Math 22 Winter 2009 Home Page

Math 22, Winter 2009 Home page

Green Sheet

I've added some updates to the study guide (Thursday night). Here is the revised final exam study guide. I've now added a question on generating functions.

Class 20, Tue., Mar. 17
We went over parts of ch. 8.4 on counting with repetition, also 9.3 on 2nd order linear recurrence relations,
and parts of 9.5 and 9.6 on generating functions (we had already covered parts of chapter 9.2.)
We'll skip sections 8.5 and 8.6, and also 9.4.
Please work on the HW from sections 9.3 and 9.5:
Ch. 9.3 #1,14,23,25,
Ch. 9.5 # 3,8,13,27,31

By the way, here's the HW from chapter 8 (see below also):
Ch. 8.1 # 3,9,19,27,28
Ch. 8.2: # 1,3,6,7,15,19,23,34-36
Ch. 8.3: # 3,14,16,19-21,31,33-35
Ch. 8.4: # 3,9,10,15,30,34,36

Class 19 and 20, Tue/Th Mar. 10,12
Sub went over sections 8.1-8.3
Had two class activity handouts on recurrence relations and Pascal and other triangles.

Class 18, Th., Mar. 5, 2009
We went over ch. 9.1
Ch. 9.1 #1,7,18,26,31

Class 17, Tue. Mar. 3, 2009
We went over the exam, also new material on trees: depth first search, rooted trees, and traversals of binary trees.
Also went over the Prufer code for labeled trees again.
New HW:
Ch. 5.3 # 3,25,28,32,33
Ch. 5.4 #13,16,21,30
Ch. 5.5 #5,16,22,28,35,41,45,53,56,64
Challenge problem from class: prove that if the edges of K₆ are colored red or blue, that there must be two one-color triangles (K₃s) - that is two red K₃s or two blue K₃s or a red and a blue K₃.

Class 16, Th. Feb. 26, 2009
We went over material from sections 4.3, 4.4 and 4.5 that we had missed, and also sections 5.1, and 5.2.
We did a quiz on "Bulgarian Solitaire."
I'll plan to check your chapter 4 homework next week, have it ready Tuesday.
We also learned about how many graphs there are!
Here's an image of the internet graph.
Here's another. Here's one of the world wide web, or at least a small section of it.
Here's the BOINC main page, for using your computer to contribute to scientific projects.
Here's a paper showing the relationship between the African game of Owari and Bulgarian Solitaire.
Here's a web page with some simple Bulgarian Solitaire puzzles.
New homework:
Ch. 4.5 # 5,11,17,19,24,27,45,63,67,73,77
Ch. 5.1 # 13,18,25,27,32,37
Ch. 5.2 # 5,9 (we will definitely discuss this one in class!), 12,13,15,19,37,39

Class 15, Tues., Feb. 24, 2009
We went over section 4.4 on coloring problems, also had exam 2.
Ch. 4.4 # 4,6,10,11,12,13,20,22,25,2932,34.35(a little difficult)

Class 14, Th. Feb. 19, 2009
Here's a study guide for exam 2 on Tuesday, may be a few other topics on the exam as well...

We went over section 4.3 on shortest paths, also other material on graphs, also more on codes.

Here's the hat puzzle:

Three players enter a room and a red or blue hat is placed on each person's head. The color of each hat is determined by a coin toss, with the outcome of one coin toss having no effect on the others. Each person can see the other players' hats but not his own.
No communication of any sort is allowed, except for an initial strategy session before the game begins. Once they have had a chance to look at the other hats, the players must simultaneously guess the color of their own hats or pass. The group shares a hypothetical $3 million prize if at least one player guesses correctly and no players guess incorrectly.

What strategy should the players adopt to insure their success rate is 3/4? How is this related to Hamming codes? What if there are 7 players?

Watch this site for more info on exam 2, scheduled for Tuesday.

