Foundations of Discrete Mathematics, Math 328, Fall 2008
This site was updated May 12, 2008
(Recent updates will usually be above the first horizontal line. The syllabus is below.)
                        
Syllabus for Foundations of Discrete Mathematics, Math 328, at Montana State University.

Goals:  You will learn to read, write, and think like an advanced mathematician. You will learn to read symbolic mathematics with comprehension, express mathematical thoughts clearly, reason logically, recognize and employ common patterns of mathematical thought, and read and write proofs. Also, you will learn the basics, including proofs, of number theory, group theory in abstract algebra, combinatorics, and probability theory. 

Instructor:  Dr. Warren Esty, 994-5354, Wilson 2-238 (East wing, South wall). Office hours are here. 

Required text:  Proof: Introduction to Higher Mathematics, third edition, by Warren W. Esty and Norah C. Esty. There is no solutions manual.
    Some other material will be placed on-line at the library reserve site, including  the material on combinatorics. 

Course Content:  The course covers several topics of "discrete mathematics," including number theory and group theory (from Esty and Esty) and combinatorics and probability theory (from another source), focusing on proofs in each area.
    There will be required reading and homework due each day. It will be listed on this website. Bookmark this site. 

Prerequisite:  Math 256 (Foundations of Higher Mathematics) or explicit consent of Dr. Esty.  Do not attempt this course without Math 256 first.
    This course is primarily for students who wish to be math teachers or math majors. 

Time and room:  10:00 am, MWF, Wilson Hall, 1-133.  

Etiquette. Proper etiquette is required. During class, students will not engage in any potentially distracting behavior such as reading a newspaper, whispering about non-math subjects, or using electronic devices. Cell phones must be turned off and unavailable. Pagers or watches that make a sound, however quietly, must have the sound off. Also, no type of earphones is allowed.

Attendence:  Attendence is expected. More than a couple unexcused abscences is unacceptable. Of course, excuses for academic reasons, illness, participation in university sporting events, and significant life events will be accepted.
    Every day in class you will learn about common mistakes and how to avoid them. It is not possible to recogonize your own errors in logic, so you must take every opportunity to see deceptive errors in reasoning explained and to get feedback about your own and your classmates errors in reasoning. Students who miss a day are missing a significant lesson that cannot easily be recovered from the text alone.

Homework.  There will be homework due almost every day. It is important that it be attempted on time. The work you hand in need not be all correct, but it must display serious effort. More than a few late homeworks is not acceptable.
    You are expected to work, on average, about two hours outside class for each class hour.
    You must read the assigned sections. Learning to read math with full comprehension is one of your goals, and you learn to read by reading.  Reading is part of those two hours.
    Bring your text to class every day. We will use it in class regularly.
 
Exams and Grading.  There will be unit exams, frequent quizzes, regular homework, class participation, and a comprehensive final.
   To receive full credit, daily homework must be handed in on time. Homework handed in late will receive half credit.
Exam dates will be announced on this site.
Homework and its due dates will be announced on this site.
Check this site frequently.
 
Conflicts.  You are required to take all exams and the final exam at the scheduled hours (unless you have another exam or class scheduled at that hour, in which case we will make arrangements). Any exceptions must be approved well in advance, and in no case will exceptions be made for two exams.


Here is how Fall 2007 began. Fall 2008 will be similar, but the HW might be different.

HW due Friday, August 31, 2007:  Section 5.1.  Problem 1, part a)  Complete Definition 4B.  part b)  Give the negation of the definition in 4B.  c)  Give the negation of the definition in 4A.   d)  How does the negation of  4A differ from the the negation of the contrapositive in 4B?  e)  In general, how does the negation of the contrapositive differ from the negation of the original conditional?
        Problem 2:  [2nd ed, B21]  Conjecture:  If gºf is 1-1 and f is onto, then f is a bijection.
        Problem 3:  [2nd ed. B13]  Conjecture:  If  gºf  is onto, then f is onto.
        Problem 4:  [2nd ed. B11]   Conjecture:  If  gºf  is onto, then g is onto.
        Problem 5:  Conjecture:  There is a 1-1 and onto map from (0, 1] to (0, 1).

Monday, Sept. 3, 2007 is Labor Day, a holiday. There are no classes Sept. 3. 

HW due Wednesday, September 5:  Read Section 5.2.  Re-read any earlier sections that might be relevant, such as 1.1 and the logic in Chapter 1.
        Section 5.2:  A7
        Problem 2:  Conjecture:  If S is a subset of T, then f(S) is a subset of f(T).   [Make your proof begin in the right place. Theorem 1.4.7, A Hypothesis in the Conclusion, is relevant.]
        Problem 3:  Conjecture:  If  f -1(S) is a subset of f -1(T), then S is a subset of T.
        Problem 4:   This conjecture is false: "If  f(S) is a subset of f(T), then S is a subset of T."  State a similar conjecture that is true by including the idea of 1-1 or onto, and then prove your revised conjecture.
    Problem 5:  The conjecture in problem 3 was false.  State a similar conjecture that is true by including the idea of 1-1 or onto, and then prove your revised conjecture.

    We will not finish Chapter 5 (although you might find it useful) because it is time to move ahead to number theory, the subject of Chapter 6.  Read 6.1 (or more) and be prepared to consider its conjectures.
    The first day you got a green handout with a list of  terms from the prerequisite Math 256. If there is even one of them you are not comfortable with, please bring it up in class or see me in my office. They are all critical.

HW due Friday, Sept. 7:  Section (6.1): [There are no significant changes between the first and second editions. This numbering serves for both.]  B1, 3, 5, 11

HW due Monday, Sept. 10.  B13, 14, 19, 24, 28.  [Not all results need to be proved from scratch. You may cite earlier results as prior.]  Also, begin a sheet outlining the principles of proof.  That is, write out, concisely in outline form, the principles that you (we) often find useful when it comes to resolving a conjecture and proving it true or false. Hand it in to show me you are thinking about this (This course is more about principles than it is about particular proofs.) . I will hand it back and you can add to it or reorganize it repeatedly throughout the semester.

You are expected to work, on average, about two hours outside class for each class hour. If the hours you spend regularly exceed this amount, please let me know.