Instructor: Assistant Professor Pete L. Clark, Ph.D., pete (at) math (dot) uga (dot) edu

Course webpage: http://www.math.uga.edu/~pete/MATH3200F2009.html (i.e., right here)

Office Hours: Boyd 502, TuTh 2-3 pm, and by appointment

Course Text (required): Mathematical Proofs: A Transition to Advanced Mathematics, by Gary Chartrand, Albert D. Polimeni and Ping Zhang, 2nd edition.

For information on grades, exams and other procedural matters, please consult the course syllabus.

These notes are provided to you for your enlightenment and edification. In some places they go further than what was presented in the lectures. You are not responsible for any of this additional material. Moreover, the notes contain some "exercises", which are not to be turned in, and some of them are quite challenging, but those of you who are seeking deeper understanding of the material may enjoy thinking about them.

HOMEWORK

Assignment 1: Due in class, Thursday August 27th

To solve: Chapter 1, exercises 1-45, 47

To be turned in: Chapter 1, exercises 3, 6, 8, 9, 10, 16, 17, 20, 24, 26, 31, 32, 41

Optional Typed Problems:

OT1.1) Draw Venn diagrams for 4 and for 5 sets. (It is not possible to use circles. You might think about trying to prove this if that sounds interesting to you.)

OT1.2) Believe it or not, there has been some recent work done on Venn diagrams with certain nice properties. For instance, it is possible to make a very pretty Venn diagram for 5 sets using congruent ellipses. Research this on the internet and write a short essay (approximately two pages) detailing some of the interesting results.

Assignment 2: Due in Class, Tuesday September 8th To solve: Chapter 2, exercises: 1-6, 8-12, 14-20, 22, 23, 25, 26, 28, 30-35, 37-41, 45, 46

To be turned in: 2, 4, 8, 10, 14, 16, 18, 20, 22, 30, 32, 34, 38, 40, 46

Typed Problems: Please do at least two of the following problems.

T2.1) Is there a partition of the empty set? (Comment: The definition in your text explicitly excludes the empty set. I am asking you to nevertheless consider whether there exists a family of sets satisfying the three properties of the partition of X when X is the empty set.)

T2.2) a) Let a,b,a',b' be objects. Show that { {a}, {a,b} } = { {a'}, {a',b'} } if and only if a = a' and b = b'.
b) Explain why the result of part a) would allow us to define the ordered pair (a,b) as { {a}, {a,b} }.
c) Do you have any reservations about this definition? (For example, is it the only possible definition? Is it helpful to explicitly define ordered pairs in this way?) Discuss.

T2.3) a) Let X,Y,Z be sets. Prove that (X union Y) intersect Z = (X intersect Z) union (Y intersect Z).
b) Let P,Q,R be statements. Prove that (P or Q) and R = (P and R) or (Q and R).
c) Can you give an argument which proves both a) and b) at the same time?

T2.4) a) Show that there exists a binary logical operator, P*Q, such that not P, (P or Q) and (P and Q), can all be constructed in terms of the operator *.
b) Of the 16 binary logical operators, how many have the property in part a)?

Assignment 3: Due in Class, Tuesday September 15th

To solve: chapter 3, all exercises (but none of the additional exercises)

To be turned in: the even-numbered problems 3.2-3.30 except 3.26. (N.B.: Feel free to disregard 3.26 until further notice.)

Typed Probems:

T3.1) a) You are shown a selection of cards, each of which has a single letter printed on one side and a single number printed on the other side. Then four cards are placed on the table. On the up side of these cards you can see, respectively, D, K, 3 and 7. Here is a rule: "Every card that has a D on one side has a 3 on the other." Your task is to select all those cards, but only those cards, which you would have to turn over in order to discover whether or not the rule has been violated.

b) You have been hired to watch, via closed-circuit camera, the bouncer at a certain 18-and-over club. In order to be allowed to drink once inside the club, a patron must display valid 21-and-over ID to the bouncer, who then gives him/her a special bracelet. In theory the bouncer should check everyone's ID, but (assume for the purposes of this problem, at least!) it is not illegal for someone who is under 18 to enter the club, so you are not concerned about who the bouncer lets in or turns away, but only who gets a bracelet. You watch four people walk into the club, but because the bouncer is so large, sometimes he obscures the camera. Here is what you can see:

The first person gets a bracelet.
The second person does not get a bracelet.
The third person displays ID indicating they are 21.
The fourth person does not display any ID.

