Problem Set 9—Proving Set Relationships

Purpose

This problem set develops your ability to prove that one set is a subset of another, equal to another, etc. It also reinforces your understanding of basic set operations.

Background

This problem set is based on material in sections 5.1 and 5.2 of our textbook. We discussed, or will discuss, this material in class on October 28 and 31.

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

Exercises 8a, 8b, and 8c in section 5.1 of our textbook (find A ∩ B, A ∪ B), and (A ∪ B)^C where A is the set of natural numbers greater than or equal to 7 and B is the set of odd natural numbers; see the textbook for more details).

Problem 2

Exercise 2b in section 5.2 of our textbook (prove that if A ⊆ B and B ⊆ C, then A ⊆ C).

Problem 3

Exercise 9 in section 5.2 of our textbook (determine whether it is true or not that for all sets A and B that are subsets of some universal set, A ∩ B is disjoint from A - B; support your answer with either a proof or a counterexample).

Problem 4

A slight extension of exercise 13c in section 5.2 of our textbook: letting A, B, and C be subsets of some universal set, determine whether it is true or not that if A ∪ C = B ∪ C then A = B; support your answer with a counterexample or proof.

Problem 5

Exercise 14 in section 5.2 of our textbook (prove that for all sets A, B, and C that are subsets of some universal set, if A ∩ B = A ∩ C and A^C ∩ B = A^C ∩ C, then B = C).

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.