Chapter 7 Infinite Sets226
1.Cis finite.2.Cis countable.3.Cis uncountable.4.Ccontains only finitely many non-integers.5.Ccontains the rational number 2/3.- There is a maximum non-integer inC.
- There is an > 0such that any two elements ofCareapart.
8.Cis countable.9.Cis uncountable.10.Chas no infinite decreasing sequencec 0 > c 1 >.- Every nonempty subset ofChas a minimum element.
12.Chas a maximum element.13.Chas a minimum element.Class Problems
Problem 7.8.
Show that the setNof finite sequences of nonnegative integers is countable.
Problem 7.9. (a)Several students felt the proof of Lemma 7.1.7 was worrisome,
if not circular. What do you think?
(b)Use the proof of Lemma 7.1.7 to show that ifAis an infinite set, thenAsurjN,
that is, every infinite set is “as big as” the set of nonnegative integers.
Problem 7.10.
The rational numbers fill the space between integers, so a first thought is that there
must be more of them than the integers, but it’s not true. In this problem you’ll
show that there are the same number of positive rationals as positive integers. That
is, the positive rationals are countable.