Math 2320 Course Information

Announcements, Lecture Notes and Outlines

  • (Feb 27) Unofficial grades are available. Click to view.

  • (Feb 9) The February 2 midterms have been graded. They can be picked up from the box outside my office door.

  • (Jan 28) The makeup midterm on Friday will be given in the regular classroom (Curtis C) at the regular class time (12:30).

  • (Jan 10) I will meet the class on Friday, January 12, at the usual time and place to discuss the process of remediation. Details of the remediation plan are now available on the web. I am meeting with the other instructors to establish a consistent plan for making up missed work.

  • (Dec 7) 5.2, 5.3, and 5.4 were covered during the last week of lectures and will be tested on the examination. We spent some time on 8.1 on Monday. There are no questions on trees on the final examination. Students may elect to write the examination next week independent of whether they have written all the quizzes and tests. Makeup arrangements for missed work will be established after the end of the CUPE strike.
  • (Nov 27) We finished 5.1 (iterative method to obtain an explicit formula) and started 5.2.
  • (Nov 25) There will be a FINAL EXAMINATION for this course given on the originally scheduled date and at the originally scheduled time. (The room may change -- you should be able to check the registrar's web pages for details later in the week.) Old exams are available from Club Infinity, Ross N537. Students who do not write this exam will have the opportunity to write an exam after the makeup period planned to follow the end of the CUPE strike.
    Please note that it is not permitted to rewrite Final Exams. This means that once you have written the exam in December you will not be permitted to write a second time at a later date.
    There will be provisions for making up work missed due to the strike for all students, whether or not they write the exam in December.

  • (Nov 24) Did 5.1, concentrating on how to set up recurrence relations which solve counting problems. I also indicated how to use recursive methods to count the number of onto functions from one finite set to another.
  • (Nov 22) Did 4.2, Pigeonhole Principle, and used it to prove the following:
    Proposition: Given n+1 integers among the numbers from 1 to 2n, at least one will be a divisor of another.
    The solution is based on placing numbers in boxes labeled with their largest odd divisor. Given any two numbers in the same box, the smaller will necessarily divide the larger. Observe that there are only n boxes.

    We also started 5.1, Recurrence Relations. I started discussing a method for determining the number of onto functions from a set with n elements to a set with n or fewer elements.

  • (Nov 20) Covered 4.6. The emphasis was on the combinatorial thinking underlying the derivation of the formulas.
    How many functions are there from a set with r elements to one with n elements?
    If you get sent out to the doughnut shop to buy a dozen assorted donuts how many different assortments (types, number of each type) can you come home with?
    How many arrangements are there of the letters in the word MISSISSIPPI?
  • (Nov 14) The midterm has been written. It has four questions.
    1. (8 points) Proof that two expressions give relatively prime results. In the vein of gcd(a, a+1) = 1.
    2. (18 points) Proof by mathematical induction of an explicit formula for a function which is defined recursively. Recursive definition of another function. Proof by mathematical induction that the two functions are equal.
    3. (6 points) Recursively defined set. Identify and give examples of some elements.
    4. (8 points) Basic counting. Number of functions between two finite sets with given properties.

  • (Nov 13) We looked at a selection of problems from Chapter 4 Section 1. I find the ability to think concretely about what you need to decide when making a list of all the possibilities the key to being able to solve the problems successfully.

  • (Nov 11) I am making available one version of last year's second midterm for your information only. Click to view in pdf format. Please note that your midterm will concentrate on mathematical induction, recursive definitions, and induction involving recursively defined functions or sets. There will be only one question on "counting". I will post an outline once I have prepared my draft.
  • (Nov 8) To view lecture notes from November 8 click here.
  • (Nov 7) To view lecture notes from November 6 click here.
  • (Nov 3) To view a version of Quiz 2 click here.
  • (Nov 1) We used what the text calls the second principle of mathematical induction to prove that any integer greater than 1 is prime or a product of primes. We had done this previously informally. We began (3.3) and did a couple of examples of recursively defined functions. As an application I showed that the number of ways to walk up 100 stairs if steps can be taken 1 or 2 at a time can be calculated using a recursively defined function. This is covered in (5.1) which is covered later.
  • (Oct 30) Click for an outline of what was covered today in class.
  • (Oct 27) Began 3.2 in Rosen. Please read over the section on your own. The quiz on November 3 will only cover Number Theory topics.

    October 26, 2000

    Please take note:

    I am a member of the YUFA bargaining unit which is not on strike.

    My courses, MATH 1090 A, MATH 2320 A, ITEC 1011 A, BETHUNE 1800 A, will continue to meet. All tests and quizzes will be given as scheduled.

