Math 327 - Discrete Mathematical Structures

### MATH 327 - Discrete Mathematical Structures

This class is listed as:

• Math 327-002 - Call #32507 - Discrete Mathematical Structures

Textbook: Discrete Mathematics with Graph Theory by Edgar G. Goodaire and Michael M. Parmenter. Prentice Hall, Third Edition, 2006.
We will cover most of the material in Chapters 0-7, and time permitting some of the material in Chapters 8-9. There are many other introductory books in discrete mathematics. Reading from other sources is always very valuable. We highly recommend How to solve it by G. Polya. Princeton Press, Second Edition, 1971.

Course Structure: There are 2 lectures per week. They will be devoted to lecturing new material, solving problems, occasionally working in groups. We will have periodic homework, and quizes to help you keep up with the material, and test your understanding of the material discussed. I expect you to read the book every week, and any additional reading material handed out.

Course Goals and Outcomes: Math 327 has two main roles in the curriculum at UNM. It prepares EECE and CS majors to take a data structures course, and it serves as a transition course for math majors between that light rigor of calculus and the strict rigor of Math 401 (Advanced Calculus) and 322 (Basic Algebra). At the end of these class you (students) are expected to:

• Be able to follow and construct simple proofs using induction, basic logic and set and function notation. In particular, you should: understand basic rules of logic, understand different methods of proof, and understand the importance of clear definitions of mathematical objects to be able to deduce properties and to manipulate them properly to achieve other goals.
• Start writing up your homework in complete sentences, not just a list of equations. In particular, you should: be able to organize information when solving a problem, such as identify the hypotheses, and know what one is required to prove, and be able to write carefully mathematical arguments and algorithms.
• Understand how two different discrete structures hold the same information. In particular, you should: know how to translate functions as formulas to functions as ordered pairs, know how to translate equivalence relations (as ordered pairs) to partitions, understand isomorphism of some object, such as small graphs or semigroups, know modern language about functions, such as one-to-one and onto, at least for functions between finite sets, and know what is the difference between a recursively defined sequence and a closed-form description of a sequence.
• Be able to work some counting problems using the basic combinatorics methods. In particular, you should: be able to identify when the additive and/or multiplicative rules apply to a counting problem, and use these rules effectively to solve the problem; be able to distinguish when a counting problem involves permutations (order matters) or combinations (order does not matter), and use these counting techniques to effectively solve the problem; understand that the same counting problem can be approached differently leading to solutions (expressions) that a priori one would not think represent the same number, but indeed they do; and be able to use this knowledge to solve simple discrete probability problems.
• Understand basic properties of the natural numbers. In particular you should: be comfortable with the concept of prime decomposition of abstract numbers, be able to manipulate at an abstract level the concept of greatest common divisor and least common multiple, know the Euclidean algorithm well enough to work problems involving the Chinese remainder theorem and simultaneous congruences.
If you have an interest in a particular discrete structure (graphs, trees, groups), let me know so that I can attempt to give you an introduction to the structure, and you can then consider taking a full class on the subject (Math 317 Graph Theory, Math 318 Combinatorics, Math 319 Introduction to Number Theory, Math 322/422 Basic Algebra).

Prerequisites: You need some mathematical maturity. We list as a prerequisite one year calculus: Math 162-163, or permission from the instructor.

Homework: The problems and exercises in the textbook are an integral part of the course. You should do as many as possible. I will occasionally provide exercises or problems that are not in the textbook. In those occasions, solutions to the problems will be posted in the webpage.
Homework will be assigned periodically, the problems in the homework will be carefully graded, and returned to you with feedback that will help you correct any errors. You are encouraged to discuss the homework with each other, but you should do the writing separately. You learn mathematics by doing, and there is no way around it, it is not enough to see your teacher or your friends solving problems, you have to try it yourself. Difficult as it may seem at the beginning, if you persist you will learn how to write a proper mathematical proof, you will learn how to read and understand other's proofs, and you will learn to appreciate and enjoy the beauty of an elegant argument.

Exams: There will be two midterms during weeks six (Midterm 1, October 2, 2008) and twelve (actually thirteen,Midterm 2 - Take home, due on November 25, 2008) and a final exam on Thursday December 18, 2008 at 7:30am.

Grades: The final grade will be determined by your performance on the homework, quizes, midterms, and the final exam. The grading policies will be discussed in class.

Important Dates: Fall 2008 deadlines, Fall 2008 finals schedule .

Americans with Disabilities Act: Qualified students with disabilities needing appropriate academic adjustments should contact me as soon as possible to ensure your needs are met in a timely manner. Handouts are available in alternative accessible formats upon request.

Last updated: October 7, 2008