DISCRETE MATHEMATICS THROUGH GUIDED DISCOVERY ...

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

The Generalized Pigeonhole Principle If a set with more than kn elements is partitioned into n blocks, then at least one block has at least k + 1 elements. 37. Prove the Generalized Pigeonhole Principle follows from the Sum Principle. 38. Draw five circles labelled Al, Sue, Don, Pam, and Jo. (a) Find a way to draw red and green lines between people (circles) so that every pair of people is joined by a line and there is neither a triangle consisting entirely of red lines or a triangle consisting of green lines. (b) Suppose Bob joins the original group of five people. Now can you draw a combination of red and green lines that have the same property as those in part (a)? Explain. →39. Show that in a set of six people, there is either a subset of at least three people who all know each other or a subset of at least three people none of whom know each other. (Here it is assumed that if Person 1 knows Person 2, then Person 2 knows Person 1.) Does the conclusion hold when there are five people in the set rather than six? Problems 38 and 39 together show that six is the smallest number R with the property that if there are R people in a room, then there is either a set of (at least) three mutual acquaintances or a set of (at least) three mutual strangers. Another way to say the same thing is to say that six is the smallest number so that no matter how you connect six points in the plane (no three on a line) with red and green lines, you can find either a red triangle or a green triangle. There is a name for this: The Ramsey Number R(m, n) is the smallest number R such that if you have R people in a room, then there is a set of at least m mutual acquaintances or at least n mutual strangers. Problem 38 hints at a geometric description of Ramsey Numbers which uses the idea of a complete graph on R vertices. A complete graph on R vertices consists of R points in the plane, together with line segments (or curves) connecting each pair vertices. As you may guess, a complete graph is a special case of something called a graph, which will be defined more carefully in Section 4.1. The points in a graph are called vertices (or 19
nodes) and the line segments are called edges. The notation Kn is used to represent a complete graph on n vertices. The geometric description of R(3, 3) may be translated into the language of graph theory by saying R(3, 3) is the smallest number R such that if you color the edges of a KR with two colors, then you can find in the picture a K3 all of whose edges have the same color. The graph theory description of R(m, n) is that R(m, n) is the smallest number R such that if you color the edges of a KR with the colors red and green, then you can find in your picture either a Km all of whose edges are red or a Kn all of whose edges are green. Because you could have said your colors in the opposite order, you may conclude that R(m, n) = R(n, m). In particular R(n, n) is the smallest number R such that if you color the edges of a KR with two colors, then your picture contains a Kn all of whose edges have the same color. The results of Problems 38 and 39 combine to prove R(3, 3) = 6. 40. Since R(3, 3) = 6, an uneducated guess might be that R(4, 4) = 8. Show that this is not the case. Hint. To get started, try to write down what it means to say R(4, 4) does not equal 8. 41. Show that among ten people, there are either four mutual acquaintances or three mutual strangers. What does this say about R(4, 3)? 42. Show that among an odd number of people there is at least one person who is an acquaintance of an even number of people and therefore also a stranger to an even number of people. Hint. Let ai be the number of acquaintances of person i. 43. Find a way to color the edges of a K8 with red and green so that there is no red K4 and no green K3. →44. Find R(4, 3). Hint. There is a relevant problem that you have not used yet. As of this writing, relatively few Ramsey Numbers are known. Some that have been found are: R(3, n) for all n < 10; R(4, 4) = 18; R(4, 5) = 25. 1.6 An Overview Now is a good time for you to go back into this chapter with a fresh outlook, informed by your working through the problems in this chapter. Because 20
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 and 8: to problems that are particularly i
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: ◦32. Prove that any function from
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
Page 59 and 60: Provided you can show that there is
Page 61 and 62: Chapter 4 Graph Theory 4.1 Undirect
Page 63 and 64: ◦142. Find the degree of each ver
Page 65 and 66: A cycle in a graph is a walk which
Page 67 and 68: and then figure out in how many dif
Page 69 and 70: can begin to see this by working th
Page 71 and 72: 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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