Page 324 - HOW TO PROVE IT: A Structured Approach, Second Edition

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310 Infinite Sets
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Definition 7.1.4. A set A is called denumerable if Z ∼ A. It is called
countable if it is either finite or denumerable. Otherwise, it is uncountable.

You might think of the countable sets as those sets whose elements can be
counted by pointing to all of them, one by one, while naming positive integers
in order. If the counting process ends at some point, then the set is finite; and
if it never ends, then the set is denumerable. The following theorem gives two
more ways of thinking about countable sets.

Theorem 7.1.5. Suppose A is a set. The following statements are equivalent:

1. A is countable.
2. Either A = ∅ or there is a function f : Z → A that is onto.
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3. There is a function f : A → Z that is one-to-one.
Proof. 1 → 2. Suppose A is countable. If A is denumerable, then there is a
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function f : Z → A that is one-to-one and onto, so clearly statement 2 is
true. Now suppose A is finite. If A = ∅ then there is nothing more to prove, so
suppose A = ∅. Then we can choose some element a 0 ∈ A. Let g : I n → A
be a one-to-one, onto function, where n is the number of elements of A.Now
define f : Z → A as follows:
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g(i) if i ≤ n

f (i) =
a 0 if i > n.
It is easy to check now that f is onto, as required.
2 → 3. Suppose that either A = ∅ or there is an onto function from Z to
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A. We consider these two possibilities in turn. If A = ∅, then the empty set
is a one-to-one function from A to Z . Now suppose g : Z → A, and g is
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onto. Then for each a ∈ A, the set {n ∈ Z | g(n) = a} is not empty, so by the
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well-ordering principle it must have a smallest element. Thus, we can define a
function f : A → Z by the formula
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f (a) = the smallest n ∈ Z such that g(n) = a.
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Note that for each a ∈ A, g( f (a)) = a,so g ◦ f = i A . But then by Theo-
rem 5.3.3, it follows that f is one-to-one, as required.
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3 → 1. Suppose g : A → Z and g is one-to-one. Let B = Ran(g) ⊆ Z .
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If we think of g as a function from A to B, then it is one-to-one and onto, so
A ∼ B. Thus, it suffices to show that B is countable, since by Theorem 7.1.3
it follows from this that A is also countable.
Suppose B is not finite. We must show that B is denumerable, which we can
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do by defining a one-to-one, onto function f : Z → B. The idea behind the