Class 22, Thu., Mar. 17, 2011
Here are the problems we worked on today in class:
Let A={x|x is congruent to 1, mod 4}, B = {a,b}.
Find the
(1) # of functions from A to B.
(2) # of functions from B to A.
(3) # of 1 to 1 functions from A to B.
(4) # of 1 to 1 functions from B to A.
(5) # of onto functions from A to B.
(6) # of onto functions from B to A.
(7) # of equivalence relations on A X A.
(8) # of subsets of AUB with 3 elements.
(9) # of ordered arrangements of AUB with 3 elements.
(10) # of functions f:AUB -> {1,2,3,...,10} which have f(1)+f(5)+f(9)+f(a)+f(b) = 5.
(11) Expand (3-4x^2)^5
(12) Expand (1-4x^2)^(-3) and show the first five terms.
(13) # of strings from AUB of length 5 using exactly 3 symbols of AUB.
Note solution to (13) which we briefly went over incorporates many of the ideas about counting:
There are C(5,3) ways to choose the 3 symbols from AUB = {1,5,9,a,b}.
There are two possible selections of numbers of symbols once those 3 are chosen, in order to have total of five:
(i) 1 of one of the 3 chosen symbols and 2 of each of the others.
(ii) 3 of one of the symbols and 1 of each of the others.
(Note that in general we could find the number of such selections using methods of counting selection with repetition, though we would still have to examine and count # of possible strings for each case.)
In both cases there are 3 ways, once the three symbols have been chosen to make these selections: in (i) there are three ways to choose the symbol that will appear in the string once, and in (ii) there are three ways to choose the symbol that will appear in the string three times.
In (i) there are 5!/(2!2!1!) ways to arrange the 5 symbols chosen.
In (ii) there are
5!/(3!1!1!) ways to arrange the 5 symbols chosen.
Thus total number is C(5,3) (3) [ 5!/(2!2!1!) + 5!/(3!1!1!)] = (10)(3)[30 + 20] = 1500.
(If this seems large, remember that the total number of strings of length 5 from 5 symbols is 5^5.)
Class 21, Tue., Mar. 15, 2011
We went over sections 9.5 and 9.6 (skip 9.4 and 8.7).
Do homework from these sections for Thursday.
Class 20, Th., Mar. 10, 2011
We went over sections 9.1-9.3, and more on derangements.
Class 19, Tue., Mar. 8, 2011
We went over several areas: inclusion/exclusion (Section 8.6), derangements, and several new problems. Here is the "100 prisoner problem:"
There
are 100 prisoners in cells numbered 1 to 100.
I mentioned in class that there is a strategy that guarantees success (in the long run) 1–ln(2) = 31.2% of the time!
If you want to see a solution, look at this link to a paper by
Peter Winkler.
In a previous class we went over the 6 people at the party problem, an example of Ramsey Theory.
Extra homework: Prove that R(3,4) = 9. That is, if the edges of the graph K9 are colored red and blue, then there is either a red K3 or
a
blue
K4, or vice versa. Hint : Consider
two
possibilities
for a random vertex v: v has 5 blue edges from it to
its neighbors, or v has 6 or more blue edges.
Class 18, Th., Mar. 3, 2011
We went over sections 8.3-8.5.
Chapter 5 homework is due Tuesday, March 8.
Here is the proof I asked you to complete for Tuesday:
If you color the edges of the complete graph on six vertices, K6, red or blue, then there will be a red C3 or a blue C3. I suggested you begin the proof by choosing a vertex u, and considering the edges joining u to the other five vertices. By the way, a corollary to this result is that a game between two players who alternately color the edges of K6 red and blue cannot end in a tie.
This is an example of Ramsey Theory.
Class 17, Tue., Mar. 1, 2011
We went over 5.4, 5.5, 8.1-8.3.
Class 16, Th., Feb. 24, 2011
Exam 2.
Class 15, Tue., Feb. 22, 2011
We went over sections 4.5, 5.1, 5.2, and briefly, 5.3.
Exam will cover chapters 3, 4, the appendices, and these parts of 5: definition of a tree and finding a spanning tree for a graph.
Here's a short study guide for exam 2.
Do this problem: find two simple graphs with 6 or fewer vertices that have the same numbers of vertices, edges, and degrees, but are not isomorphic.
Class 14, Thu., Feb. 17, 2011
We went over sections 4.3 and 4.4.
Exam 2 will be on Thursday, Feb. 24, not Tuesday, Feb. 22.
Homework for chapter 3 will be due on Thursday, Feb. 24, after the exam.
Do this problem: find two simple graphs with 6 or fewer vertices that have the same numbers of vertices, edges, and degrees, but are not isomorphic.
Class 13, Tue., Feb. 15, 2011
We went over code homework and graph theory ideas from 4.1 and 4.2. Also, find two simple graphs with 6 or fewer vertices that have the same numbers of vertices, edges, and degrees, but are not isomorphic.
Class 12, Thu., Feb. 10, 2011
We finished chapter 3, and began chapter 4.
