DISCRETE MATHEMATICS THROUGH GUIDED DISCOVERY ...

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

up to the next integer and adjust your payment so your balance comes out to zero. (d) What should the monthly payment be to pay off the loan over a period of 30 years? •74. A tennis club has 2n members and it wants to pair the members in twos for singles matches. (a) Find a recurrence for the number of different ways to divide all the 2n members into sets of two. Be sure to give the initial value. (b) Give a recurrence for the number of ways to divide 2n people into sets of two for tennis games in which the first server is determined. (c) In each of the previous two parts, use your recurrences to write the number of ways as a product. 75. Draw n mutually intersecting circles in the plane so that each one crosses each other one exactly twice and no three intersect in the same point. Find a recurrence for the number rn of regions into which the plane is divided by n circles. (One circle divides the plane into two regions, the inside and the outside.) Find the number of regions with n circles. For what values of n can you draw a Venn diagram showing all the possible intersections of n sets using circles to represent each of the sets? Hint. Suppose n − 1 circles have been drawn in such a way that they define rn−1 regions. When you draw a new circle, each time it crosses a new circle it finishes dividing one region into two parts and starts dividing a new region into two parts. 2.3 The General Product Principle Although the Product Principle in Chapter 1 can be applied directly to solve problems such as Problems 5 and 6, the reasoning can be cumbersome. An easier way to work this type of counting problem is to think in terms of making a sequence of choices as in the next problem. •76. Suppose you make a sequence of m choices, where • the first choice can be made in k1 ways, and • for each way of making the first i − 1 choices, the i-th choice can be made in ki ways. Explain why this is an inductive process. In how many different ways may you make your sequence of m choices? At this time you need not 35
prove your answer is correct, but simply write your answer in the next box. The General Product Principle Suppose you make a sequence of m choices, where • the first choice can be made in k1 ways, and • for each way of making the first i − 1 choices, the i-th choice can be made in ki ways. Then the total number of different ways to make this sequence of m choices is: + 77. Return to Chapter 1 and re-do Problems 5, 6, and 19 by applying the General Product Principle. In each case, write the problem as a sequence of choices, count ki for each choice, and then apply the principle. •78. Use the General Product Principle to count the number of bijections on the set [k]. Explain. (It might be a good idea to work through some of the cases k = 2, 3, and 4 before doing the general case.) •79. Let S(2) denote the statement of the General Product Principle for a sequence of m = 2 choices. Let S denote the set of all different ways to make these two choices. (a) Write out S(2) explicitly. (b) Remember that the first choice can be made in any of k1 ways and let I1, I2, ..., I be the specific items that can be chosen for the k1 first choice. Letting Bj be the subset of S for which the first choice made was the item Ij, find the size of Bj for each j. (c) Explain why B1, ..., Bk1 is a partition of S. (d) What principle allows you to conclude that the size of S equals k1k2? •80. Use mathematical induction to prove the General Product Principle. 2.3.1 Counting the number of functions Problem 72 is now revisited from the perspective of the General Product Principle. 36
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 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: a recurrence can have infinitely ma
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
Page 73 and 74: (b) Draw another connected graph wi
Page 75 and 76: Figure 4.8: A graph. 1 2 4.6 Findin
Page 77 and 78: 202. Use Stirling’s Formula to sh
Page 79 and 80: other restrictions are placed on th
Page 81 and 82: of x n is the number of ways to cho
Page 83 and 84: generating polynomial for the numbe
Page 85 and 86: Express this coefficient in the for
Page 87 and 88: chosen from the singleton set {i} .
Page 89 and 90:
coefficient of x k in � ∞� ai
Page 91 and 92:
In the last problem you represented
Page 93 and 94:
The hardest part of generalizing yo
Page 95 and 96:
248. A more elegant proof can be ob
Page 97 and 98:
if and only if f(x) �= i for ever
Page 99 and 100:
•262. Let G be a graph with verte
Page 101 and 102:
Chapter 7 Distribution Problems 7.1
Page 103 and 104:
◦271. In how many ways may you st
Page 105 and 106:
•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
Page 111 and 112:
Part II REVIEW MATERIAL 108
Page 113 and 114:
Exercise A.2. Here are some questio
Page 115 and 116:
(a) Draw the digraph of the functio
Page 117 and 118:
an n-element set has 2 n subsets. O
Page 119 and 120:
of B1 is 2 k−1 . Consider the fun
Page 121:
Appendix C More on Equivalence Rela
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DISCRETE MATHEMATICS THROUGH GUIDED DISCOVERY ...

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