Problem Set 4—Set Basics

Purpose

This problem set develops your ability to reason with basic facts about sets. In particular, by the time you finish this problem set you should be able to describe sets using both the roster method and set builder notation; reason about open sentences and truth sets; recognise when two sets are equal, when one set is a subset of another, and when a set is empty; and use ideas of sets in proofs.

Background

This problem set is based on material in section 2.3 of our textbook. We discussed, or will discuss, this material in class on February 1 and 3.

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 4c in section 2.3 of our textbook (use the roster method to specify the set of integers n such that √n is a natural number and n < 50).

Problem 2

Use set builder notation to specify the set of all integers that are multiples of 5.

Problem 3

Let P(x) be the predicate |x| < π, and suppose the universal set is ℝ. Give another description of the truth set of P (i.e., besides “|x| < π”), give two examples of its members, and give two examples of real numbers that are not members.

Problem 4

Exercises 6c and 6d in section 2.3 of our textbook (describe in English, and roster notation where appropriate, the sets {x∈ℝ | x² = 16} and {x∈ℝ | x²+16 = 0}).

Problem 5

For each of the following pairs of sets, determine whether the sets are equal, and whether either is a subset of the other. Give sound arguments, but not necessarily formal proofs, that justify your answers:

Pair 1

{4n+1 | n∈ℤ} and the odd integers

Pair 2

ℝ and {x∈ℝ | √x∈ℤ}

Pair 3

The set of prime numbers and {7n | n∈ℕ ∧ n≥2}

Problem 6

An extension of exercise 7c (under “Explorations and Activities”) in section 2.3 of our textbook: either formally prove, or find a counter-example that disproves, closure of the set {1, 4, 7, 10, 13, … } (a) under addition, and (b) under multiplication.

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.