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Beginning Activity Beginning Activity 1: Introduction to Infinite Sets
A4US

In Section 9.1, we defined a finite set to be the empty set or a set A such that A≈Nk for some natural number k. We also defined an infinite set to be a set that is not finite, but the question now is, “How do we know if a set is infinite?” One way to determine if a set is an infinite set is to use Corollary 9.8, which states that a finite set is not equivalent to any of its subsets. We can write this as a conditional statement as follows:

If A is a finite set, then A is not equivalent to any of its proper subsets.

or more formally as

For each set A, if A is a finite set, then for each proper subset B of A, A≉B.

1.

Write the contrapositive of the preceding conditional statement. Then explain how this statement can be used to determine if a set is infinite.

2.

Let D+ be the set of all odd natural numbers. In Beginning Activity 1 from Section 9.1, we proved that N≈D+.

(a)

Use this to explain carefully why N is an infinite set.

(b)

Is D+ a finite set or an infinite set? Explain carefully how you know.

3.

Let b be a positive real number. Let (0,1) and (0,b) be the open intervals from 0 to 1 and 0 to b, respectively. In Task 9.2.c of Progress Check 9.2, we proved that (0,1)≈(0,b).

(a)

Use a value for b where 0<b<1 to explain why (0,1) is an infinite set.

(b)

Use a value for b where b>1 to explain why (0,b) is an infinite set.