Page 335 - HOW TO PROVE IT: A Structured Approach, Second Edition

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Countable and Uncountable Sets 321
In other words, in decimal notation we have f (A) = 0.d 1 d 2 d 3 ... . For example,
if E is the set of all positive even integers, then f (E) = 0.37373737 ... .If P
is the set of all prime numbers, then f (P) = 0.37737373337 ....
+ +
To see that f is one-to-one, suppose that A ∈ P (Z ), B ∈ P (Z ), and A =
B. Then there is some n ∈ Z such that either n ∈ A and n /∈ B,or n ∈ B and
+
n /∈ A. But then f (A) and f (B) cannot be equal, since their decimal expansions
th
differ in the n digit. Thus, f is one-to-one.

Exercises

∗ 1. (a) Prove that the set of all irrational numbers, R \ Q, is uncountable.
(b) Prove that R \ Q ∼ R.
2. Let F : S n × A → S n+1 be the function defined in the proof of Theo-
rem 7.2.4. Show that F is one-to-one and onto.
3. Let P ={X ∈ P (Z ) | X is finite}. Prove that P is denumerable.
+
4. Prove the following more general form of Cantor’s theorem: For any set
∗
A, A ∼ P (A). (Hint: Imitate the proof of Theorem 7.2.5.)
5. For the meaning of the notation used in this exercise, see exercise 21 of
Section 7.1.
A
A
A
(a) Prove that for any sets A, B, and C, (B × C) ∼ B × C.
A B
(b) Prove that for any sets A, B, and C, (A×B) C ∼ ( C).
A
(c) Prove that for any set A, P (A) ∼ {yes, no}. (Note that if A is finite
and |A|= n then, by exercise 21(c) of Section 7.1, it follows that
|A| n
|P (A)|=|{yes, no}| = 2 . Of course, you already proved this, by
a different method, in exercise 10 of Section 6.2.)
+
+
(d) Prove that Z + P (Z ) ∼ P (Z ).
6. Suppose A is denumerable. Prove that there is a partition P of A such that
P is denumerable and for every X ∈ P, X is denumerable.
∗
7. Prove that if A and B are disjoint sets, then P (A ∪ B) ∼ P (A) ×
P (B).
8. Suppose A ⊆ R , b ∈ R , and for every list a 1 , a 2 , ..., a n of finitely many
+
+
distinct elements of A, a 1 + a 2 + ··· + a n ≤ b. Prove that A is countable.
(Hint: For each positive integer n, let A n ={x ∈ A | x ≥ 1/n}. What can
you say about the number of elements in A n ?)
9. Suppose F ⊆{ f | f : Z → R} and F is countable. Prove that there is
∗ +
a function g : Z → R such that F ⊆ O(g). (See exercise 16 of Section
+
5.1 for the meaning of the notation used here.)
10. Prove that the set of all grammatical sentences of English is denumer-
able. (Hint: Every grammatical sentence of English is a finite sequence