Problem Set 10—The Algebra of Set Operations

Purpose

This problem set develops your ability to reason with and about algebraic rules involving set operations. It also develops your ability to reason about Cartesian products and their algebra.

Background

This problem set is based on material in sections 5.3 and 5.4 of our textbook. We discussed, or will discuss, this material in class on November 4 and 7.

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 3 from section 5.3 of our textbook (prove that (A∩B)^C = A^C ∪ B^C).

Problem 2

Exercise 5 from section 5.3 of our textbook (form and prove, in two ways, a conjecture about the relationship between A - (B∩C) and (A-B) ∪ (A-C); see book for more details).

Problem 3

Let A = {-1,0,1} and B = {a,b}. Find the Cartesian product A × B.

Problem 4

Exercise 8 from section 5.4 of our textbook (prove that A×B = B×A if and only if 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.