Often mathematicians use expressions such as "almost
every positive integer is composite" or "almost all real
numbers are irrational." In each case almost every
or almost all means all but a "negligible" fraction,
but how we define that fraction (and negligible) depends on
the underlying set.
In the positive integers (the usual case in this
Let P(n) be a predicate (a statement about the
integer n such as "n is prime").
Let #P(N) be the number of positive integers
less than N which satisfy P(n). For example,
if P(n) = "n is even," then #P(N)
In other sets:
If the ratio #P(N)/N gets arbitrarily
close to 1 as N gets big (that
is, lim #P(N)/N = 1 as N approaches infinity),
then we say "for almost every n, P(n)."
For example, if P(n) is "n is composite,"
then #P(N) is about (1 - 1/log N) N
(by the prime number theorem), so #P(N)/N
is about 1-1/log N. This clearly gets close to
1 as N gets large; so, almost every positive
integer is composite.
In other sets the concept "almost every" is defined
in different ways.
For example, in the real numbers a common way to define
"almost every" is to specify all but a set of measure zero
(usually using the Lesbegue measure). These ideas are
beyond the scope of this glossary.