1. CHAPTER 1 Counting
2. 1.1 Basic Counting
3. The Sum Principle
4. Abstraction
5. Summing Consecutive Integers
6. The Product Principle
7. Two-Element Subsets
8. Important Concepts, Formulas, and Theorems
9. Problems
10. 1.2 Counting Lists, Permutations, and Subsets
11. Using the Sum and Product Principles
12. Lists and Functions
13. The Bijection Principle
14. k-Element Permutations of a Set
15. Counting Subsets of a Set
16. Important Concepts, Formulas, and Theorems
17. Problems
18. 1.3 Binomial Coeficients
19. Pascal’s Triangle
20. A Proof Using the Sum Principle
21. The Binomial Theorem
22. Labeling and Trinomial Coeficients
23. Important Concepts, Formulas, and Theorems
24. Problems
25. 1.4 Relations
26. What Is a Relation?
27. Functions as Relations
28. Properties of Relations
29. Equivalence Relations
30. Partial and Total Orders
31. Important Concepts, Formulas, and Theorems
32. Problems
33. 1.5 Using Equivalence Relations in Counting
34. The Symmetry Principle
35. Equivalence Relations
36. The Quotient Principle
37. Equivalence Class Counting
38. Multisets
39. The Bookcase Arrangement Problem
40. The Number of k-Element Multisets of an n-Element Set
41. Using the Quotient Principle to Explain a Quotient
42. Important Concepts, Formulas, and Theorems
43. Problems
44. CHAPTER 2 Cryptography and Number Theory
45. 2.1 Cryptography and Modular Arithmetic
46. Introduction to Cryptography
47. Private-Key Cryptography
48. Public-Key Cryptosystems
49. Arithmetic Modulo n
50. Cryptography Using Addition mod n
51. Cryptography Using Multiplication mod n
52. Important Concepts, Formulas, and Theorems
53. Problems
54. 2.2 Inverses and Greatest Common Divisors
55. Solutions to Equations and Inverses mod n
56. Inverses mod n
57. Converting Modular Equations to Normal Equations
58. Greatest Common Divisors
59. Euclid’s Division Theorem
60. Euclid’s GCD Algorithm
61. Extended GCD Algorithm
62. Computing Inverses
63. Important Concepts, Formulas, and Theorems
64. Problems
65. 2.3 The RSA Cryptosystem
66. Exponentiation mod n
67. The Rules of Exponents
68. Fermat’s Little Theorem
69. The RSA Cryptosystem
70. The Chinese Remainder Theorem
71. Important Concepts, Formulas, and Theorems
72. Problems
73. 2.4 Details of the RSA Cryptosystem
74. Practical Aspects of Exponentiation mod n
75. How Long Does It Take to Use the RSA Algorithm?
76. How Hard Is Factoring?
77. Finding Large Primes
78. Important Concepts, Formulas, and Theorems
79. Problems
80. CHAPTER 3 Reflections on Logic and Proof
81. 3.1 Equivalence and Implication
82. Equivalence of Statements
83. Truth Tables
84. DeMorgan’s Laws
85. Implication
86. If and Only If
87. Important Concepts, Formulas, and Theorems
88. Problems
89. 3.2 Variables and Quantifiers
90. Variables and Universes
91. Quantifiers
92. Standard Notation for Quantification
93. Statements about Variables
94. Rewriting Statements to Encompass Larger Universes
95. Proving Quantified Statements True or False
96. Negation of Quantified Statements
97. Implicit Quantification
98. Proof of Quantified Statements
99. Important Concepts, Formulas, and Theorems
100. Problems
101. 3.3 Inference
102. Direct Inference (Modus Ponens) and Proofs
103. Rules of Inference for Direct Proofs
104. Contrapositive Rule of Inference
105. Proof by Contradiction
106. Important Concepts, Formulas, and Theorems
107. Problems
108. CHAPTER 4 Induction, Recursion, and Recurrences
109. 4.1 Mathematical Induction
110. Smallest Counterexamples
111. The Principle of Mathematical Induction
112. Strong Induction
113. Induction in General
114. A Recursive View of Induction
115. Structural Induction
116. Important Concepts, Formulas, and Theorems
117. Problems
118. 4.2 Recursion, Recurrences, and Induction
119. Recursion
120. Examples of First-Order Linear Recurrences
121. Iterating a Recurrence
122. Geometric Series
123. First-Order Linear Recurrences
124. Important Concepts, Formulas, and Theorems
125. Problems
126. 4.3 Growth Rates of Solutions to Recurrences
127. Divide and Conquer Algorithms
128. Recursion Trees
129. Three Different Behaviors
130. Important Concepts, Formulas, and Theorems
131. Problems
132. 4.4 The Master Theorem
133. Master Theorem
134. Solving More General Kinds of Recurrences
