Discrete Mathematics

Math 245
Fall 2011
Schedule Number: 21851
Meeting: MWF 1:00 - 1:50
NE 60
San Diego State University

Final Exam: Friday, Dec. 16, 10:30-12:30.

Professor: Mike O'Sullivan
Email: m.osullivan@math.sdsu.edu
Office: GMCS #579, ext. 594-6697
Office Hours: MW 2-4.
        TuTh afternoons I may also be in my office and available.
        Other times: by appointment.


S. Epp, Discrete Mathematics with Applications, 4th Ed.
(2nd Ed. or 3rd Ed. may also be used, but you are responsible for attention to any differences.)

My lecture notes, available on Blackboard.

Detailed Information

REVIEW SHEETS:    First Exam,    Second Exam,    Third Exam and some notes,    Fourth Exam,    Fifth Exam,    Final Exam.   

Course Description

Discrete mathematics is an exciting and rapidly growing area of mathematics which has important applications in computer science and in many high technology areas. For example, "secure" internet communication, efficient storage of data (e.g. jpeg) and robust communication networks are developed using techniques from discrete mathematics.

This course serves two main populations, students from mathematics and students from computer science. The course also has two distinct goals. One is to teach the basics of set theory, logic, combinatorics and graph theory. The other is to convey the rigorous use of terminology and proof which is essential to mathematics: that is, clarity and precision in definitions and statements of fact, and logical methods for establishing that a statement is true. The fundamental mathematics taught in this course is useful in computer science and electrical engineering. The analytical skills developed are critical to understanding computer languages, to the development of good programming skills, and, more generally, to appreciating scientific method. The experience with abstract concepts, proof writing and the use of formal definitions should help students in all disciplines to articulate their ideas more clearly.


Here is a rough idea of the amount of time I expect to spend on each topic, but the order in which we will cover the material may be a bit different. I am also open to suggestions if the class would like to spend more time on certain topics or cover items not listed here. A day by day schedule (see above) will be maintained to keep you informed of upcoming and past lectures.

§2.1-4 Logic and logical arguments. 4 classes
§3.1-3 Predicates and quantifiers. 4 classes
§4.1-7 Proof: direct proof, division into cases,proof by contraposition,
proof by contradiction and disproof by counterexample.
6 classes
Some number theory: Statements of the division theorem, unique factorization.
Proof of the irrationality of sqrt(2), proof of the infinitude of primes.
§5.1-4, 6, 7 Sequences, mathematical induction 4 classes
Recursively defined sequences: 4 classes
Finding explicit formulas,
establishing the formulas by induction.
§6.1-3 Sets: subsets, union, intersection. 4 classes
Venn diagrams. Algebra of set operations.
Cartesian product, power set.
§7.1-3 Functions: one-to-one and onto functions. 4 classes
Invertible functions.
Composition of functions.
§8.1-3,5 Relations. Reflexive, symmetric and transitive relations. 6 classes
Equivalence relations. Partially ordered sets
§9.1-5, 7 Counting: the addition rule and the multiplication rule.
The pigeonhole principle.
6 classes
Permutations and combinations.


Grading for this course is based on two levels of expectations.
Level 1 involves mastering computational problems, ability to recall and use terminology and to give short responses to questions. The maximum grade of B+ is based on the following.

Out of 1000 points, 900 and above is the cutoff for a B+, 850 is the cutoff for a B, and grades (using plus/minus scores) descend in 50 point increments to a 600 cutoff for a D+.

Level 2 involves writing proofs, and doing problems that require some organization and explanation. If you attain a grade of 800 or above on Level 1 you may be considered for a higher grade, based on additional work.

Partial credit will be stingily assigned. Cutoff for an A is 80%, B+ is 75%, and B is 70%.

Most students find this a tough class. You will greatly magnify your chances of succeeding if you work regularly and attentively! Start the assignments early and do most of the exercises in each section we cover. If you get stuck, please ask me in class. Your questions often lead to a valuable class discussion. You are also quite welcome during my office hours, but come prepared.