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 elements of T . (Part (e) is slightly different.) You then can draw an arrow from the vertex for x ∈ S to the vertex for y ∈ T if and only if x is related to y. Such arrows are called (directed) edges. Because there is an inherent order in a relation, every edge is an arrow and not just a line segment. When the relation is a function, f : S → T , one arrow is drawn from each x ∈ S to its corresponding f(x) ∈ T . Familiarize yourself with this technique by working through the digraphs in Figure 1.3. Note that in part (e) the function is from a set S to itself and the picture has been simplified by drawing only one set of vertices representing the elements of S. Digraphs can often be more enlightening if you experiment to find an attractive placement of the vertices rather than putting them in a row. There is a simple test for whether a digraph of a relation from S to T is a digraph of a function from S to T : •17. Returning to the digraphs in Figure 1.3, determine which are functions. Then formulate a precise sentence stating what properties the arrows and vertices in a digraph must possess so that the digraph represents a function f from S to T . •18. (a) In how many ways can you pass out nine different candies to three children? Set up your solution as a problem about counting functions. (b) In how many ways can you pass out the candy if each child must get at least one piece? (c) Exactly three pieces? •19. Suppose you have n distinguishable balls. How many ways can you paint each of them with one color, chosen from red, black, green and blue? The most mathematically elegant solutions to the last problems probably involve using functions. Until now, your experience with functions probably has only involved a formula in calculus. In contrast, in discrete mathematics a function can be an algorithm or it might be given by a verbal description. From now on, for any positive integer n the notation [n] will be used for the set {1, . . . , n}. For example, [4] equals the set {1, 2, 3, 4}. This symbol is not used in all branches of mathematics, but for us [n] will always mean the set {1, . . . , n}. 13
•20. If B1, B2, . . . , Bm is a partition of a finite set S, define a relation f from S to [m] by f(s) = i ⇐⇒ s ∈ Bi . (a) Use the definition of partition to explain completely why f is a function with domain S. (b) This correspondence defines a relation from the set of all m-block partitions of S to the set of all functions from S to [m]. Is this relation a function whose domain is the set of all m-block partitions of S? Either find a counterexample or prove the relation is always a function. •21. A function f : S → T is an onto function (also called a surjection) if each element of T is f(x) for at least one x ∈ S. (a) Choose finite sets S and T and a function from S to T that is one-to-one but not onto. Draw the digraph of your function. (b) Choose finite sets S and T and a function from S to T that is onto but not one-to-one. Draw the digraph of your function. •22. Visual inspection of the digraph of a function can reveal whether or not the function is one-to-one or is onto. In each of the following parts devise a test similar to the one for testing when a digraph is the digraph of a function. (a) Give a written statement of a test for whether or not the digraph of a function is the digraph of a one-to-one function. (Remember that in order to be a one-to-one function, a relation must be a function.) (b) Write a statement of a test for whether or not the digraph of a function is the digraph of an onto function. If you think you might need more work on functions, consider working through Appendix A outside of class. It covers functions and also digraphs in more detail. •23. A function from a set X to a set Y which is both one-to-one and onto is usually called a bijection (but is sometimes called a 1-1 correspondence). What does the digraph of a bijection from a set S to a set T look like? 14
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: The domain and co-domain of the fun
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
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
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can begin to see this by working th
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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
Page 107 and 108:
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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