Miscellaneous Information

• The information sheet  for Math 2020 can be found  here:  Math 2020 Syllabus 
• The graduate assistant and grader for the course is Amber Russell.  She will also be available for additional office hours. The contact information is: Amber Russell  Office: 105 Lockett  Email: arussell@math.lsu.edu  Office Hours: 10:00-11:00 Monday  1:30-2:30 Wednesday.
• Do not purchase the text found in the LSU Bookstore for Math 2020!. I have decided to use a different text:
  Edward A. Bender and S. Gill Williamson.  A Short Course in Discrete Mathematics, Dover Publications, 2005. ISBN
  0-486-43946-1.  This book can be obtained from Amazon.com ($11.66) or from Dover Publications ($14.95). Alternatively, it can be downloaded for free from the author's website: Discrete Mathematics.  Since the book is relatively inexpensive, I recommend purchasing the book since it is generally easier to carry around a small book than the version obtained by printing it yourself.  Nevertheless, you may wish to print the first section of Unit BF since we will be covering that the first week of class.
• Solutions are given in the text for all of the exercises.  I will give suggested exercises from the text.  You should try to work these without first looking at the solution section. Because of the presence of these solutions, I will not collect or grade exercises from the text, but any of these exercises could easily appear on an exam, where you will be required to work it on your own, including complete explanations.  Graded homework assignments will in the weekly summary section.  You should turn in for grading only those that are requested.
  

Weekly Summaries and Homework Assignments

  We will start off by covering Boolean functions and logic (Units BF and Lo in Bender-Williamson).  Read Section 1 of Unit BF.

Suggested exercises from the text:  Section BF1: 1- 15 (Odd).  Recall that solutions to all of the exercises in the text are included in the solutions section of the text.

This week we will cover Unit Lo on logic.  We will concentrate on the various forms of the implication statement, predicate logic (that is statements with the quantifiers "for all" and "there exists".  You should read and study this unit.  Here are some suggested exercises from the text

Section Lo-1:  2, 5, 7, 8, 11, 12, 14
Section Lo-1:  1, 2, 6, 8, 9, 13, 16

The first exercise set to be turned in is here: Exercise Set 1  (Due: January 30) Exercise Set 1 Solutions (Numbers 1 (a), (c), 2, 4, 5, 6 were graded.)

This week we are learning the principle of mathematical induction and how to use it in proofs.  The text material is Section 1 of Chapter IS.  The presentation is somewhat terse, so I am attaching some additional materials here for your use.  The first is a set of lecture notes on induction from a class at MIT:  Induction Lecture 1.  These notes contain most of the examples done in class.  The second supplement is an interactive lesson on induction from a university in the UK:   Induction Presentation. The third supplement is a "fill in the blank" form for writing induction proofs.  You may wish to print in out and use it as a template for writing your induction proofs.   Induction Proof Template. 

The second exercise set to be turned in is here: Exercise Set 2  (Due: February 8) Exercise Set 2 Solutions  Notice how the template provided above was used in writing up the induction proofs for Exercises 2, 3, and 4.

This week we will use induction to prove the fundamental theory of arithmetic. That is, the statement that every natural number greater than 1 can be written (essentially uniquely) as a product of primes.  We will then continue with some basic properties of integers: Division algorithm, Greatest Common Divisor, Least Common Multiple, the Euclidean Algorithm for computing the GCD, the fact that there are infinitely many primes, modular arithmetic.  All of this material comes together in the RSA algorithm for crytopgraphy.  This material is in Unit NT of the text.

The first exam will be on Thursday, February 15.  The exam syllabus, together with some problems of the type and difficulty that you might expect on the exam can be found here: Exam I Review Sheet.  Here are solutions (or at least answers) to the Exam I review  problems: Exam I Review Sheet Solutions 

Exam I (February 15)
Exam I, Exam I Solutions,

This week we started on the theory of divisibility found in Chapter NT and the supplement handed out in class.  You should study Pages 54, 55, 59, 68-70 in the text, together with the similar material provided in the handout. 

