# About the Author. Dependency Chart. Chapter 1: Logic and Sets 1. Chapter 2: Relations and Functions, Boolean Algebra, and Circuit Design

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1 Preface About the Author Dependency Chart xiii xix xxi Chapter 1: Logic and Sets 1 1.1: Logical Operators: Statements and Truth Values, Negations, Conjunctions, and Disjunctions, Truth Tables, Conditional Statements (Implications), Converses and Contrapositives, Logical Equivalence and Biconditionals, Hierarchy of Logical Operators, Some Useful Logical Equivalences, Logical Implication, Proofs and Counterexamples, Logical Puzzles,, Computer 1.2: Logical Quantifiers: Predicates and Universes, Universal and Existential Quantifiers, Negations of Quantifiers, Nested Quantifiers, 1.3: Sets: Sets and Their Elements, Unions and Intersections, Venn Diagrams, Subsets and the Empty Set, Complements and Differences of Sets, Set Theoretic Identities, Unions and Intersections of Set Families, Power Sets, Cartesian Products of Sets, The Historical Development of Logic and Sets,, Computer Chapter 2: Relations and Functions, Boolean Algebra, and Circuit Design : Relations and Functions: Binary Relations, Functions, Function Images and Pre-images, One-to-One, Onto, and Bijective Functions, Inverse Functions, 2.2: Equivalence Relations and Partial Orderings: Equivalence Relations, Congruence Modulo a Positive Integer, Equivalence Classes and Their Representatives, Strings, Partial Order(ings), Hasse Diagrams, Poset Isomorphisms, v

2 vi 2.3: Boolean Algebra and Circuit Design: Boolean Operations, Variables, and Functions, Boolean Algebra Identities, Sums, Products, and Complements of Boolean Functions, Sums of Products Expansions (Disjunctive Normal Form), Duality, Logic Gates and Circuit Designs, Karnaugh Maps, Chapter 3: The Integers, Induction, and Recursion : Mathematical Induction: The Principle of Mathematical Induction: Basic Form, The Principle of Mathematical Induction: General Form, Strong Mathematical Induction, Finite Geometric Series, 3.2: Recursion: Infinite Sequences, Recursion and Recursively Defined Sequences, The Fibonacci Sequence, Recursive Sequences of Higher Degree, Explicit Solution Methods for Linear Recursion Formulas, Appendix: Recursive Definitions and Structural Induction Computer 3.3: Some Topics in Elementary Number Theory: Divisibility, Primes, The Prime Number Theorem, Greatest Common Divisors, Relatively Prime Integers, The Division Algorithm, The Euclidean Algorithm, Congruent Substitutions in Modular Arithmetic, Fermat s Little Theorem, Euler s Theorem, Orders and Primitive Roots,, Computer Appendix: Probabilistic Primality Tests Chapter 4: Number Systems : Representations of Integers in Different Bases: Representation of Integers in a Base b, Hex(adecimal) and Binary Expansions, Addition Algorithm with Base b Expansions, Subtraction Algorithm with Base b Expansions, Multiplication Algorithm in Base b Expansions,, Computer 4.2: Modular Arithmetic and Congruences: Modular Integer Systems, Modular Inverses, Fast Modular Exponentiation, Congruences, The Extended Euclidean Algorithm, Solving Linear Congruences, The Chinese Remainder Theorem, Pseudo-Random Numbers: The Linear Congruential Method,, Computer 4.3: Matrices: Matrix Addition, Subtraction, and Scalar Multiplication, Matrix Multiplication, Matrix Arithmetic, Definition of an Invertible

3 vii (Square) Matrix, The Determinant of a Square Matrix, Inverses of 2 2 Matrices, The Transpose of a Matrix, Modular Integer Matrices, The Classical Adjoint (for Matrix Inversions), Application of Modular Matrices: The Hill Cryptosystem,, Computer 4.4: Floating Point Arithmetic: Exact Arithmetic, Floating Point Arithmetic Systems, Unit Roundoff (Machine Epsilon), Underflows, Overflows,, Computer 4.5: Public Key Cryptography: An Informal Analogy for a Public Key Cryptosystem, The Quest for Secure Electronic Key Exchange, One-Way Functions, Review of the Discrete Logarithm Problem, The Diffie Hellman Key Exchange, The Quest for a Complete Public Key Cryptosystem, The RSA Cryptosystem, The El Gamal Cryptosystem, Knapsack Problems, The Merkle Hellman Knapsack Cryptosystem, Government Controls on Cryptography,, Computer Chapter 5: Counting Techniques, Combinatorics, and Generating Functions : Fundamental Principles of Counting: The Multiplication Principle, the Complement Principle, The Inclusion-Exclusion Principle, The Pigeonhole Principle, The Generalized Pigeonhole Principle, 5.2: Permutations, Combinations, and the Binomial Theorem: The Difference between a Permutation and a Combination, Computing and Counting with Permutations and Combinations, the Binomial Theorem, Multinomial Coefficients, The Multinomial Theorem, 5.3: Generating Functions: Generating Functions and Power Series, Arithmetic of Generating Functions, The Generalized Binomial Theorem, Using Generating Functions to Solve Recursive Sequences, Using Generating Functions in Counting Problems, Appendix: Application to Weighted Democracies Computer Chapter 6: Discrete Probability and Simulation : Introduction To Discrete Probability: Experiments, Sample Spaces, and Events, Experiments with Equally Likely Outcomes, Kolmogorov s Axioms, Probability Rules, Conditional Probability, The Multiplication Rule, Conditioning and Bayes Formula, Independent Events, Discrete Problems with Infinite Sample Spaces, 6.2: Random Numbers, Random Variables, and Basic Simulations: Probabilities as Relative Frequencies, Random Numbers and Random Variables, Binomial Random Variables, Continuous Random Variables,

