Handout #16
CS103
April 16, 2010
Robert Plummer
Infinite Sets and Countability
Infinite Sets
In a
finite
set, we can always designate one element as the first member, another as the second member,
etc.
If there are k elements in the set, then these can be listed in the order we have selected:
s
1
,
s
2
,
...,
s
k
Therefore, a finite set is one that can be enumerated by the natural numbers from 1 up to some integer k.
More precisely, set A is finite if there is a positive integer k such that there is a one-to-one correspondence
between A and the set of all natural numbers less than k.
A
one-to-one correspondence
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
1
, and a second element s
2
, but we have no
limit k.
So the list of chosen elements may look like this:
s
1
,
s
2
,
s
3
, .
..
Such an infinite set is called
denumerable
or
countably infinite
.
Both finite and denumerable sets are
countable
sets 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 natural numbers,
denoted
N
.
Thus, the set
N
is, in a sense, the most basic of all infinite sets.
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
N
.
The counting scheme is the function that maps
N
to some other infinite set.
Example 1
The set of positive even integers {2, 4, 6, 8, .
..} is denumerable because there is an obvious one-to-one
correspondence between these integers and the natural numbers (f(n) = 2n).
The set of
all
integers is
denumerable because we can list them as follows: {0, 1, -1, 2, -2, 3, -3 .
..} and define the following one-
to-one correspondence: f(n) = n/2 if n is even, -((n-1)/2) if n is odd.
But at first blush, these examples
seem to go against common sense.
We might think to ourselves that the set of positive even integers must
be 1/2 the size of the set of natural numbers and the set of all integers must be twice the size of the set of
natural numbers.
We have certainly defined functions for these two sets that are both one-to-one and onto,
so by the basic definitions given above,
N
even
and
Z
have the same cardinality as
N
.
Surprising, but true!