Why Read This Book? - Index of

140 Chapter 4 Functions
the repetition in f and is therefore one-to-one. 13 By creating f1 in this way, we
would have that A is finite. The point is stated in the following.
Theorem 4.7.2 A set A is infinite if and only if for every nonnegative integer n
and every function f : Nn → A, f is not onto.
Loosely speaking, if a set is infinite, then no matter how large n is, there are not
enough elements in Nn to tag all the elements of the set.
Before we get into the interesting results of infinite sets, let’s point out where
it will lead. Strangely, just because two sets are both infinite, it does not follow that
they have the same cardinality, in the sense that there is a one-to-one correspondence
between them. Some infinite sets are actually bigger than others. This makes
for some real surprises and motivates us to discuss different orders of infinity. In
fact, it is possible to generate an infinite sequence of infinite sets A1,A2,...where
|An| is a higher order of infinity than |An−1|. It’s mind boggling. Not only is there
more than one size of infinity, but also there are infinitely many infinities. A natural
question then arises: Of the infinitely many different orders of infinity, which
one represents the number of distinct infinities that exist? We will look at only
two sizes of infinity. Here is our first.
Definition 4.7.3 Suppose A is an infinite set. Then A is said to be countably
infinite provided there exists a one-to-one function from the positive integers N
onto A, and we say that A has cardinality ℵ0 (ℵ is the Hebrew letter aleph). If A
is finite or countably infinite, we say that A is countable. IfA is not countable, we
say that it is uncountable.
Example 4.7.4 By the identity function, the natural numbers are countably
infinite. �
EXERCISE 4.7.5 Show that the whole numbers are countably infinite.
The equivalence relation from Exercise 4.4.9 allows us to say that the natural
numbers are a representative set from the equivalence class of all countably infinite
sets. Let’s see some other sets that are countably infinite and prove some theorems
about countable sets in general. Remember, to show a set A is countably infinite,
your task is to construct a one-to-one function from the positive integers onto A.
Sometimes an easy way to build such a function is by systematically declaring the
values of f(1), f(2), f(3), and so on, which is the same as ordering the elements
13 We begin by creating a subset S ⊆ Nn so that f : S → A is one-to-one by applying the axiom of choice
(discussed on page 118). For each a ∈ A, we could take a single element of f −1 (a), and then let
S consist of all these chosen pre-images. Then we could apply Theorem 4.5.2 to conclude the existence
of a bijection g : S → Nm for some m