Day 14, Mat385

Last time

Next time

• Announcements
  • Our first exam is Wednesday, and will cover the material we've studied from chapters 1-3. Today we'll review (here are the notes from the review we did).

  • For the exam you'll be allowed a one-page (front and back) sheet.

    You will not need (nor may you use) a calculator.

  • You'll probably want to review your quizzes, and their keys.

  • The Quiz:
    • The Quiz
    • The Key
    • Comments
      1. Nice work Toshi!

         This is the recurrence relation for the Towers of Hanoi problem, that I worked out in class awhile back.

      2. A really interesting moment: one student showed that there can be multiple different linear recurrence relations that give rise to the same sequence!

         Can you think of another one? How many are there?

      3. I did this one in class, on the board (it was one of your homework problems).

         If you didn't understand what I was doing, then why didn't you ask a question?

  • I see that there are some of you who have used the Learning Plus tutoring: any comments for the good of the cause?

• Last time:
  • We began with an example of a linear, first-order, non-constant coefficient recurrence relation (#12, p. 196): \[ \begin{array}{ll} {S(1)=2}&{}\cr {S(n)=3(n+1)S(n-1)}&{n \ge 2} \end{array} \]
  • We finished off 3.3: Analysis of Algorithms, with a wrap-up of Euclid's GCD (Greatest Common Divisor), whose upper bound calculation leads to a divide-and-conquer recurrence, with worst case \[ E(n) \le 2\log_2(n). \]

• Today: Review!

  Let's look back over a few likely candidates for problems on the exam....

  • Here are some exams in the past, covering the expanse of the material from these three chapters (and perhaps a little bit more or less.

  • First of all, let's take a few minutes to look over the problems of section 3.3, in particular a couple of the homework "story problems". In each of these cases, you can see in the code the recurrence relation:

    pp. 214--, #9, 10, 13, 14, 38, 40

    • Polynomial evaluation (#9, p. 213): do you see two recurrence relations?
    • Bubble sort (#13, p. 214): Ditto... let's solve it!

  • Now we'll go over some sample problems (in reverse order) from our course so far. I won't claim that these cover everything that we studied, and won't guarantee that material not covered within these problems won't appear; but it gives you a good idea of the kinds of things that I expect you to be able to do for the exam.
    • Recursion:
      • Section 3.2, #25, p. 200 (pp. 197--, #1, 6, 12, 15, 21, 23, 24, 41, 48 )
      • Section 3.1, #58, p. 177: give a recursive definition of all binary strings containing an even number of 1s. (pp. 171--, #1, 3, 6, 16, 18, 21, 24, 26, 42, 50, 58)

    • Proofs:
      • Section 2.2: some induction problem, e.g. #44, p. 125. (pp. 123--, 4, 7, 16, 22, 27, 28, 31, 32, 41, 50, 61, 55, 56 )
      • Section 2.1: a contraposition/contradition problem, e.g. #44, p. 109 (prove by contradiction) p. 107--, #4, 10, 15, 19, 20, 32, 44, 61, 66, 71, 74

    • Formal logic:
      • Section 1.4: Proof with predicates (p. 69--, #7, 9, 10, 13, 15, 18, 22, 23, 26, 28, 31, 34, 42)
        • 34
        • 21 (temp hyp)

      • Section 1.3: Predicate formulation (p. 50--, #1, 5, 8, 9(a,b), 13(b,f,j), 17, 21(a,f), 22(a-d), 25(a-d), 27(a,b), 24(a-c), 34(a,c), 36(a,c)))
        • #30
        • #36

      • Section 1.2: Direct proof, perhaps with language. (p. 35--, #6, 11, 12, 15, 18, 20, 27, 35, 37, 44, 45, 49, 55, 56 (use statement letters I, K, J, O, H).)
        • #50

      • Section 1.1:

        (p. 16--, #2, 9, 11(d,e), 15(c,d,e), 17(b,c), 19, 23, 41(a,c), 47, 48, 49, 57)

        • Writing negations (e.g. 12a-c, p. 18)
        • Turning English into wffs (e.g. 13ab, p. 18)
        • Logical equivalences (e.g. 29, p. 20)
        • Tautology test (e.g. 41a; foreshadows proof by contradiction)

• The Curious History of Venn Diagrams: A look at the curious history of Venn diagrams and how they blend logic with geometry (Scientific American)

• Some Fun, Motivating Examples from Propositional and Predicate Calculus