    Students should be aware that under Senate Guidelines they cannot be penalized if they choose not to cross picket lines to attend lectures, write tests or quizzes, submit assignments.

    Once the strike is over, arrangements will be made for makeup tests and quizzes and to submit assignments which were due during the period of the strike. I will post to the appropriate course web pages lecture information to help students who miss classes keep up with the course material covered.

    Some paternal advice:

    This strike may very well last three or more weeks. If you lose touch with your studies during this time interval it may be very difficult to adequately catch up afterwards. Make every attempt to do whatever reading is assigned, to work through whatever notes are made available, and to do (even when you might wait to submit them) any course assignments.

    During the last strike, students suffered loss of credits, loss of money (for courses they ended up dropping), and reduced GPAs because they misjudged their ability to catch up once the strike ended.

    Stay informed:

  • CUPE 3903
  • York University Administration
  • YUFA
  • Announcements:

  • (Oct. 19) The test papers will be returned in class tomorrow. Here are some data on the scores.
    TEST GRADES from 40
         0     9    10    13    13    13    13    14    15    15    16    17    17 
        17    17    17    17    19    19    19    19    20    20    21    21    21 
        21    21    21    21    22    22    22    22    23    23    23    23    23 
        23    24    24    24    24    24    24    25    25    25    25    25    25 
        25    25    26    26    26    26    27    27    27    27    27    27    27 
        27    28    28    28    29    29    29    29    30    30    30    30    30 
        30    31    31    31    32    32    32    32    33    33    34    34    34 
        34    34    34    34    34    34    34    34    36    36    38    40    40 
     MTB > describe TEST GRADES
                     N     MEAN   MEDIAN   TRMEAN    STDEV   SEMEAN
     C2            104   25.163   25.000   25.340    7.172    0.703
                   MIN      MAX       Q1       Q3
     C2          0.000   40.000   21.000   30.000
     MTB > histogram TEST GRADES
     Histogram of TEST GRADES   N = 104
     Midpoint   Count
            0       1  *
            5       0
           10       2  **
           15      14  **************
           20      17  *****************
           25      32  ********************************
           30      20  ********************
           35      15  ***************
           40       3  ***
  • (Oct. 11) The test on Friday will have the form indicated in today's lecture. There are five questions.
    1. (8 points) An abstract proof to prove a function is one-to-one or onto.
    2. (6 points) A proof about the cardinality of a particular set.
    3. (12 points) Proving for some particular f and g that f is O(g) but that g is not O(f).
    4. (6 points) Comparison using worst-case time complexity of two procedures each of which is implemented using different algorithms where the worst-case time complexity of the algorithms to be used is known.
    5. (8 points) Basic proof using the definition of |   b.
  • (Oct. 3) A version of last year's Test 1 with Solutions is available in postscript format. Print using lpr from any reasonable *nix system or view and/or print from ghostview or gv. MSwindows users should use gsview.
  • (Sept. 23) A version of last Fall's Quiz 1 and Solutions are available in pdf format. Use Adobe Acrobat Reader to view and print them. The other versions are similar. PLEASE NOTE that on this quiz I took |N = {1,2,3,4,...}.

    Lecturer: Eli Brettler
    Office: South 508 Ross
    Telephone: 736-2100 Extension 66321
    Normal Office hours: MW 11-12:30 and by appointment

    Lectures: MWF 12:30 - 13:30 CLH C

    Tutorials: Wednesday 10:30 - 11:20, Thursday 12:30 - 1:20 N 501 Ross

    Text: Kenneth H. Rosen, Discrete Mathematics and its Applications, Fourth Edition

    Coverage: Chapters 1.6 - 1.8, 2.1 - 2.5, 3.2 - 3.4, 4.1 - 4.3, 4.6 - 4.7 5.1 - 5.5, Chapter 8

    Homework: Homework problems will be given but will not be collected for grading. Click for a selection of homework problems .


    QuizzesSept. 29, Nov. 315%
    Class TestsOct. 13, Nov. 1740%
    Final ExamExamination Period45%

    Note: There will be no makeups for missed tests or quizzes. If you miss a test or quiz for medical reasons and provide a medical certificate, the weight of the test will be transferred to the final exam. Otherwise, the mark for the missed work will be 0.

    Answers and Solutions:

    For the pdf versions you need the free Acrobat reader.

    Note: The last date to drop the course without academic penalty is November 10. It is extremely important to realistically assess your course performance prior to this date. .

    Eli Brettler
    Last modified: Tue Feb 27 09:17:37 EST 2001