You realize that you need to go down to the club to check some IDs. Precisely whose ID's do you need to check to verify that the bouncer is obeying the law?

Optional Typed Problems (Turn in Before 9/22/09)

OT3.2) a) Write down clearly the parity rules for addition and multiplication that were discussed and used in class, and verify (i.e., prove!) all of them using the 2k / 2l+1 technique seen in class and in the textbook.

b) Are there similar parity rules for exponentiation? E.g., is an even/odd number raised to an even/odd power always even/odd? Discuss.

Assignment 4: Due in Class, Thursday September 24th

To solve: Chapter 4, all even exercises except 4.10, 4.12, 4.14, 4.16.

To be handed in: Problems 4.2, 4.4, 4.6, 4.8, 4.28, 4.30, 4.38, 4.40, 4.44

Typed problems:

T4.1) Write a one page essay describing your experience with the course so far, especially any concerns or suggestions that you may have. For instance, the pace of the course, the difficulty of specific topics, and/or the amount of homework are all good topics. How would you feel about being asked to present solutions to problems on the board during class time?

Assignment 5: Due in Class, Tuesday October 6th

To solve: Chapter 5, all even exercises (none of the additional exercises).

To be handed in: Problems 5.2, 5.4, 5.8, 5.14, 5.18, 5.20, 5.22, 5.24, 5.26, 5.28, 5.29, 5.30, 5.31, 5.32, 5.36

Typed problems: Chapter 5, Exercises 5.38, 5.45, 5.46

Assignment 6: Due in class, Thursday October 15th

To be read: Chapter 7 Quiz, pp. 167-168
To solve but not hand in: Problems 7.1, 7.3, 7.4, 7.5, 7.7, 7.8, 7.9, 7.11, 7.12, 7.13, 7.15
To be handed in: Problems 6.6, 6.10, 6.11, 6.14, 6.15, 6.17, 7.2, 7.6, 7.10, 7.14, 7.16

Typed Problems:

T6.1) Exercise 6 on p. 10 of the induction handout.

T6.2) Exercise 8 on p. 12 of the induction handout.

Optional Typed Problems (turn in any time before the end of the semester):

OT6.1) Exercise 2 on p.6 of the induction handout.

OT6.2) Exercise 3 on p. 6 of the induction handout.

OT6.3) Exercise 4 on p.7 of the induction handout.

OT 6.4) Find the smallest real number a such that for all real numbers x, ax >= x.

OT 6.5) Explore variants of Proposition 7 (p. 8 of the induction handout) to partial sums of other p-series.

Assignment 7: Due in class, Tuesday, November 17th

To solve: Chapter 6, Exercises 6.18, 6.19, 6.20, 6.21, 6.22, 6.23, 6.26, 6.28, 6.30, 6.31, 6.32, 6.34, 6.35, 7.(18+3n) for 0 <= n <= 16, 7.67

To be handed in: 6.18, 6.20, 6.22, 6.26, 6.28, 6.30, 6.32, 6.34, 7.18, 7.24, 7.30, 7.36, 7.42, 7.48, 7.54, 7.60, 7.66

Typed problems: none due this week

Optional Typed Problems:

OT7.1: Let x,y,z be any integers such that x2 + y2 = z2. Show that 60 | xyz. Show also that 60 cannot be replaced by any larger integer.

Assignment 8: Due in class, Thursday, November 19th

To solve: Chapter 8: Exercises 1-27 To be handed in: Chapter 8: Even-numbered exercises from 2-26

Spring 2009 Review Materials:

I will let you know if we cover anything differently this time around so as to require less, more, or different review materials.

Review problems for first midterm exam

Commment: Please disregard Problem 12 on congruences. We have not yet covered this material, and you are not responsible for it on the first midterm.

Review problems for second midterm exam
Review problems for third midterm exam

First midterm exam
First midterm exam, Fall 2009

Second midterm exam
Second midterm exam, Fall 2009

Third midterm exam
Third midterm exam, Fall 2009

Some interesting -- and perhaps relevant -- links:

Why You Should Be A Math Major

Why Major in Mathematics?

What Can a Math Degree Do For You?, Jaqueline Jensen, Sam Houston State University.

Mathematics of Venn Diagrams

Wikipedia article on Venn diagrams.

Are Venn Diagrams Limited to Three or Fewer Sets?, by Amy N. Myers.

A Survey of Venn Diagrams, by Frank Ruskey and Mark Weston.

Wikipedia article on ordered pairs.

Some logical weaponry: the Sheffer stroke, the Peirce arrow, the Quine dagger