135. Extending the Master Theorem
136. Important Concepts, Formulas, and Theorems
137. Problems
138. 4.5 More General Kinds of Recurrences
139. Recurrence Inequalities
140. The Master Theorem for Inequalities
141. A Wrinkle with Induction
142. Further Wrinkles in Induction Proofs
143. Dealing with Functions Other Than n[sup(c)]
144. Important Concepts, Formulas, and Theorems
145. Problems
146. 4.6 Recurrences and Selection
147. The Idea of Selection
148. A Recursive Selection Algorithm
149. Selection without Knowing the Median in Advance
150. An Algorithm to Find an Element in the Middle Half
151. An Analysis of the Revised Selection Algorithm
152. Uneven Divisions
153. Important Concepts, Formulas, and Theorems
154. Problems
155. CHAPTER 5 Probability
156. 5.1 Introduction to Probability
157. Why Study Probability?
158. Some Examples of Probability Computations
159. Complementary Probabilities
160. Probability and Hashing
161. The Uniform Probability Distribution
162. Important Concepts, Formulas, and Theorems
163. Problems
164. 5.2 Unions and Intersections
165. The Probability of a Union of Events
166. Principle of Inclusion and Exclusion for Probability
167. The Principle of Inclusion and Exclusion for Counting
168. Important Concepts, Formulas, and Theorems
169. Problems
170. 5.3 Conditional Probability and Independence
171. Conditional Probability
172. Bayes’ Theorem
173. Independence
174. Independent Trials Processes
175. Tree Diagrams
176. Primality Testing
177. Important Concepts, Formulas, and Theorems
178. Problems
179. 5.4 Random Variables
180. What Are Random Variables?
181. Binomial Probabilities
182. A Taste of Generating Functions
183. Expected Value
184. Expected Values of Sums and Numerical Multiples
185. Indicator Random Variables
186. The Number of Trials until the First Success
187. Important Concepts, Formulas, and Theorems
188. Problems
189. 5.5 Probability Calculations in Hashing
190. Expected Number of Items per Location
191. Expected Number of Empty Locations
192. Expected Number of Collisions
193. Expected Maximum Number of Elements in a Location of a Hash Table
194. Important Concepts, Formulas, and Theorems
195. Problems
196. 5.6 Conditional Expectations, Recurrences, and Algorithms
197. When Running Times Depend on More than Size of Inputs
198. Conditional Expected Values
199. Randomized Algorithms
200. Selection Revisited
201. QuickSort
202. A More Careful Analysis of RandomSelect
203. Important Concepts, Formulas, and Theorems
204. Problems
205. 5.7 Probability Distributions and Variance
206. Distributions of Random Variables
207. Variance
208. Important Concepts, Formulas, and Theorems
209. Problems
210. CHAPTER 6 Graphs
211. 6.1 Graphs
212. The Degree of a Vertex
213. Connectivity
214. Cycles
215. Trees
216. Other Properties of Trees
217. Important Concepts, Formulas, and Theorems
218. Problems
219. 6.2 Spanning Trees and Rooted Trees
220. Spanning Trees
221. Breadth-First Search
222. Rooted Trees
223. Important Concepts, Formulas, and Theorems
224. Problems
225. 6.3 Eulerian and Hamiltonian Graphs
226. Eulerian Tours and Trails
227. Finding Eulerian Tours
228. Hamiltonian Paths and Cycles
229. NP-Complete Problems
230. Proving That Problems Are NP-Complete
231. Important Concepts, Formulas, and Theorems
232. Problems
233. 6.4 Matching Theory
234. The Idea of a Matching
235. Making Matchings Bigger
236. Matching in Bipartite Graphs
237. Searching for Augmenting Paths in Bipartite Graphs
238. The Augmentation-Cover Algorithm
239. Efficient Algorithms
240. Important Concepts, Formulas, and Theorems
241. Problems
242. 6.5 Coloring and Planarity
243. The Idea of Coloring
244. Interval Graphs
245. Planarity
246. The Faces of a Planar Drawing
247. The Five-Color Theorem
248. Important Concepts, Formulas, and Theorems
249. Problems
250. APPENDIX A: Derivation of the More General Master Theorem
251. More General Recurrences
252. Recurrences for General n
253. Removing Floors and Ceilings
254. Floors and Ceilings in the Stronger Version of the Master Theorem
255. Proofs of Theorems
256. Important Concepts, Formulas, and Theorems
257. Problems
258. APPENDIX B: Answers and Hints to Selected Problems
259. Bibliography
260. Index

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