DISCRETE MATHEMATICS THROUGH GUIDED DISCOVERY ...

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

most of the intellectual content of these notes is in the problems, it is natural that definitions of concepts will often be within problems. 1 For example, Problem 4 is meant to suggest that the question asked in Problem 3 was really a problem of counting all the ordered pairs consisting of a bread choice and a filling choice. The notation A × B is usually used to represent the set of all ordered pairs whose first member is in A and whose second member is in B, and A × B is called the Cartesian product of A and B. Therefore you can think of Problem 3 as asking you for the size of the Cartesian product of M and N, where M is the set of all bread types and N is the set of all possible fillings; that is, the number of different kinds of sandwiches equals the number of elements in the Cartesian product M × N. •7. The idea of a function is ubiquitous in mathematics. A function f from a set S to a set T is a relationship between the two sets that associates to each element x in the set S exactly one member f(x) in the set T . The ideas of function and relationship will be revisited in more detail and from different points of view from time to time. (a) Using f, g, . . . to stand for various functions, list all the different functions from the set {1, 2} to the set {a, b}. For example, you might start with the function f given by f(1) = a and f(2) = b . (b) Let us look at the last part in a different way. Instead of asking for a list of all the functions, suppose you simply asked how many functions are there from the set {1, 2} to the set {a, b}. Now devise a way to count the number of functions without writing an exhaustive list. (c) How many functions are there from the 3-element set {1, 2, 3} to the 2-element set {a, b}? (d) How many functions are there from the 2-element set {a, b} to the 3-element set {1, 2, 3}? (e) How many functions are there from any 3-element set to any 12-element set? (f) Re-do Problem 6(a) by constructing a function from the 3-element set of positions in the triple-decker to the set of 12-element set of flavors. Give an explicit verbal description of your function. 1 When you come across an unfamiliar term in a problem, most likely it was defined earlier, and you should be able to find the term listed in the index. 7
This idea of using functions to count is very powerful, and is one of the foundations of combinatorics. It also illustrates the added insight that can be gained by looking at a problem from more than one point of view. Take time right now to check to see if this has already happened in earlier problems. •8. A function f is called a one-to-one function (often called an injection) if f is a function which has the property that whenever x is different from y, then f(x) is different from f(y). (a) How many one-to-one functions are there from a 2-element set to a 3-element set? Hint. Since 2 and 3 are fairly small numbers, you might first list all the functions, and then decide which are one-to-one. But you should then work out a method for counting them without an exhaustive listing. (b) How many one-to-one functions are there from a 3-element set to a 2-element set? (c) How many one-to-one functions are there from a 3-element set to a 12-element set? (d) Re-do Problem 6(b) by showing that a solution amounts to counting all the one-to-one functions between two sets. In order to completely make this connection, you must explicitly define these two sets as well as a function between them that describes the possible ice cream cones. •9. A group of three hungry members of the team in Problem 6 notices it would be cheaper to buy three pints of ice cream to share among the three of them. (And they would also then get more ice cream!) (a) In how many ways may they choose three pints of two different flavors? How is this problem different from Problem 6? (b) In how many ways may they choose three pints with at least two different flavors? (c) In how many ways may they choose three pints of ice cream with no restrictions on repeating flavors? In the last part of Problem 9 you probably found it helpful to break the question into certain cases that you could solve by previous methods. After doing that you could then figure out the answer by using the answers from all the cases. Because this is a fairly common strategy, some special terminology is helpful. Two sets are said to be disjoint if they have no 8
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: While working the next problems, be
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 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
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◦142. Find the degree of each ver
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A cycle in a graph is a walk which
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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
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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