Page 326 - HOW TO PROVE IT: A Structured Approach, Second Edition

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312 Infinite Sets
Proof. Let n =|A| and m =|B|. Then A ∼ I n and B ∼ I m . Notice that
if x ∈ I m then 1 ≤ x ≤ m, and therefore n + 1 ≤ x + n ≤ n + m,so x +
n ∈ I n+m \I n . Thus we can define a function f : I m → I n+m \ I n by the
formula f (x) = x + n. It is easy to check that f is one-to-one and onto, so
I m ∼ I n+m \ I n . Since B ∼ I m , it follows that B ∼ I n+m \ I n . Applying part
2 of Theorem 7.1.2, we can conclude that A ∪ B ∼ I n ∪ (I n+m \ I n ) = I n+m .
Therefore A ∪ B is finite, and |A ∪ B|= n + m =|A|+|B|.

Exercises

1. Show that the following sets are denumerable.
∗
(a) N.
(b) The set of all even integers.
2. Show that the following sets are denumerable:
(a) Q × Q.
√
(b) Q( 2). (See exercise 6(b) of Section 6.5 for the meaning of the
notation used here.)
3. In this problem we’ll use the following notation for intervals of real num-
bers. If a and b are real numbers and a < b, then

[a, b] ={x ∈ R | a ≤ x ≤ b}
(a, b) ={x ∈ R | a < x < b}
(a, b] ={x ∈ R | a < x ≤ b}
[a, b) ={x ∈ R | a ≤ x < b}.

(a) Show that [0, 1] ∼ [0, 2].
(b) Show that (−π/2,π/2) ∼ R. (Hint: Use a trigonometric function.)
(c) Show that (0, 1) ∼ R.
(d) Show that (0, 1] ∼ (0, 1).
4. Justify your answer to each question with either a proof or a counter-
∗
example.
(a) Suppose A ∼ B and A × C ∼ B × D. Must it be the case that
C ∼ D?
(b) Suppose A ∼ B, A and C are disjoint, B and D are disjoint, and
A ∪ C ∼ B ∪ D. Must it be the case that C ∼ D?
5. Prove that if A ∼ B then P (A) ∼ P (B).
6. (a) Prove that for all natural numbers n and m,if I n ∼ I m then n = m.
∗
(Hint: Use induction on n.)
(b) Prove that if A is finite, then there is exactly one natural number n
such that I n ∼ A.