaning in algebra
[edit]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".Template:R 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.Template:R