Please finish chapter 3 homework and do ch. 4.1 homework.
Class 11, Tue., Feb. 8, 2011
We went over the "group of units" or non-zero dividers in the modular system Zm. One of the groups we looked at is known as the Klein four group (It was the one made up of 1,5,7, and 11, mod 12.) We did a problem on linear congruences.
We also took a first look at Hamming codes, and saw actual Hamming distances on a cube made of straws!
Here are some properties of the phi function from last class:
Phi(n) = the number of elements of {1,2,3,...,n–1} that are relatively prime to n.
Phi(p) = p–1, for p prime.
Phi(p^n) = p^n –p^(n–1), for p prime.
Phi(pq) = (p–1)(q–1), for p and q prime.
Phi(mn) = phi(m) phi(n), for m and n relatively prime.
a^phi(n) = 1, for a relatively prime to n.
Here's a hat puzzle related to codes:
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?
Class 10, Thu., Feb. 3, 2011
We went over the Euclidean algorithm, modular exponentiation, Fermat's Little Theorem, Euler's theorem, the phi function (also called the Euler totient function), and how they all apply to the RSA code. Be able to solve the congruence ax = = b (mod m).
We also went over more of section 3.4 and also 3.5. Do homework through 3.5.
Class 9, Tue., Feb. 1, 2011
We went over most of sections 3.3 and 3.4, and learned to solve the congruence ax = = b (mod m) (The double equal sign is used for the congruence symbol!). Here's a pretty good page on linear congruences that covers what we did today.
We also learned about Fermat's Little Theorem, used in the RSA code, and learned about a magic trick that uses mod 9. How does the trick work?
Do homework through 3.4.
Class 8, Thu., Jan. 27, 2011
We went over the exam, orders of infinity (not from the text), and proofs (Appendix A.3) Do homework through section A.3, and read ahead in chapter 3.
We went over infinite sets and Cantor's diagonal argument.
We also went over the equivalence of the well-ordering principle and mathematical induction.
Class 7, Tue., Jan. 25, 2011
We had exam 1 and went over Appendix sections A.1 and A.2.
Do homework from these sections.
Class 6, Thu., Jan. 20, 2011
We went over homework problems and also section 3.2.
Turn in chapter 2 homework following the exam on Tuesday.
Here's the page on the Extended Euclidean Algorithm. The method I showed you at the end of class is under the "Table Method" section at the page.
Class 5, Tue., Jan. 18, 2011
We went
over parts of sections 3.2 and 3.3, and went over homework.
Please continue to work on 2.3 #40-42 and 2.6 #45.
We also worked on the take-away game and the "pile" problem. Prove the pile formula using strong induction.
Turn in HW for chapter 2 on Tuesday, Jan. 25.
We introduced the RSA code (Wikipedia page).
Class 4, Thu., Jan. 13, 2011
We went over sections 2.5 2.6, and 3.1. Turn in HW for chapter 2 on Tuesday.
We had quiz 1 on Stirling's formula, estimating powers of 2, and finding the shortest "Hamiltonian Cycle" by the brute force method (a problem known as the Traveling Salesperson Problem).
We also went over "Big Oh" notation, and I mentioned the NP-Complete problem.
Do homework through section 3.1.
Class 3, Tue., Jan. 11, 2011
We went over homework on equivalence relations, and introduced partial
orders (section 2.3), including the lexicographic order and topological
sorting. We also went over material from section 2.4 on functions,
including one-to-one and onto functions.
Do homework through end of section 2.4. Also do the following problem:
(I should have been using Z to indicate the integers!)
Find a function from Z to Z which is onto and one-to-one.
Find a function from Z to Z which is onto but not one-to-one.
Find a function from Z to Z which is not onto but is one-to-one.
Find a function from Z to Z which is neither onto nor one-to-one.
Here's a link for the National Science Foundation Research Experience for Undergraduates program. Take a look at summer programs for undergraduates who want to do scientific research in the summer.
Class 2, Thu., Jan. 6, 2011
We went over sections 1.4, 2.1, and 2.2.
Do homework through 2.2.
Turn in chapter 1 homework on Tuesday.
Here's more detail on Horner's method from Wikipedia.
Here's the Horner's Method powerpoint I showed.
Class 1, Tue., Jan. 4, 2011
We went over sections 1.1, 1.2, 1.3, and the Hungarian Algorithm from chapter 6.
Do homework through 1.3.
Here are Vi Hart's Math Doodle videos.
Homework Winter 2011
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
Appendix A.3 # 3,11,15,16,18,20,23,26
Ch. 3.4 # 1,9,11,23,33
Ch. 3.5 # 5,9,15,19,29,31,35,40
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 # 5,11,17,19,24,27,45,63,67,73,77Ch. 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: # 3,9,10,18,24,34
Ch. 8.5: # 3,9,10,15,30,34,36
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,
Skip Ch. 9.4 # 11,13,15,19,25
Ch. 9.5 # 3,8,13,27,31
Ch. 9.6 # 1,3,11,15,21,27
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