Mathematical Reasoning- Writing and Proof, Version 2.1, 2014a

464 Chapter 9. Finite and Infinite Sets
6. Use the formula in Part (5) to
(a) Calculate f.1/ through f .10/. Are these results consistent with the
pattern exhibited at the start of this activity?
(b) Calculate f .1000/ and f .1001/.
(c) Determine the value of n so that f.n/ D 1000.
In this section, we will describe several infinite sets and define the cardinal
number for so-called countable sets. Most of our examples will be subsets of some
of our standard number systems such as N, Z, andQ.
Infinite Sets
In Preview Activity 1, we saw how to use Corollary 9.8 to prove that a set is infinite.
This corollary implies that if A is a finite set, then A is not equivalent to any of its
proper subsets. By writing the contrapositive of this conditional statement, we can
restate Corollary 9.8 in the following form:
Corollary 9.8. If a set A is equivalent to one of its proper subsets, then A is infinite.
In Preview Activity 1, we used Corollary 9.8 to prove that
The set of natural numbers, N, is an infinite set.
The open interval .0; 1/ is an infinite set.
Although Corollary 9.8 provides one way to prove that a set is infinite, it is sometimes
more convenient to use a proof by contradiction to prove that a set is infinite.
The idea is to use results from Section 9.1 about finite sets to help obtain a contradiction.
This is illustrated in the next theorem.
Theorem 9.10. Let A and B be sets.
1. If A is infinite and A B, thenB is infinite.
2. If A is infinite and A B,thenB is infinite.
Proof. We will prove part (1). The proof of part (2) isexercise(3) onpage473.