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 ".
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:
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:
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:
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:
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:
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]
This can also be expressed in the statement: 'Almost all prime numbers are odd.'
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 ".
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:
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:
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:
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:
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:
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]
This can also be expressed in the statement: 'Almost all prime numbers are odd.'