In
set theory
Set theory is the branch of mathematical logic that studies Set (mathematics), sets, which can be informally described as collections of objects. Although objects of any kind can be collected into a set, set theory – as a branch of mathema ...
, when dealing with
sets of
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 sets are involved).
For example:
*The set
is almost
for any
in
, because only finitely many
natural number
In mathematics, the natural numbers are the numbers 0, 1, 2, 3, and so on, possibly excluding 0. Some start counting with 0, defining the natural numbers as the non-negative integers , while others start with 1, defining them as the positive in ...
s are less than ''
''.
*The set of
prime number
A prime number (or a prime) is a natural number greater than 1 that is not a Product (mathematics), product of two smaller natural numbers. A natural number greater than 1 that is not prime is called a composite number. For example, 5 is prime ...
s is not almost
, because there are infinitely many natural numbers that are not prime numbers.
*The set of
transcendental numbers are almost
, because the
algebraic real numbers form a
countable subset of the set of real numbers (which is uncountable).
*The
Cantor set is uncountably infinite, but has
Lebesgue measure zero.
So almost all real numbers in (0, 1) are members of the
complement of the Cantor set.
See also
*
Almost periodic function - and Operators
*
Almost all
*
Almost surely
*
Approximation
*
List of mathematical jargon
References
Mathematical terminology
Set theory
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