4 viii Uniform Random Variables, Setting up a Simulation, Generating Random Permutations and Random Subsets, Expectation and Variance of a Random Variable, Independence of Random Variables, Linearity of Expectation, Properties of Variances, Poisson Random Variables,, Computer Chapter 7: Complexity of Algorithms : Some Algorithms for Searching and Sorting: The Linear Search Algorithm, The Binary Search Algorithm, The Selection Sort Algorithm, the Bubble Sort Algorithm, The Quick Sort Algorithm, The Merge Sort Algorithm, A Randomized Algorithm for Computing Medians,, Computer 7.2: Growth Rates of Functions and the Complexity of Algorithms: A Brief and Informal Preview, Big-O Notation, Combinations of Big-O Estimates, Big-Omega and Big-Theta Notation, Complexity of Algorithms, Optimality of the Merge Sort Algorithm, the Classes P and NP,, Computer Chapter 8: Graphs, Trees, and Associated Algorithms : Graph Concepts and Properties: Simple Graphs, General Graphs, Degrees, Regular Graphs, and the Handshaking Theorem, Some Important Families of Simple Graphs, Bipartite Graphs, Degree Sequences, Subgraphs, Isomorphism of Simple Graphs, the Complement of a Simple Graph, Representing Graphs on Computers, Directed Graphs (Digraphs), Some Graph Models for Optimization Problems,, Computer 8.2: Paths, Connectedness, and Distances in Graphs: Paths, Circuits and Reachability in Graphs, Paths, Circuits, and Reachability in Digraphs, Connectedness and Connected Components, Distances and Diameters in Graphs, Eccentricity, Radius, and Central Vertices, Adjacency Matrices and Distance Computations in Graphs and Directed Graphs, Edge and Vertex Cuts in Connected Graphs/Digraphs, Characterization of Bipartite Graphs Using Cycles,, Computer 8.3: Trees: Basic Concepts about Trees, Rooted Trees and Binary Trees, Models with Rooted Trees, Properties of Rooted Trees, Ordered Tree Traversal Algorithms, Binary Search Trees, Representing Rooted Trees on Computers,, Computer Appendix: Application of Rooted Trees to Data Compression and Coding; Huffman Codes

5 ix Chapter 9: Graph Traversal and Optimization Problems : Graph Traversal Problems: Euler Paths and Tours and the Origin of Graph Theory, Euler Paths and Tours in Digraphs, Application of Eulerian Digraphs: De Bruijn Sequences, Hamilton Paths and Tours, Application of Hamiltonian Graphs: Gray Codes, Sufficient Conditions for a Graph to Be Hamiltonian, Necessary Conditions for a Graph to Be Hamiltonian,, Computer 9.2: Tree Growing and Graph Optimization Algorithms: Minimum Spanning Tree for an Edge-Weighted Graph, Tree Growing Meta- Algorithm, Prim s Algorithm for Minimum Spanning Trees, Dijkstra s Algorithm for Shortest Distances in an Edge-Weighted Connected Graph, Depth-First Searches and Breadth-First Searches, The Traveling Salesman Problem, Insertion Heuristics for the Traveling Salesman Problem, Performance Guarantees for Insertion Heuristics for the Traveling Salesman Problem,, Computer 9.3: Network Flows: Flow Networks, Cuts in Flow Networks, The Maximum Flow/Minimum Cut Theorem, The Ford-Fulkerson Maximum Flow/Minimum Algorithm, Applications of Maximum Flows, Maximum Matchings in Bipartite Graphs, Hall s Marriage Theorem,, Computer Chapter 10: Randomized Search and Optimization Algorithms : Randomized Search and Optimization: An Overview: Random Search Algorithms, Hill Climbing Algorithms, The k-opt Local Search Algorithm for Traveling Salesman Problems, Randomized Hill Climbing Algorithms, A Brief Discussion on Some Other Randomized Heuristic Algorithms, Tabu Search, Ant Colony Optimization, Simulated Annealing, and Computer 10.2: Genetic Algorithms: Motivating Example, The Basic Genetic Algorithm, Cloning and Inversions, Application to Ramsey Numbers, Theoretical Underpinnings, and Computer Appendix A: Pseudo Code Dictionary 781 Appendix B: Solutions to All for the Reader 787 Appendix C: Answers/Brief Solutions to Odd Numbered 859

6 x References 957 Index of Theorems, Propositions, Lemmas, and Corollaries 963 Index of Algorithms 967 Index 969

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