Page 320 - HOW TO PROVE IT: A Structured Approach, Second Edition

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7

Infinite Sets

7.1. Equinumerous Sets

In this chapter, we’ll discuss a method of comparing the sizes of infinite sets.
Surprisingly, we’ll find that, in a sense, infinity comes in different sizes! By
now, you should be fairly proficient at reading and writing proofs, so we’ll
give less discussion of the strategy behind proofs and leave more proofs as
exercises.
For finite sets, we determine the size of a set by counting. What does it mean
to count the number of elements in a set? When you count the elements in a set
A, you point to the elements of A in turn while saying the words one, two, and
so forth. We could think of this process as defining a function f from the set
{1, 2,..., n} to A, for some natural number n. For each i ∈{1, 2,..., n},we
let f (i) be the element of A you’re pointing to when you say “i.” Because every
element of A gets pointed to exactly once, the function f is one-to-one and onto.
Thus, counting the elements of A is simply a method of establishing a one-
to-one correspondence between the sets {1, 2,..., n} and A, for some natural
number n. One-to-one correspondence is the key idea behind measuring the
sizes of sets, and sets of the form {1, 2,..., n} are the standards against which
we measure the sizes of finite sets. This suggests the following definition.

Definition 7.1.1. Suppose A and B are sets. We’ll say that A is equinumerous
with B if there is a function f : A → B that is one-to-one and onto. We’ll write
A ∼ B to indicate that A is equinumerous with B. For each natural number n,
let I n ={i ∈ Z | i ≤ n}. A set A is called finite if there is a natural number n
+
such that I n ∼ A. Otherwise, A is infinite.

You are asked in exercise 6 to show that if A is finite, then there is exactly one
n such that I n ∼ A. Thus, it makes sense to define the number of elements of a
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