DISCRETE MATHEMATICS THROUGH GUIDED DISCOVERY ...

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

Chapter 1 Beginning Combinatorics In most textbooks, an introductory explanation is followed by problems which use the information just given. Since much of the learning in guided discovery occurs as you work on problems, it is natural that definitions of concepts be given within the problems and that the text material often occurs after you have already worked problems developing the concepts. Because of this, be sure to read over all the problems including any you don’t work right away. Table 1.1: The meaning of the tags on the problem numbers. • essential for the course ◦ motivational material + summary → especially interesting As you flip through the pages of these notes you will see that most of the problems are marked by a symbol to the left of the number. The problems that are marked with a bullet (•) are essential for understanding later material and are where the main ideas of the book are developed. (Your instructor may leave out some of these problems because he or she plans not to cover future problems that rely on them.) Other problems are marked with open circles (◦) which indicate that they are designed to provide a motivation for important concepts by hinting at or partially developing ideas that can be helpful in solving subsequent problems. A few problems that summarize ideas that have come before are marked with a plus sign (+). You will also find some problems marked with an arrow (→). These point 3
to problems that are particularly interesting; some of them are difficult but not all are. Frequently these problems are not intrinsically difficult but only might seem hard in light of what has come before and they will be easier when you return to them after working on more problems. 1.1 What is Combinatorics? Combinatorial mathematics arises from studying combining objects into arrangements. For example, we might be combining sports teams into a tournament, samples of tires into groups for testing on cars, students into classes to compare approaches to teaching a subject, or members of a tennis club into pairs to play tennis. There are many questions that can be asked about such arrangements of objects. Here the primary focus will be questions about how many ways objects can be combined into arrangements of the desired type. These are called counting problems. One way to count is to enumerate a complete list of all the objects with the desired properties, and another way is to count how many objects have the properties without actually making a complete list. Enumerative combinatorics is usually interested in the second approach, although obtaining a solution might involve writing a partial list. Sometimes combinatorial mathematicians ask if a certain arrangement is possible. For instance, if there are ten baseball teams and each team has to play each other team once, can the whole series be scheduled if the fields are available at enough hours for forty games? Sometimes combinatorial mathematicians ask if all the arrangements able to be made have a certain desirable property. For example, do all ways of testing five brands of tires on five different cars compare each brand with each other brand on at least one common car? Counting problems (and problems of the other sorts described above) arise throughout physics, biology, computer science, statistics, and many other subjects. In order to demonstrate all these relationships detours would have to be taken into all of these subjects. Instead, although there will be some important applications, the discussions will usually be phrased around either your everyday experience or your mathematical experience so that you won’t have to learn a new context before learning the mathematics. 4
Page 1 and 2: DISCRETE MATHEMATICS THROUGH GUIDED
Page 3 and 4: Contents I COURSE NOTES 2 1 Beginni
Page 5: Part I COURSE NOTES 2
Page 9 and 10: While working the next problems, be
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
Page 61 and 62:
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
Page 67 and 68:
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
Page 73 and 74:
(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
Page 93 and 94:
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
Page 101 and 102:
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
Page 107 and 108:
Index Kn, 20, 58 n!, Stirling’s f
Page 109 and 110:
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
Page 119 and 120:
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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