New Homework:
Ch. 4.3 # 7,11,13,17

Class 13, Tue. Feb. 17, 2009
We went over some questions on Hamming codes, also some material on quantifiers.
I'll take up the handout on quantifiers as a quiz on Thursday.
I'll check homework for chapters 3 and the appendix on Thursday.

We started chapter 4, and covered material from 4.1 and 4.2.
New homework:
Ch. 4.1 # 1,7,8,16,18,21,22,24,26,31,35,42,47,48,53,54
Ch. 4.2 #19,23,31-34,40,41,50,52,61,63

Can you find two graphs with the same numbers of vertices and edges, and the same degree sequences, yet which are non-isomorphic graphs? This can be done with as few as 5 or 6 vertices.

Can you write the predicate form of "God helps those who help themselves," assuming it means "God helps all and only those who help themselves"?

Class 12, Th. Feb. 12, 2009
We went over section 3.6 and also Appendix A.3.
Appendix A.3 # 3,11,15,16,18,20,23,26
Ch. 3.6 # 1,5,9,13,17,29,31,33,35,40,41,44
We also had a brief intro to predicate calculus, specifically the "for all" and "there exists, so-called quantifiers.
Please do the homework on this handout on quantifiers.

If you are interested in mathematics and the arts, here's an announcement about mathematical plays about one-sidedness. to be performed in Berkeley this coming Monday.

Class 11, Tue., Feb. 10, 2009
We went over logic more, continue working on Appendices A.1 and A.2.
We also went over section 3.5, do the HW from that section.
Do this Smullyan problem using truth tables:
(5) The anthropologist meets two inhabitants, A and B, and A says, "B and I are the same type." What can be deduced about A and B?

Class 10, Th., Feb. 5, 2009
We went over the exams, also rest of Ch. 3.4 and parts of Appendices A.1 and A.2
Homework:
Appendix A.1 # 1,7,12,16-18,22,25,29,33,35
Appendix A.2 # 1,9,15,21,30-34
Ch. 3.4 # 1,9,11,23,33
Be prepared to turn in problems from the handout on modular arithmetic on Tuesday.

Class 9, Tue., Feb. 3, 2009
We went over the extended Euclidean Algorithm, how to do fast exponentiation, how to solve linear congruences with the Euclidean algorithm, the RSA code, and began to look at binary codes (sections 3.4 - 3.6).
Start on the HW for 3.4, and finish the problems in this handout on modular arithmetic.
Here is the 2002 AKS primality test, which runs in O(log^6+ε(n)) operations - not exponential, but not terribly fast either!
Here's Wikipedia on primality tests in general.
And here's how to use Fermat's Little Theorem for primality testing. (We've gone over most of the ideas at this page.)
Here's how to do fast modular exponentiation.

Class 8, Thu., Jan. 29, 2009
We went over modular arithmetic in great detail, including Fermat's Little Theorem, the Phi function (also known as Euler's totient function), and Euler's Theorem.
Read through this handout on modular arithmetic, and work as many of the problems as you can. We will finish going over this material on Tuesday.
Work the homework for chapter 3.3:
Ch. 3.3 # 1,5,11,15,31-37

Class 7, Tue., Jan. 27, 2009
We had our first exam, and went over a bit more about strong induction and modular arithmetic.
Carry over the induction quiz till Thursday and turn it in then.
Please read ahead in section 3.3 and also appendices A.1, A.2, and A.3.

Class 6, Thu., Jan. 22, 2009
We worked on induction problems, and then an induction quiz.
Please complete the induction quiz and turn in on Tuesday. Notice that I changed the notation, so that S(n) now refers to the nth statement, as in the text, and F(n) is the result of starting with n objects.

Here's a sample problem: let the universal set be {1,2,3,4,5}, and let A={1,2,3}. Give a counterexample to the statement
(A–B) ∪ (B–A) = (A ∪ B) – A. By the way, the left hand side of this equation is often called the symmetric difference, A∆ B.

