Last Problem Set—Infinite Sets

Purpose

This problem set reinforces your understanding of countable and uncountable sets, and how to prove that a set is one or the other category.

Background

This exercise is based on sections 9.1 through 9.3 of our textbook. We discussed, or will discuss, this material in class on December 7, December 9, and December 12.

Problem 4 continues the project of challenging you with Prof. Nicodemi’s real analysis problems, and so does not draw on the material we are discussing now in class.

Activity

Solve the following problems. Formal proofs should be word-processed (i.e., not hand-written) and should follow the guidelines in Sundstrom’s text.

Problem 1

Exercise 5c in section 9.1 of our textbook (prove that if A ∩ B is an infinite set, then A is an infinite set).

Problem 2

Exercise 2b in section 9.2 of our textbook (prove that the set of integers that are multiples of 5 is countably infinite).

Problem 3

Exercise 5 in section 9.3 of our textbook, supporting your answer with a proof (determine whether the set of infinite sequences of 0s and 1s is countable or uncountable).

Problem 4

Recall the following definitions from problem set 13:

Definition 1.

Let A be a subset of the real numbers ℝ and let c ∈ ℝ. We call c a controller of A if x ≤ c for every x ∈ A.

Definition 2.

A set A ⊂ ℝ is called controlled if there exists a controller of A.

Definition 3.

Let A ⊆ ℝ be a controlled, non-empty set. A real number b is called the best controller of A if

1. b is a controller of A and
2. b ≤ c where c is any controller of A.

Part A

Suppose that A is a nonempty controlled set and that c ∈ A is a controller of A. Prove that c is the best controller of A.

Part B

Suppose that A is a nonempty controlled set and that b is its best controller. Let y be any real number such that y < b. Prove there is an element a ∈ A such that y < a ≤ b.

Follow-Up

I will grade this exercise in a face-to-face meeting with you. During this meeting I will look at your solution, ask you any questions I have about it, answer questions you have, etc. Please bring a written solution to the exercise to your meeting, as that will speed the process along.

Sign up for a meeting via Google calendar. Please make the meeting 15 minutes long, and schedule it to finish before the end of the “Grade By” date above.