January 9, 2009
Infinite Sets and Countability
Thanks to Maggie Johnson for some portions of this handout.
set, we can always designate one element as the first member, another as the second
If there are k elements in the set, then these can be listed in the order we have
Therefore, a finite set is one that can be enumerated by the positive integers from 1 up to some
More precisely, set A is finite if there is a positive integer k such that there is a one-to-
one correspondence (bijection) between A and the set of all natural numbers less than k.
defined it when discussing functions, a
between the elements of a set
P and the elements of a set Q means that it is possible to pair off the elements of P and Q such
that every element of P is paired off with a distinct element of Q.
If a set is infinite, we may still be able to select a first element s
, and a second element s
, but we
have no limit k.
So the list of chosen elements may look like this:
Such an infinite set is called
Both finite and denumerable sets are
because we can count, or enumerate the elements in the set.
Being countable, however, does not
always mean that we can give a value for the total number of elements in the set; it just means we
can say "Here is a first one, here is a second one, etc.".
More formally, a denumerable set is one
where we can define a one-to-one correspondence between the elements in the set and the set of
Thus, the set
is, in a sense, the most basic of all infinite sets.
The reason for this is that the
one-to-one correspondence can be used to "count" the elements of an infinite set.
If f is a one-to-
one and onto function from
to some infinite set A, then f(1) can be designated as the first
element of A, f(2), the second, and so forth.
Because f is one-to-one, no element is ever counted
twice; and because f is onto, every element of A is counted eventually.
To prove denumerability, we need only exhibit a counting scheme (i.e., if starting from a
particular element, we can sequentially list all the elements in the list), because such a listing will
yield a one-to-one correspondence between the elements in the set and
The counting scheme
is the function that maps
to some other infinite set.