MC 36 -- Relation-Based Structures
Spring 1999
Moira McDermott

Office: OHS 313
Office phone: 933-7478
E-mail: mmcdermo

Web Page: http://www.gac.edu/~mmcdermo/mc36s99.html
I will post announcements, course information and assignments here.

Office hours: MTR 2:00-3:00, and always by appointment.

I will be in my office and available for questions, discussion, and general conversation at the above times. If you can't come during any of these times, please call or e-mail and make an appointment. Please do not hesitate to come see me--in fact, I strongly encourage you to do so.

Prerequisites: There are no formal prerequisites for this course, but I will assume that you have some exposure to thinking abstractly.

Text: Discrete and Combinatorial Mathematics (3rd Edition) by Ralph P. Grimaldi.

Course Objectives:

To develop the ability to work with relations, functions and some common relation-based structures such as graphs, trees and lattices. These topics are particularly relevant to computer science.

To develop the ability to think abstractly, make conjectures, and understand and construct rigorous mathematical proofs.

Evaluation: Your final grade will be assigned using the following percentages as a guide:

Homework	25%
Proof Portfolio	5%
Paper	10%
Exams	60%

Academic Integrity: The academic honesty policy can be found on page 31 of the 1998-1999 college catalogue. I call your attention to the following excerpt: "In all academic exercises, examinations, papers, and reports, students shall submit their own work. Footnotes or some other acceptable form of citation must accompany any use of another's words or ideas."

Exams: We will have two exams during the semester and a cumulative final exam. The two exams during the semester will each have an in-class portion and a take-home portion. The in-class portion of the exams will require you to demonstrate an understanding of important definitions and theorems, and to produce only very short, straightforward proofs. More involved proofs will be reserved for the take-home portions, when you have time to reflect and revise.

The in-class exams are tentatively scheduled for March 12 and April 23.
The final exam will be given May 22, 10:30-12:30 p.m., in OHS 321.

Class Format:

A good lecture is usually systematic, complete, precise -- and dull; it is a bad teaching instrument.
-- Paul Halmos

The best way to learn anything is to discover it by yourself... . What you have been obliged to discover by yourself leaves a path in your mind which you can use again when the need arises.
-- George Polya

We learn by thinking and doing, not by watching and listening. Learning is an active process: it is something we must do, not have done to us. Class time will be a mixture of lectures, discussions, problem solving and presentation of proofs. At various times you will be asked to present problems, reflect on the reading and generate questions for your classmates. It is essential that you come to class prepared to do the day's work. In particular, you should read the text and attempt homework before coming toclass. Class meetings are not intended to be a complete encapsulation of the
course material. You will be responsible for learning some of the material on your own.

Homework:

I hear, and I forget;
I see, and I remember;
I do, and I understand.
- Proverb

There are two types of homework in this course, daily homework and extended problem sets. Daily homework is intended to give you practice working with new definitions and techniques. Extended problem sets are intended to help you hone your investigative and proof-writing skills. You are encouraged to collaborate with other students. Talking over ideas and
reading and commenting on each other's proofs are essential activities of working mathematicians. However, you must
acknowledge your collaborators, and write up your solutions and proofs individually. If a conversation with another person (or your outside reading) provides you with a key idea, intellectual integrity demands that you credit the source of the idea.

Writing:

Thought and expression of thought are so closely interrelated that to
require the latter will engender the former.
-- George Gopen and David Smith

MC36 is a ``W'' course. Writing assignments will take several forms. First, homework assignments will often require written proofs. These proofs will be checked for logical and grammatical accuracy, as well as for style and exposition. It is important to be able to express your mathematical thoughts in writing, using clear, well organized paragraphs comprised of English sentences. This means more than separating your equations with a few well-placed ``Thus it follows that...'' or ``Plugging (a) into (b) shows that ...'' During the course we will work on writing mathematical prose effectively and clearly.
In addition, you will be expected to write one ``perfect proof'' for each type of proof that we study. You may resubmit revised versions of these proofs, but you will only receive credit when I designate that you have achieved perfection.
You will also be be expected to write one expository paper. Details of the paper, including deadlines for drafts and revisions, will be described later in the course.

Syllabus: I expect to cover the following topics.

Counting: 1.1-1.4
Logic and proofs: 2.1 - 2.5
Set theory: 3.1 - 3.3, 8.1
Mathematical Induction: 4.1-4.5
Relations and Functions: 5.1-5.6, 7.1 - 7.4
Introduction to Graph Theory: 11.1 - 11.6
Trees: 12.1 - 12.4, 13.1, 13.2
Additional topics as time permits

Some notes on taking an upper-level MCS course:

In this course, many of you will be making the transition from assignments that are primarily computational to those that require original, creative thinking. Be persistent and realize that homework problems will take time and inspiration. You might not be able to solve every homework problem each week.

Be prepared to work hard and to start assignments early. It often helps to start with an example or two or a simpler version of the problem assigned, so that you really get a feel for what it is that you're trying to prove. If the proof or solution does not come to you after a while, it is best to put the problem aside and take a break. I get my best mathematical ideas while I am out doing something else.

For those of you with some experience in upper level courses, start to work on making your arguments clear and concise and on making judicious use of notation and pictures.

Homework Guidelines:

Please follow these guidelines in writing up your assignments.

Please be neat. Leave enough space for me to write comments. [Note: If you are concerned about wasting paper, feel free to turn in your homework on scrap paper.]
Show enough work so that any student in the class can follow your solution. Just writing the answer is never enough.
Proofs should be written in complete English sentences. Proof read what you have written to make sure it makes sense.
Don't try to fake a proof. It is much better to openly acknowledge a gap in your proof than to pretend it isn't there.