The Department of Computer Science & Engineering 
CSE 191:
DISCRETE STRUCTURES Spring 2009 
http://www.cse.buffalo.edu/~rapaport/191/S09/syl.html
Last Update: 27 April 2009
Note: or material is highlighted 
This course covers various topics in "discrete" (as opposed to "continuous") mathematics.
The differential and integral calculus that you study in MTH 141142 covers "continuous" mathematical topics in the sense that it analyzes data whose values can be real numbers. The realnumber line has no gaps in it.
On the other hand, this course covers "discrete" mathematical topics in the sense that it analyzes data whose values are "separated", like the integers. The integer number line has gaps.
(Compare an analog clock—one with hands—with a digital clock: With the former, every point on the circle that is the "face" of the clock represents a time that one of the clock's hands can point to—there are no gaps. With the latter, not every time between any two times is represented (e.g., there's no time between 11:50 and 11:51). Analog clocks are "continuous"; digital clocks are "discrete".)
This course provides some of the mathematical foundations and skills that you will need in your further study of computer science and engineering. The central concept of computer science is that of an algorithm. Algorithms are discretemathematical objects. To understand an algorithm, you need to appreciate that it is a formal mathematical entity, not a program in a particular language; it is based on the discretemathematical notion of recursion. To design an algorithm, you need to know logic, set theory, relations, functions, graph theory, and other discrete structures. To verify that an algorithm works correctly, you need mathematical rigor and good proof techniques—in particular, mathematical induction. These are the areas covered by this course.
We will begin with a study of logic (propositional and firstorder predicate logics), a subject which is at the foundation of mathematics and computer science. Logic can be considered as what AI researchers call a language for knowledge representation and reasoning. As a language, it enables us to talk precisely about anything in mathematics, just as a programming language enables us to talk precisely about computational procedures.
But we need objects to talk about. We will see that we can represent any mathematical or computational object in terms of a single data type: sets (along with their members and the setmembership relation [∈] between them); so, we will study set theory.
Then we'll use the language of logic and the set datatype to investigate relations among objects, including recursive relations (which are at the heart of computer science), as well as investigating functions (which are a special kind of relation)—and computable functions are what computer science is all about.
Finally, we'll use logic, set theory, and relations to discuss graphs and trees, yet another very general and useful data type.
CSE 113 or 115 or permission of the instructor.
No programming will be required, but you will be expected to understand various highlevel programminglanguage algorithmic techniques, structures, and terminology from an introductory computerscience or programming course.
Teaching Assistants:
CLASS  INSTR.  REG. #  DAYS  HOURS  LOCN 

Lecture  Rapaport  326525  MWF  11:0011:50 a.m.  NSC 220 
Recitation R1  Evanko  387915  T  8:008:50 a.m.  Capen 260 
Recitation R2  Evanko  309977  W  12:00 noon  12:50 p.m.  Capen 10 
Recitation R3  Chen  200557  Th  3:003:50 p.m.  Norton 213 
Recitations will begin meeting the week of January 19.
which is supposed to be packaged together with:
Grossman, Jerrold (2007), Student Solutions Guide to Accompany Discrete Mathematics and Its Applications, 6th Edition (New York: McGrawHill); combined ISBN = 0073503177.
Note: I have adjusted some of the dates and assignments below to reflect what we actually did in class, rather than on what I had planned or hoped to do:)
DAY  MONTH  DATE  TOPICS  SUBTOPICS  § in Rosen  HW 

M  Jan  12  DISCRETE MATH  Intro. to course; What is discrete math? 
pp. xxxxii  HW #1 assigned 
W  14  LOGIC 
What is discrete math? (cont'd);
Propositional Logic 
1.1  
F  16  Propositional Logic (cont'd)  1.11.2  
M  19  No class: Martin Luther King Day 

T  20  1st meeting of Recitation R1 

W  21 
Propositional Logic (cont'd);
Propositional Equivalences
1st meeting of 
1.11.2 
HW #1 due
HW #2 assigned 

Th  22  1st meeting of Recitation R3 

F  23 
Propositional Equivalences (cont'd)
Last drop/add day 
1.2  
M  26 
HOW TO STUDY MATH;
Propositional Equivs (cont'd); 
1.21.3  
W  28  Preds & Qfrs (cont'd)  1.3 
HW #2 due
HW #3 assigned 

F  30  Preds & Qfrs (cont'd)  1.3  
M  Feb  2 
Nested Qfrs;
Peano's Axioms for Arithmetic 
1.4;
p.A5 

W  4 
Peano's axioms (cont'd); Rules of Inf 
p.A5; 1.5 
HW #3 due;
HW #4 assigned  
F  6 
Rules of Inference (cont'd); Proofs 
1.51.6  
M  9  Rules of Inf & Pfs (cont'd)  1.51.6  
W  11  Rules of Inf & Pfs (cont'd)  1.51.6  
F  13  Pfs (cont'd)  1.61.7 
HW #4 due TODAY!;
HW #5 assigned 

M  16  Pfs (cont'd)  1.7  
W  18  Pfs (cont'd)  1.7  
F  20  Pfs (cont'd)  1.7 
HW #5 due; Virtual HW #6 assigned 

