DISCRETE MATHEMATICS THROUGH GUIDED DISCOVERY ...

More documents

Recommendations

Info

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

Chapter 2 The Principle of Mathematical Induction 2.1 Inductive Processes Proof by mathematical induction is a more subtle method of reasoning than what first meets the eye. It is also much more widely applicable than you might have guessed based on your previous experience with the technique. In this chapter you’ll use mathematical induction to prove the Sum Principle and the Product Principle, as well as other counting techniques you used in the first chapter. This chapter assumes you’ve had some prior experience with proof by mathematical induction. If that assumption is not true for you, you should first work through Appendix B before beginning this section. Most likely the first examples of proof by induction you worked involved proving identities such as or n� i=1 n� i = i=1 n(n + 1) 2 (2.1) 1 n = . (2.2) i · (i + 1) n + 1 There’s a common thread to these arguments: Looking at (2.1) more closely, you are asked to prove a sequence of statements which are indexed by positive integers n. The first four statements in this sequence of statements are 1 = 1 · 2 2 ; 1 + 2 = 2 · 3 2 ; 1 + 2 + 3 = 3 · 4 2 23 ; 1 + 2 + 3 + 4 = 4 · 5 2 .
This illustrates one reason why someone might think of using induction to prove (2.1): The identity gives a sequence of statements indexed by integers n that are greater than a certain size (in this case for all integers n ≥ 1). In other words, the identity yields a family of statements S(n) parametrized by integers n ≥ 1. This means that for each integer n ≥ 1, S(n) is the statement n� i = i=1 n(n + 1) 2 That is, S(n) is the statement that the sum �n i=1 i always equals n(n+1)/2 for every integer n ≥ 1. Any result that can be proved by induction must be able to be written as a sequence of statements which can be parameterized by integers greater than some base integer b. Because this initial requirement is satisfied by (2.1), you can see that induction might be able to be used to prove this result. Next is an explanation of why induction can be successfully used to prove (2.1). For the moment, let us suspend belief and suppose you had no idea whether or not (2.1) is true for any n ≥ 1, no less that it is true for all n ≥ 1. How can you proceed? In the spirit of scientific reasoning, you might think of checking that S(n) is true for some small integers n. Maybe you would even use a computer to verify the statement S(n) for all values of n up to a quite large size. If you do this, you will find that S(n) is in fact true for every value of n ≥ 1 which you check, and this gives you some confidence that the statement is true in general for all n ≥ 1. Of course this is not yet a proof, but rather simply a justification for continuing to try to prove the statement. In order to get induction to work, an underlying inductive process must be discovered. Namely, how is the statement S(N) for any integer N related to the statements that precede it, statements that you may already have checked are true? For this particular sequence of statements, the relationship is simple: For any N ≥ 2, the truth of any statement S(N) relies only on the truth of the immediately preceding statement S(N − 1). The reason for this is that the sum on the left-hand side in the statement S(N) is built up from the left-hand sum in the statement S(N − 1) by adding the integer N to the sum. The uncovering of an inductive process for this problem is further evidence that a proof by mathematical induction might be able to be constructed. For similar reasons, the second sequence of statements (2.2) can most likely be proved by induction. Indeed, the underlying inductive process is really the same. First identify the sequence of statements S(1), S(2), . . . , S(n), . . .. 24 .
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: niques which you have used and shou
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
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
show all

DISCRETE MATHEMATICS THROUGH GUIDED DISCOVERY ...

Create successful ePaper yourself

Delete template?

Save as template?