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.
Lecture Notes on Mathematical Induction:
click here
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.
Lecture Notes on Relations and Functions:
click here
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?
c) Any comments?
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
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
But wikipedia does not do justice to this unsolved problem in the psychology of logical reasoning. A
google search reveals a rich literature on the subject.