Page 342 - HOW TO PROVE IT: A Structured Approach, Second Edition

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328 Infinite Sets
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8. Let A 1 = Z , and for all n ∈ Z let A n+1 = P (A n ).
(a) Prove that for all n ∈ Z and m ∈ Z ,if n < m then A n ≺ A m .
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(b) The sets A n , for n ∈ Z , represent infinitely many sizes of infinity.
Are there any more sizes of infinity? In other words, can you think
of an infinite set that is not equinumerous with A n for any n ∈ Z ?
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9. The proof of the Cantor–Schr¨oder–Bernstein theorem gives a method
for constructing a one-to-one and onto function h : A → B from one-
to-one functions f : A → B and g : B → A. Use this method to find
a one-to-one, onto function h :(0, 1] → (0, 1). Start with the functions
f :(0, 1] → (0, 1) and g :(0, 1) → (0, 1] given by the formulas:
x
f (x) = , g(x) = x.
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10. Let E ={R | R is an equivalence relation on Z }.
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(a) Prove that E P (Z ).
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(b) Let A = Z \{1, 2}andlet P bethesetofallpartitionsofZ .Define
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f : P (A) → P by the formula f (X) ={X ∪{1}, (A \ X) ∪{2}}.
Prove that f is one-to-one.
(c) Prove that E ∼ P (Z ).
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11. Let T ={R | R is a total order on Z }. Prove that T ∼ P (Z ). (Hint:
Imitate the solution to exercise 10.)
12. (a) Provethatif A hasatleasttwoelementsand A × A ∼ A thenP (A) ×
P (A) ∼ P (A). (Hint: Use exercise 7 of Section 7.2.)
(b) Prove that R × R ∼ R.
13. An interval is a set I ⊆ R with the property that for all real numbers
x, y, and z,if x ∈ I, z ∈ I, and x < y < z, then y ∈ I. An interval is
nondegenerate if it contains at least two different real numbers. Suppose
F is a set of nondegenerate intervals and F is pairwise disjoint. Prove
that F is countable. (Hint: By Lemma 7.3.4, every nondegenerate interval
contains a rational number.)
*14. For the meaning of the notation used in this exercise, see exercise 21 of
Section 7.1.
R
(a) Prove that R ∼ P (R).
Q
(b) Prove that R ∼ R.
R
(c) (For students who have studied calculus) Let C ={ f ∈ R | f is
continuous}. Prove that C ∼ R. (Hint: Show that if f and g are con-
tinuous functions and ∀x ∈ Q( f (x) = g(x)), then f = g.)