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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 \(A \approx \mathbb{N}_k\) for some natural number \(k\text{.}\) 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 \(A\text{,}\) if \(A\) is a finite set, then for each proper subset \(B\) of \(A\text{,}\) \(A \not\approx B\text{.}\)

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 \(\mathbb{N} \approx D^+\text{.}\)

(a)

Use this to explain carefully why \(\mathbb{N}\) 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 \(b\text{,}\) respectively. In Task 9.2.c of Progress Check 9.2, we proved that \(( 0, 1 ) \approx ( 0, b )\text{.}\)

(a)

Use a value for \(b\) where \(0 \lt b \lt 1\) 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.