Topics in Finite and Discrete Mathematics

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Cambridge University Press, Jul 31, 2000 - Mathematics - 265 pages
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Written for a broad audience of students in mathematics, computer science, operations research, statistics, and engineering, this textbook presents a short, lively survey of several fascinating non-calculus topics in modern applied mathematics. Coverage includes probability, mathematical finance, graphs, linear programming, statistics, computer science algorithms, and groups. A key feature is the abundance of interesting examples not normally found in standard finite mathematics courses, such as options pricing and arbitrage, tournaments, and counting formulas. The only prerequisite is a course in pre-calculus, although the added sophistication attained from studying calculus would be useful.
 

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Contents

Preliminaries
1
12 Summation
4
13 Mathematical Induction
8
14 Functions
17
15 The Division Algorithm
23
16 Exercises
28
Combinatorial Analysis
34
23 Permutations
36
63 Applications of the Maximum Flow Problem
160
632 The Tournament Win Problem
163
633 The Transshipment Problem
166
634 An Equipment Selection Problem
167
64 Shortest Path in Digraphs
170
65 Exercises
175
Linear Programming
180
72 Transforming to the Standard Form
184

24 Combinations
40
25 Counting the Number of Solutions
45
26 The InclusionExclusion Identity
47
27 Using Recursion Equations
52
28 The Pigeonhole Principle
61
29 Exercises
63
Probability
70
32 Probability Experiments Having Equally Likely Outcomes
74
33 Conditional Probability
77
34 Computing Probabilities by Conditioning
80
35 Random Variables and Expected Values
85
36 Exercises
94
Mathematics of Finance
97
42 Present Value Analysis
100
43 Pricing Contracts via Arbitrage
104
432 Other Examples of Pricing via Arbitrage
107
44 The Arbitrage Theorem
111
45 The Multiperiod Binomial Model
116
451 The BlackScholes Option Pricing Formula
120
46 Exercises
121
Graphs and Trees
124
52 Trees
127
53 The Minimum Spanning Tree Problem
131
54 Cliques and Independent Sets
134
55 Euler Graphs
142
56 Exercises
144
Directed Graphs
150
721 Minimization and WrongWay Inequality Constraints
185
722 Problems with Variables Unconstrained in Sign
186
73 The Dual Linear Programming Problem
188
74 Game Theory
194
75 Exercises
199
Sorting and Searching
203
83 The Quicksort Algorithm
206
84 Merge Sorts
209
85 Sequential Searching
210
86 Binary Searches and Rooted Trees
212
87 Exercises
218
Statistics
220
93 Summarizing Data Sets
223
932 Sample Variance and Sample Standard Deviation
225
94 Chebyshevs Inequality
227
95 Paired Data Sets and the Sample Correlation Coefficient
229
96 Testing Statistical Hypotheses
232
97 Exercises
233
Groups and Permutations
237
102 Permutation Graphs
243
103 Subgroups
244
104 Lagranges Theorem
249
105 The Alternating Subgroup
254
106 Exercises
259
Index
263
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