Here are some suggested exercises from the text

Section NT-1:  1 - 6, 13, 14, 15, 17
Section NT-2:  1 - 8

Additionally, the following exercises from the handout are due on Tuesday, March 6:

Page 9:  1 (a), (c); 2, 3 (a), (c), (3); 10, 12, 21
Page 18: 16, 18, 24 Parts (1)-(8)

Solutions to Exercise Set 3: Exercise Set 3 Solutions  

We started the study of congruence mod n and the resulting modular arithmetic, and we will continue with this material next week. This material is treated very briefly on Pages 58-60 of the text, but a more detailed supplement was provided in class on Thursday, March 8.  If you missed class you should be sure to request a copy of this handout so that you will be able to do the homework assignments.  On Thursday, we covered Chapter 8 on the solution of linear congruences in this handout.  Next week we will be covering Chapters 9 and 10 in these notes.  Here is a homework assignment on the material from Chapter 8:  

Homework Assignment 4: Exercise Set 4  (Due: March 20)
Solutions to Exercise Set 4: Exercise Set 4 Solutions  

The exam scheduled for March 20 has been rescheduled for Thursday, March 29, 2007.

Homework Assignment 5: Exercise Set 5  (Due: March 27)
Answers to Exercise Set 5: Exercise Set 5 Answers  Only answers are provided for this assignment, but the review sheet has similar exercises worked out in detail, and a few of the exercises from Set 5 were done in class.

This week we continued the study of number theory by proving Fermat's little theorem, Euler's theorem, and their use in computing high powers of integers modulo a given integer m. On Tuesday of next week we will complete the proof of the formula for Euler's Phi function by proving and using the Chinese remainder theorem.

This week we will continue with using Euler's formula to compute high powers of an integer modulo a given integer m.  We will prove the Chinese remainder theorem concerning solutions of simultaneous linear congruences and use it to complete the formula for Euler's Phi function.

The second exam will be on Thursday,  March 29.  The exam syllabus, together with some problems of the type and difficulty that you might expect on the exam can be found here: Exam 2 Review Sheet.  Exam II Review Sheet Solutions We will go over some of the review problems in class on Tuesday, March 27.  

Exam II (March 29)
Exam 2, Exam 2 Solutions,

Before the next exam we will cover as much as possible of Chapter SF (Sets and Functions) and Chapter EO (Equivalence and Order).  This week we should finish most of Chapter SF.  Here is an exercise set on this material.

Homework Assignment 6: Exercise Set 6  (Due: April 19)
Solutions to Exercise Set 6: Exercise Set 6 Solutions  

The third exam will be on Thursday,  April 26 .  The exam syllabus, together with some problems of the type and difficulty that you might expect on the exam can be found here: Exam 3 Review Sheet. Exam 3 Review Sheet Solutions  We will go over some of the review problems in class on Tuesday, lApril 24.  

Exam III (April 26)

Exam 3, Exam 3 Solutions,

The Final Exam will be on Wednesday, May 9 from 5:30--7:30 P.M. in Lockett 134. The exam will be comprehensive, so you should collect all of the earlier homework assignments, exams and review sheets to use in preparing for the final exam. You can expect the problems to be like the problems on the three exams and homework problems. The final exam sample problems sheet contains a few problems that you can use to practice on. While these problems are typical of the level of difficulty that you can expect, you should not assume that every possible problem that could
be on the exam is represented here. However this sheet, studied in conjunction with the review sheets for earlier exams, and the earlier exams themselves, should provide good preparation for the exam. Here is the sample problem sheet: Final Exam Sample Problem Sheet. A substantial number of these exercises were solved in class on the last day of class.  No further solutions will be written up.

The Final Exam will be on Wednesday, May 9, 2007 from 5:30  -- 7:30 PM  in the usual classroom, Lockett 134.