I started the google group for the class, and emailed an invitation to everyone on the official class roster.

We are still investigatiing ac ≡ bc (mod m) implies a ≡ b (mod m).
Let d = GCD(c,m).
What conditions on a,b,c, and/or m allow us to say that ac ≡ bc (mod m) implies a ≡ b (mod m)?
What conditions on a,b,c, and m allow us to say that ac ≡ bc (mod m) implies a ≡ b (mod m/d)?
This is an important problem to solve, since we must be able to answer questions like this to know whether the simplest "linear congruence," ax ≡ b (mod m) has a solution. (A linear congruence is the equivalent, in a sense, of the linear equation ax=b for algebra of real numbers.)
Keep investigating!

One of the problems I suggested you know how to solve, is to use Stirling's formula to find an estimate in scientific notation, for example, of P(100,50) = 100!/50!
Hint: Let R be the estimate given by Stirling's formula, calculate Log(R) using rules of logarithms, then recalculate to give you an estimate in scientific notation of 10^(Log R) = R.

Will give you some Big Oh homework shortly.

For a visual proof of the sum that the first n cubes equals the square of the nth triangular number (1+2+3+... + n), see the Wolfram site; you can download a free viewer to play around with the interactive applet. For another visual proof, due to Nilakantha from Kerala in India, around 1500, see this site. The author of the site, David Bressaud, also shows some other proofs from around the world, and prior to the development of calculus.

Work the problems in this handout for Tuesday: Modular arithmetic intro.

New Homework:
Ch. 3.1 # 4,5,9,10,20,32,33,39,41,42,47,,48,51
Ch. 3.2 # 1,5,11,19,23

Class 5, Tue., Jan. 20, 2009
We went over equivalence relations, also sections 2.5 and 2.6 on induction, and began section 3.1 on congruence.

Can you answer this question from class: if ac≡bc (mod m), is a ≡ b (mod m)?
Try several examples of small numbers m, from 2 to 7.

Please print out this handout, and bring them to class on Thursday: Modular arithmetic intro

Here is the next spate of HW, try to finish 2.5 and 2.6 by Thursday:
Ch. 2.5 # 6,8,12,17,19,26
Ch. 2.6 # 5,16,27,37,43,45,47,48
Ch. 3.1 # 4,5,9,10,20,32,33,39,41,42,47,,48,51Ch. 3.2 # 1,5,11,19,23

Class 4, Thu., Jan. 15, 2009
We went over material from sections 2.3, on partial orders, and 2.4, on functions. We also went over more on take-away games, as related to the equivalence relation of congruence.
Do homework for sections 2.3 and 2.4.

Do these further Raymond Smullyan problems:
(4) The anthropologist meets two inhabitants, A and B, and A says, "At least one of us is a knave." What are A and B?
(5) The anthropologist meets two inhabitants, A and B, and A says, "B and I are the same type." What can be deduced about A and B?
(6) The anthropologist comes across two inhabitants, A and B, and asks one of them whether the other is a knight and gets a yes or no answer. Then he asks the other the same question, and gets a yes or no answer. Were the two answers necessarily the same?

Further homework:
Find a function that is onto and 1 to 1.
Find a function that is onto but not 1 to 1.
Find a function that is not onto but is 1 to 1.
Find a function that is neither onto nor 1 to 1.

Class 3, Tue., Jan. 13, 2009
We went over Big Oh notation, also some HW problems, and sections 2.1 and 2.2. Here's another site on big oh. And here's another!
I also showed you some puzzles by Raymond Smullyan, from his new book, Logical Labyrinths. (Recent quote by Smullyan: "Why should I be worried about dying? It's not going to happen in my lifetime!")

