DISCRETE MATHEMATICS THROUGH GUIDED DISCOVERY ...

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

In the last few problems, you began with a set of lists and first you had to decide when two lists were equivalent representations of the objects you were trying to count. Then you drew the directed graph for that particular relation of equivalence. Go back to your digraphs and check that each vertex has an arrow to itself. This is what is meant when it is said that a relation is reflexive. Also, check that whenever you have an arrow from one vertex to a second, there is an arrow from the second back to the first. This is what is meant when a relation is said to be symmetric. There is another property of those relations you have graphed. Namely, whenever you have an arrow from L1 to L2 and an arrow from L2 to L3, then there is an arrow from L1 to L3. This is what is meant when a relation is said to be transitive. A relation on a set is called an equivalence relation on the set when it satisfies all three of these properties. 91. If R is a relation on a finite set, how can the digraph be used to check whether or not the relation is reflexive? To check symmetry? To check transitivity? 92. The matrix of a relation R on an n-element set {x1, . . . , xn} is defined to be the n × n matrix A whose (i, j) entry equals the number of edges from xi to xj in the associated digraph. For at least three relations on the set [4], find the adjacency matrix A and calculate A 2 . Use this information to formulate a conjecture about what is recorded in the entries of A 2 . 93. Let B be the matrix obtained from the matrix A 2 by changing every positive entry to 1. Explain how the matrix A−B can be used to determine whether the relation is transitive. If the relation is not transitive, explain how the matrix A − B can be used to find a counterexample to transitivity. Check that each relation of equivalence in Problems 88, 89 and 90 satisfies the three properties, and so each is an equivalence relation. Carefully visualize the same three properties in the relations of equivalence that you use in the remaining problems of this chapter. Work through Appendix C if you would like more practice with checking whether or not a relation is an equivalence relation. You undoubtedly have noticed that for each of the equivalence relations you’ve considered so far, the digraph is divided into clumps of mutually connected vertices. In the next section you will show that this “clumping” property holds for all equivalence relations, and that it is exactly this property which makes equivalence relations a powerful tool for counting. 41
3.2 Equivalence Classes Next is an investigation of the “clumping” behavior you observed in the digraphs of the equivalence relations in the last section. •94. Describe the equivalence relation inherent in seating n people around a round table in such a way that they are equally spaced around the table. Check that it is an equivalence relation on the set of all n! lists of n people. This equivalence relation induces a partition of the set of lists. What size blocks occur? How many non-equivalent seating arrangements are there? Compare this problem with Problem 90. •95. In this problem, consider the question of making necklaces by arranging n distinguishable beads on a string. Assume that once all n beads are placed on the string its ends are carefully knotted so the knot cannot be seen and that the beads are equally spaced on the string. Regard two necklaces as equivalent if when both are placed on a table, a person can pick up one of the necklaces, move it around in space, and put it back down so that it exactly matches the other necklace. Describe this as an equivalence relation on some set. What size blocks of necklaces occur? How many non-equivalent necklaces can be constructed using n distinguishable beads? •96. Verify explicitly that your relationship of equivalent necklaces is reflexive, symmetric, and transitive, and so is an equivalence relation. Some standard notation is useful for working with equivalence relations: When R is a relation on the set X , the notation xRy will indicate that x, y ∈ X are related under R. Using this notation, the relation • R is called reflexive if xRx for every x ∈ X. • R is called symmetric if xRy holds whenever yRx holds. • R is called transitive if whenever both xRy and yRz, then xRz as well. • R is an equivalence relation if R is reflexive, symmetric and transitive. Suppose that R is an equivalence relation on a set X and for each x ∈ X, define the set Cx by Cx = {y ∈ X : yRx}. That is, Cx is the subset of X consisting of all elements of X which are equivalent to the element x under 42
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: ◦89. Now again suppose you are ch
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
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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
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
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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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