DISCRETE MATHEMATICS THROUGH GUIDED DISCOVERY ...

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

Table 3.1: The 3-element permutations of {a, b, c, d, e} organized by rows according to which 3-element set they permute. abc acb bac bca cab cba abd adb bad bda dab dba abe aeb bae bea eab eba acd adc cad cda dac dca ace aec cae cea eac eca ade aed dae dea ead eda bcd bdc cbd cdb dbc dcb bce bec cbe ceb ebc ecb bde bed dbe deb ebd edb cde ced dce dec ecd edc such a way that each row of the table lists all permutations of a certain 3-element subset of {a, b, c, d, e}. Since each 3-element permutation appears exactly once, the rows of the table determine a partition of the set S. From Problem 100 you know any partition is the set of equivalence classes of some equivalence relation. Find the equivalence relation for this partition of S. •109. Rather than restricting to n = 5 and k = 3, it is possible to partition the set of all k-element permutations of an n-element set (which can be assumed to be the set [n] ) into equivalence classes. Let S be the set of all k-element permutations of [n], and for s1, s2 ∈ S, define s1 R s2 ⇐⇒ s1 has the same elements as s2 . (a) Prove R is an equivalence relation on S. (b) How many elements are in any equivalence class? (c) What is the size of S? (You found this earlier. In which problem?) (d) Write a carefully worded sentence that describes a bijection between the set of equivalence classes of R and the set of k-element subsets of [n]. (e) What formula does this give you for the number � � n k of k-element subsets of an n-element set? In the last problem sequence you proved the following formula. 47
Calculation of Binomial Coefficients For any k, n ≥ 0 with k ≤ n, the number of k-element subsets of an n-element set is � � n n! = k k! (n − k)! . •110. Use the above formula to find numerical answers to Problem 106. •111. You can write n as a sum of n ones. (a) How many plus signs did you use in the sum? (b) In how many ways can you write n as a sum of a list of k positive numbers? Such a list is called a composition of the integer n into k parts. (c) Find the total number of compositions of n, that is, into any number of parts. Compare this with your solution to Problem 61. →112. The answer in Problem 111(b) can be expressed as a binomial coefficient. This means it should be possible to interpret a composition into k parts as a subset of some set. Find a bijection between compositions of n into k parts and certain subsets of some set. Be sure to explain explicitly what your function is; that is, tell how to get the subset from the composition. →113. Give a recurrence for the number of ways to divide 4n people into sets of four for games of bridge. (Do not worry about how they sit around the bridge table or who is the first dealer.) 114. A town has n street lights running along the north side of Main Street. The poles on which they are mounted need to be painted so that they do not rust. In how many ways may they be painted with red, white, blue, and green if an even number of them are to be painted green? Hint. First try for small n. 115. This is the third time you’ve seen this problem. Here, binomial coefficients should be used to construct a solution. A tennis club has 2n members, and you want to pair up the members by twos for singles matches. (a) How many ways can you list all possible pairs of these 2n tennis players, where each list consists of n unordered pairs? (b) Define an equivalence relation on the set of lists from part (a) which identifies lists which yield the same pairings for singles matches if you do not care who serves first. 48
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 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: way in this course.) Sometimes �
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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