DISCRETE MATHEMATICS THROUGH GUIDED DISCOVERY ...

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$Math 664 Homework #2: Solutions 1. Let Î© = R, F = {A â R : either A ...$

1.2 Basic Counting Principles ◦1. Five schools plan to send their baseball team to a tournament in which each team must play each other team exactly once. How many games must be played? •2. Now some number n of schools plan to send their baseball teams to a tournament in which each team must play each other team exactly once. Think of the teams as numbered 1 through n. (a) How many games does Team 1 have to play? (b) How many additional games (other than the one with Team 1) does Team 2 have to play? (c) How many additional games (other than those with the first i − 1 teams) does Team i have to play? (d) In terms of your answers to the previous parts of this problem, what is the total number of games that must be played? Hint. If you have trouble doing this problem, work on n = 6 before studying the general n. •3. One of the schools sending its team to the tournament has to travel some distance, and so the school is making sandwiches for team members to eat along the way. There are three choices for the kind of bread and five choices for the kind of filling. How many different kinds of sandwiches are available? An ordered pair (a, b) consists of two members (which are often called coordinates) that are labeled here as a and b. Then a is called the first member of the pair and b is the second member of the pair. What is an ordered triple? You almost certainly used ordered pairs, at least implicitly, to solve the first three problems. At the time, did you recognize you were doing so? + 4. (a) If M is a set with m elements and N is a set with n elements, how many ordered pairs are there whose first element is a member of M and whose second element is a member of N? (Note that when such a question is asked in any problem in these notes, you are required both to answer the question and to provide a justification for the answer you give.) (b) Explain carefully how Problem 3 can be viewed mathematically as asking you to count the number of ordered pairs from two specific sets. 5
While working the next problems, be sure to ask yourself whether the problem amounts to counting the number of ordered pairs or ordered triples or ordered n-tuples of elements chosen from appropriate sets. What is an ordered n-tuple? ◦5. Since a sandwich by itself is pretty boring, students from the school in Problem 3 are offered a choice of a drink (from among five different kinds), a sandwich (with the choices as in Problem 3), and a fruit (from among four different kinds). In how many ways may a student make a choice of a lunch, if every lunch is a choice of these three items? Now that you have worked with your group on a few problems, you understand more about the setup of the course. For instance, you’ve been working on the problems in groups and your instructor has principally been listening to what is going on in the groups. Any guidance from your instructor has primarily been in the form of asking a question or occasionally giving a hint—a question or hint that might at first seem unrelated to the problem at hand. The reason for this is that in this course your instructor is a guide. Problems in the notes (along with perhaps cryptic hints from your instructor) are designed to lead you and your group to discover for yourselves and to prove for yourselves. There is considerable pedagogical evidence that this can lead to deep learning and understanding. In addition, it is satisfying and fun to discover things for yourself. •6. The coach of the team in Problem 3 knows of an ice cream shop along the way where she plans to stop to buy each team member a tripledecker cone. The store offers 12 different flavors of ice cream, and triple-decker cones are made only in homemade waffle cones. (Here repeated flavors are allowed; in fact, a triple-decker with three scoops of the same flavor is even possible. Be sure to count Strawberry, Vanilla, Chocolate as different from Chocolate, Vanilla, Strawberry, etc.) (a) How many possible triple-decker cones will be available to the team members? (b) How many triple-deckers have three different kinds of ice cream? You probably have noticed some standard mathematical words and phrases (for instance, set, ordered pair, function) are creeping into the problems. One of the goals of these notes is to show how the solution of most counting problems uses standard mathematical objects. As was said earlier, since 6
Page 1 and 2: DISCRETE MATHEMATICS THROUGH GUIDED
Page 3 and 4: Contents I COURSE NOTES 2 1 Beginni
Page 5 and 6: Part I COURSE NOTES 2
Page 7: to problems that are particularly i
Page 11 and 12: This idea of using functions to cou
Page 13 and 14: Figure 1.2: The union of four disjo
Page 15 and 16: The domain and co-domain of the fun
Page 17 and 18: •20. If B1, B2, . . . , Bm is a p
Page 19 and 20: In the last problem you used the fa
Page 21 and 22: ◦32. Prove that any function from
Page 23 and 24: nodes) and the line segments are ca
Page 25 and 26: niques which you have used and shou
Page 27 and 28: This illustrates one reason why som
Page 29 and 30: observation is the key to describin
Page 31 and 32: een used above is rather restrictiv
Page 33 and 34: to [10]. Give an inductive process
Page 35 and 36: 66. Are more moves required if the
Page 37 and 38: a recurrence can have infinitely ma
Page 39 and 40: prove your answer is correct, but s
Page 41 and 42: •86. In combinatorics, an (ordere
Page 43 and 44: ◦89. Now again suppose you are ch
Page 45 and 46: 3.2 Equivalence Classes Next is an
Page 47 and 48: are in different relative positions
Page 49 and 50: way in this course.) Sometimes �
Page 51 and 52: Calculation of Binomial Coefficient
Page 53 and 54: 118. Let S(k, n) denote the number
Page 55 and 56: Notice that a lattice path from (0,
Page 57 and 58: The suggestion intended by the hint
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Provided you can show that there is
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Chapter 4 Graph Theory 4.1 Undirect
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◦142. Find the degree of each ver
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A cycle in a graph is a walk which
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and then figure out in how many dif
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can begin to see this by working th
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185. Use the definition of the Rams
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(b) Draw another connected graph wi
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Figure 4.8: A graph. 1 2 4.6 Findin
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202. Use Stirling’s Formula to sh
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other restrictions are placed on th
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of x n is the number of ways to cho
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generating polynomial for the numbe
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Express this coefficient in the for
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chosen from the singleton set {i} .
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coefficient of x k in � ∞� ai
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In the last problem you represented
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The hardest part of generalizing yo
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248. A more elegant proof can be ob
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if and only if f(x) �= i for ever
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•262. Let G be a graph with verte
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Chapter 7 Distribution Problems 7.1
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◦271. In how many ways may you st
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•282. (a) Prove the Bell Numbers
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Index Kn, 20, 58 n!, Stirling’s f
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one-to-one function, 8 onto functio
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Part II REVIEW MATERIAL 108
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Exercise A.2. Here are some questio
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(a) Draw the digraph of the functio
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an n-element set has 2 n subsets. O
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of B1 is 2 k−1 . Consider the fun
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Appendix C More on Equivalence Rela
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DISCRETE MATHEMATICS THROUGH GUIDED DISCOVERY ...

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