HW0 due Wednesday, May 3 Do problems 8 and 20 in the section Mathematical Statements and come to class prepared to discuss those. Further questions will be assigned at Wednesday's class. HW1 due Wednesday, May 10 Section: Mathematical Statements Exercises: 3. change part d) to the contrapositive. 10. for each part, first define some phrases with P,Q, etc., then write each part with symbols only 15. first rewrite the table by changing T to F and F to T for each entry, then do all parts. 17. do each part, then swap existential for universal and vice versa and do each part again. 18. 19. Section: Sets 4. change the first set to {1,2,3,4,5,6} 6. change 20 to 25 in the description of the first set 18. 19. 21. 26. HW2 due Wednesday, May 17 Section: Functions Exercises: 8,11,15,18,19 (note: |X| and |Y| can be infinite), 28. Section: Additive and Multiplicative Principles Exercises: 6 (but change |A|=9 and |B|=6), 10 (but also include multiples of 11), 11, 13 HW3 due Wednesday, May 31 Section: Binomial Coefficients Exercises: 7, 8, 9 but find coeffficient of x^5. Section: Stars and Bars Exercises: 9, 10 but use f : {1,2,3,4} --> {0,1,2,3,4,5} Other: 1. How many draws of 7 tiles from a scrabble bag can spell the word BANANAS? 2. How many draws of 7 tiles from a scrabble bag can spell the word ZYZZYVA? 3. How many draws of 7 tiles from a scrabble bag can spell the word FREEBEE? HW4 due Wednesday, June 7 Scrabble Problems: 1. Find an alphabetized sequence of 7 letters (with possible repeats but no blanks) which can produce at least 5 different seven letter words (from the word list). 2. Find an alphabetized sequence of 6 letters (with possible repeats) and one blank, which can produce at least 10 different seven letter words (from the word list). 3. Find the probability that a draw of 7 tiles from the scrabble bag of 100 tiles contains two blanks. Other: 4. Find the probablility in each part when two fair dice are rolled: a) that the sum on the dice is at least five but not 10. b) that at least one of the two numbers is odd. c) that the sum on the dice is less than 7, given that at least one of the numbers is 4. 5. Find the probability in each part when a fair coin is flipped 4 times in sequence. a) that there are exactly two H's b) that T is always followed by H if a T occurs. c) that there is at least one T, given that there are at least two H's HW5 due Wednesday, June 14 Scrabble Problems: 1. Find an alphabetized sequence of 6 letters (with possible repeats) and one blank, which can produce at least 20 different seven letter words (from the word list). 2. Find the probability that a draw of 7 tiles from the scrabble bag of 100 tiles has all tiles of the same letter. 3. Find the probability that a draw of 7 tiles from the scrabble bag of 100 tiles has exactly four tiles of one letter and three tiles of another letter, but no other letters. Other: 1. Given the experiment of flipping a fair coin six times, find the probability that there are at most two or more than three Heads. 2. Same experiment, find the probability that there are four Heads given that there are at least two Heads. 3. Same experiment, and let X be random variable equal to the number of Heads in the outcome. What is the expected value E(X)? 4. Same X. What is the expected value of X^2, or E(X^2)? 5. Given the experiment of rolling two fair dice, let A be the event that the sum is greater than the square of the difference of the two numbers. (For example the outcome (1,2) qualifies, but (1,6) does not.) Find A by listing all of its elements as ordered pairs. 6. Same A. Find P(A), the probability that the event A happens. 7. Find an event B which is independent of A, so that if C is the intersection of A and B, then P(C) = P(A)*P(B). HW 6 due Wednesday, June 28 1. Suppose that two dice are rolled, with one fair and one biased. The biased die has P(1)=P(2)=1/3, and all other numbers are equally likely. a) What is P(3) for the biased die? b) Write all the outcomes as ordered pairs in a grid and indicate outcomes with the same Probability by circling and writing the probability to the side of the circled group of outcomes. c) Let X be the sum on the two dice, and find the probability P(X=7). d) Find the mean and variance of X, ie. find E(X) and V(X) = E(X^2) - E(X)^2. 2. Let G be the domineering game with starting board given by the triangular grid of squares obtained by taking a 3x3 grid and removing the 3 squares in the bottom right corner. (This grid can also be obtained by connecting the dots in the plane: (0,0),(0,1),(0,2),(0,3),(1,0),(1,1),(1,2),(1,3) (2,1),(2,2),(2,3),(3,2),(3,3) with line segments parallel to x and y axes. a) Find L's (Left's) best first move and explain why it is. b) Find R's (Right's) best first move and explain why it is. c) Two of L's first possible moves result in isomorphic positions. Which moves? Explain why. (Recall that two positions are isomorphic if they have the same game tree structure.) d) Draw the game tree for G. (Remember to exclude branches on same side which are isomorphic) HW 7 due Monday, July 10 Submit a zip of your folder containing required files for #1 and pdf for #2. Programming problem: 1. Write a program to take a positive integer parameter N and simulate that many random draws from a scrabble bag of letters. For each draw compute whether the set of letters can produce a 7-letter word from the list on the website, and keep a running total of those, say q. Output the ratio q/N as a decimal, where q is the number of draws that produced a 7-letter word. Turn in your code, a README file, and also sample output which includes at least several runs with N=1000. Graph Problems: 2. Graph Theory Definitions Section Exercises: i) #4. Find a different isomorphism than the one in the solution. ii) #4. Change Graph 2 by replacing edge {c,d} by new edge {a,e}. Draw this graph and determine if it is isomorphic to Graph 1. If they are isomorphic, give the isomorphism. If not, explain. iii) #5. In part a) label the function F (to distinguish from vertex called f). Answer the question and explain why or why not. In part b) call the function G. iv) #6. For the bipartite graph, draw your example which attains the maximum number of edges, then give reasons why this should be the maximum. v) #7.