Problem Set 3—Sets

Purpose

This problem set reinforces your understanding of basic set concepts and notations for defining sets.

Background

This problem set is mainly based on material in section 2.3 of our textbook. We discussed that material in class on February 2 and 5.

This problem set introduces a new feature that I will include in certain future problem sets: “proofs out of context.” The idea here is that this course is central to the mathematics major because most later courses expect you to read and write proofs, definitions, and similar things about whatever that later course studies — in other words, to be able to use the material you learned in this course outside of the context in which you learned it. So to get you used to doing that, I will inject problems into some problem sets that apply concepts covered in this course to branches of math not otherwise used as examples in the textbook or lectures. Problem 3 on this problem set is the first example.

Activity

Problem 1

Part A

Use the roster method to describe each of the following sets:

1. The set of even natural numbers greater than 1 but less than 1001.
2. The set of real numbers x such that 2x + 3 = 11.
3. The set of positive integer multiples of 4.

Part B

Let S be the set { 4n | n ∈ ℕ }. For each set in Part A, determine whether it is equal to S, whether it is a subset of S, whether S is a subset of it, or whether there is no equality or subset relation between it and S (note that more than one of these relations may hold at the same time). You should have informal reasons for your decisions, but do not need formal proofs.

Problem 2

Part A

Use set builder notation to describe each of the following sets:

1. The set of integer multiples of 7.
2. The interval [-3,8) on the real number line.
3. The set of integers n such that 2n is either greater than 3 or less than -11.
4. The set of integers that can be computed by multiplying 7 by a positive even integer.

Part B

Let R be the set { 7, 14, 21, … }. For each set in Part A, determine whether it is equal to R, whether it is a subset of R, whether R is a subset of it, or whether there is no equality or subset relation between it and R (note that more than one of these relations may hold at the same time). You should have informal reasons for your decisions, but do not need formal proofs.

Problem 3 (Proofs out of Context)

Define an alphabet to be a set of letters or other symbols.

If A is an alphabet, define a string over A to be any sequence of symbols from A.

Example: the set {A,E,I,O,U} is an alphabet; some strings over this alphabet include the sequences

UIA
EIEIO
A

etc.

Part A

Using either roster notation or set builder notation, describe the smallest alphabet A such that

MATH221

is a string over A.

Part B

Are there any alphabets besides your answer to Part A that MATH221 is also a string over? If so, give an example; if not explain why not.

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.

I will use the following guidelines to grade this problem set:

• What I expect (8 points). Between your written answers and verbal explanations, I expect you to show that you understand (1) how to use roster notation to define sets, (2) how to use set builder notation to define sets, (3) how to informally determine whether one set is a subset of another, and (4) how to informally determine when two sets are equal.
• Half of what I expect (4 points). Some plausible, but not the only, examples of solutions that would meet half my expectations include that you fully understand 2 of the 4 items listed as expected, and don’t understand the others at all, OR you partially understand all 4 items.
• Exceeding expectations (typically 1 point added to what you otherwise earn). Demonstrating that you have significantly engaged with math beyond what is needed to solve the given problems exceeds what I expect.