Page 325 - HOW TO PROVE IT: A Structured Approach, Second Edition

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Equinumerous Sets 311
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definition is simply to let f (n) be the n element of B, for each n ∈ Z . (Recall
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that B ⊆ Z , so we can use the ordering of the positive integers to make sense
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of the idea of the n element of B.) For a more careful definition of f and the
proof that f is one-to-one and onto, see exercise 14.
If A is countable and A = ∅, then by Theorem 7.1.5 there is a function
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f : Z → A that is onto. If, for every n ∈ Z , we let a n = f (n), then the fact
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that f is onto means that every element of A appears at least once in the list
a 1 , a 2 , a 3 ,... . In other words, A ={a 1 , a 2 , a 3 ,...}. Countability of a set A is
often used in this way to enable us to write the elements of A in a list, indexed
by the positive integers. In fact, you might want to think of countability for
nonempty sets as meaning listability. Of course, if A is denumerable, then the
function f can be taken to be one-to-one, which means that each element of A
will appear only once in the list a 1 , a 2 , a 3 ,... . For an example of an application
of countability in which the elements of a countable set are written in a list,
see exercise 17.
Theorem 7.1.5 is also sometimes useful for proving that a set is denumerable,
as the proof of our next theorem shows.

Theorem 7.1.6. Q is denumerable.
Proof. Let f : Z × Z → Q be defined as follows:
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f (p, q) = p/q.

Clearly f is onto, since by definition all rational numbers can be written as
fractions, but note that f is not one-to-one. For example, f (1, 2) = f (2, 4) =
1/2. Since Z ∼ Z, by Theorem 7.1.2 we have Z × Z ∼ Z × Z , and since
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we already know that Z × Z + is denumerable, it follows that Z × Z + is
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also denumerable. Thus, we can choose a one-to-one, onto function g : Z →
Z × Z . By Theorem 5.2.5, f ◦ g : Z → Q is onto, so by Theorem 7.1.5,
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Q is countable. Clearly Q is not finite, so it must be denumerable.
Although our focus in this chapter is on infinite sets, the methods in this
section can be used to prove theorems that are useful for computing the cardi-
nalities of finite sets. We end this section with one example of such a theorem,
and give several other examples in the exercises (see exercises 18–28).

Theorem 7.1.7. Suppose A and B are disjoint finite sets. Then A ∪ B is finite,
and |A ∪ B|=|A|+|B|.