M  23  Pfs (cont'd)  1.7  
W  25 
SET THEORY

Sets
Announce MidTerm Exam! 
2.1  
F  27 
Sets (cont'd); Set Opns 
2.12.2  HW #6 answers will be posted  
M  Mar  2!  Review for MidTerm Exam  
W  4 
MIDTERM EXAM (will cover §§1.11.7) 
clock 

F  6 
Review of MidTerm Exam
+ midsemester course evaluation 

MF  913  Spring Break  
M  16  Set operations  2.2  HW #7 assigned (!)  
W  18 
FUNCTIONS 
Set Opns (cont'd); Functions 
2.2 2.3 

F  20  Functions (cont'd)  2.3 
HW #7 due; HW #8 assigned 

M  23  Functions (cont'd)  2.3  
W  25  RECURSION 
Functions (cont'd); Sequences; Mathematical Induction; 
2.3 2.4; 4.1 

F  27 
Math Ind'n (cont'd)
Last R Day 
4.1 
HW #8 due;
HW #9 assigned 

M  30  Math. Ind'n (cont'd)  4.1  
W  Apr  1  Recursive Definitions  4.3  
F  3  Rec. Defs (cont'd)  4.3 
HW #9 due; HW #10 assigned 

M  6 
Structural Induction; Recurrence Relations 
4.3; 7.1 

W  8  Recurrence rel'ns (cont'd)  7.17.2  
F  10  Recurrence relns (cont'd)  7.2 
HW #10 due;
HW #11 assigned 

M  13  RELATIONS 
Relations; nry Relns; 
8.1; 8.2 

W  15  GRAPH THEORY 
Equivalence Relns; Representing Relns 
8.5; 8.3 

F  17 
Rep'g Relns (cont'd); Graphs:
Euler paths & circuits 
8.3; 9.1 §9.5 to p.640 
HW #11 due;
HW #12 assigned 

M  20 
Graphs (cont'd): Traveling Salesman Problem; Planar Graphs & Euler's formula; 4 Color Thm 
§9.6, esp. pp.653655; §9.7 to p.663; §9.8 

W  22 
Trees:
rooted trees; E=V–1 
10.1  
F  24  Tree traversal algorithms  10.3 
HW #12 due; Virtual HW #13 assigned 

M  27  Last Class: Summary & Review 

TW  2829  Reading Days  
Th  30 
FINAL EXAM:
11:45 a.m.  2:45 p.m. Knox 109 
"Teachers open the door, but you must enter by yourself."
— Chinese Proverb
"You can lead a horse to water, but you can't make him drink." — American Proverb "You can lead a horse to water, but you must convince him it is water before there is any chance he will drink." — Albert Goldfain "Education is not filling a bucket, but lighting a fire" — William Butler Yeats "Reading is to the mind what exercise is to the body." — Sir Richard Steele Therefore... "The more you read, the more intelligent you are. It's really that simple." — Ethan Hawke But... "To read critically is to read skeptically. The reader [should] ask...not only, 'Do I understand what this means?' but 'Do I buy it?' " — Kenneth S. Goodman 

If you try to hand yours in after they have been collected (e.g., at the end of lecture, in my mailbox, in the TA's mailbox, etc.), it will not be accepted. To repeat:
This is so that the HW can be discussed in the class period when it is due.
at the top righthand side of each page.
Announcements may also be posted to the course website or the class email list.
You will automatically be placed on the UBLearns email list for the course.
I will use this list as my main means of communicating with you out of class. And you can use it to communicate with the rest of us.
You may send questions and comments that are of general interest to the entire class using the UBLearns email list. You can also send email just to me, at:
Be sure to send your mail from your buffalo.edu account and to fill in the subject line, beginning with "CSE 191", so that my mailer doesn't think that it's spam.
If you send email just to me that I deem to be of general interest, I will feel free to remail it to the email list along with my reply unless you explicitly tell me that you want to remain anonymous, in which case I may choose to remail it to the email list preserving your anonymity.
The emails will be archived at http://www.cse.buffalo.edu/~rapaport/191/S09/EMAIL/.
Recitation Assignments (including attendance, HWs, quizzes) 
40% 
MidTerm Exam  20% 
Final Exam  40% 
Total  100% 

For information on my philosophy of grading, see my web document on "How I Grade"
Any incompletes that I might give,
in a lapse of judgment :), will have to be made up by the end of the


For more information on Incomplete policies, see the Undergraduate Catalog "Explanation of Grades" (scroll down to "Incomplete Grades").
Note that my policy on when a grade of Incomplete must be completed differs from the University policy!
Although it is acceptable to discuss general approaches with your fellow students, the work you turn in must be your own.
It is the policy of this department that any violation of academic integrity will result in an F for the course, that all departmental financial support including teaching assistantships, research assistantships, or scholarships be terminated, that notification of this action be placed in the student's confidential departmental record, and that the student be permanently ineligible for future departmental financial support.
If you need help doing the assignments, see your TA or Prof. Rapaport.
Please be sure to read the UB webpage
and the CSE webpages
which spells out all the details of this, and related, policies.
For some hints on how to avoid plagiarism when writing essays for courses, see my website "Plagiarism".