Here they are:
An anthropologist visits an island where there are two kinds of people: knights, who always tell the truth; and knaves, who always lie.
(1) The anthropologist comes across three inhabitants, A, B, and C. He asks A, "Are you a knight or a knave?" A answers, but indistinctly so the anthropologist can not understand what he says. He then asks B, "What did A say?"B replies, "He said that he is a knave." At this point, C piped up and said, "Dont' believe that; it's a lie." Was C a knight or a knave?
(2) According to another version of the above story, the anthropologist instead asks A how many of the three are knaves. Again A answers indistictly, so he asks B what A has said. B then says that A has said that exactly two are knaves. Then, as before, C says that B is lying. Is it now possible to determine whether C is a knight or a knave?
(3) Next the anthropologist meets two inhabitants, D and E. D says, "Both of us are knaves." What is D and what is E?

We also talked about Russell's paradox.

Do the HW for sections 2.1 and 2.2.

Class 2, Th., Jan. 8, 2009
We went over 1.1, 1.2, 1.3, and some of 1.4.
HW: All problems below thru section 1.4
Here's a derivation of Stirling's formula.
Here's a description of Babylonian mathematics.
Here's more on Pascal's Triangle.
Here's more on Horner's Method.

Class 1, Tue., Jan. 6, 2008
Substitute (Bambi Moise) went over sections 1.1 and 1.2.

Here's the list of all HW problems. Work all HW through section 1.2 for Tuesday.

Ch. 1.1 # 7,15,19
Ch. 1.2 # 6,16,18,25,26,31,32
Ch. 1.3 # 1,9,13,14,19,20,25,29
Ch. 1.4 # 9,11,17,23-27,31,32
Ch. 2.1 #1,13,15,17,26,31,33,39
Ch. 2.2 Odd problems #1-19,18,20, 22, 23,29
Ch. 2.3 1,3,5,9,13,18,19,21,32,40,41,42
Ch. 2.4 #1,3,5,9,13,18,19,21,32,40,41,42
Ch. 2.5 # 6,8,12,17,19,26
Ch. 2.6 # 5,16,27,37,43,45,47,48
Ch. 3.1 # 4,5,9,10,20,32,33,39,41,42,47,,48,51
Ch. 3.2 # 1,5,11,19,23
Ch. 3.3 # 1,5,11,15,31-37
Appendix A.1 # 1,7,12,16-18,22,25,29,33,35
Appendix A.2 # 1,9,15,21,30-34
Ch. 3.4 # 1,9,11,23,33
Ch. 3.5 # 5,9,15,19,29,31,35,40
Appendix A.3 # 3,11,15,16,18,20,23,26
Ch. 3.6 # 1,5,9,13,17,29,31,33,35,40,41,44
Ch. 4.1 # 1,7,8,16,18,21,22,24,26,31,35,42,47,48,53,54
Ch. 4.2 #19,23,31-34,40,41,50,52,61,63
Ch. 4.3 # 7,11,13,17
Ch. 4.4 # 4,6,10,11,12,13,20,22,25,2932,34.35(a little difficult)
Ch. 4.5 # 5,11,17,19,24,27,45,63,67,73,77
Ch. 5.1 # 13,18,25,27,32,37
Ch. 5.2 # 5,9 (we will definitely discuss this one in class!), 12,13,15,19,37,39
Ch. 5.3 # 3,25,28,32,33
Ch. 5.4 #13,16,21,30
Ch. 5.5 #5,16,22,28,35,41,45,53,56,64
Ch. 8.1 # 3,9,19,27,28
Ch. 8.2: # 1,3,6,7,15,19,23,34-36
Ch. 8.3: # 3,14,16,19-21,31,33-35
Ch. 8.4: # 3,9,10,15,30,34,36

Ch. 8.5: # 5,9,13,17,26,29-38
Ch. 8.6 #5,9,17,26,29-38

Ch. 9.1 #1,7,18,26,31
Ch. 9.2 # 1,2,5,11,27,34
Ch. 9.3 #1,14,23,25,
Ch. 9.4 # 11,13,15,19,25
Ch. 9.5 # 3,8,13,27,31