Page 334 - HOW TO PROVE IT: A Structured Approach, Second Edition

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320 Infinite Sets
be an arbitrary positive integer and prove D = f (n). Now recall that we chose
D carefully so that we would be able to prove D = f (n), and the reasoning
behind this choice hinged on whether or not n ∈ f (n). Perhaps the easiest way
to write the proof is to consider the two cases n ∈ f (n) and n /∈ f (n) separately.
In each case, applying the definition of D easily leads to the conclusion that
D = f (n).

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Proof. Suppose f : Z → P (Z ). We will show that f cannot be onto by
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finding a set D ∈ P (Z ) such that D /∈ Ran( f ). Let D ={n ∈ Z | n /∈ f (n)}.
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Clearly D ⊆ Z ,so D ∈ P (Z ). Now let n be an arbitrary positive integer.
We consider two cases.
Case 1. n ∈ f (n). Since D ={n ∈ Z | n /∈ f (n)}, we can conclude that
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n /∈ D. But then since n ∈ f (n) and n /∈ D, it follows that D = f (n).
Case 2. n /∈ f (n). Then by the definition of D, n ∈ D. Since n ∈ D and
n /∈ f (n), D = f (n).
Since these cases are exhaustive, this shows that ∀n ∈ Z (D = f (n)), so
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D /∈ Ran( f ). Since f was arbitrary, this shows that there is no onto function
f : Z → P (Z ). Clearly P (Z ) = ∅, so by Theorem 7.1.5, P (Z )is
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uncountable.
The method used in the proof of Theorem 7.2.5 is called diagonaliza-
tion because of the diagonal arrangement of the boxed answers in Figure 1.
Diagonalization is a powerful technique that can be used to prove many the-
orems, including our next theorem. However, rather than doing another di-
agonalization argument, we’ll simply apply Theorem 7.2.5 to prove the next
theorem.

Theorem 7.2.6. R is uncountable.
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Proof. We will define a function f : P (Z ) → R and show that f is
one-to-one. If R were countable, then there would be a one-to-one function
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g : R → Z . But then g ◦ f would be a one-to-one function from P (Z )to
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Z and therefore P (Z ) would be countable, contradicting Cantor’s theorem.
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Thus, this will show that R is uncountable.
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To define f, suppose A ∈ P (Z ). Then f (A) will be a real number between
0 and 1 that we will specify by giving its decimal expansion. For each positive
th
integer n, the n digit of f (A) will be the number d n defined as follows:

3 if n /∈ A
d n =
7 if n ∈ A.