Preface Part I: Preliminaries Column 1: Cracking the Oyster Column 2: Aha! Algorithms [Sketch] Column 4: Writing Correct Programs [Sketch] Column 5: A Small Matter of Programming [Sketch] Part II: Performance Column 7: The Back of the Envelope Column 8: Algorithm Design Techniques [Sketch] Part III: The Product Column 14: Heaps [Sketch] Column 15: Strings of Pearls Epilog to the First Edition Epilog to the Second Edition Appendix 2: An Estimation Quiz Appendix 3: Cost Models for Time and Space Appendix 4: Rules for Code Tuning Solutions for Column 1 Column 5 Column 7 Column 15 Index About The Book Why a Second Edition? To Readers of the First Edition About the First Edition Errata Supporting Material Source Code Web Sites Relevant to the Book Animation of Sorting Algorithms Tricks of the Trade Teaching Material

1 The How, When, and Why of Mathematics Spotlight: George Polya Tips on Doing Homework 2 Logically Speaking 3 Introducing the Contrapositive and Converse 4 Set Notation and Quantifiers Tips on Quantification 5 Proof Techniques Tips on Definitions …… 10 Relations Tips on Reading Mathematics 11 Partitions Tips on Putting It All Together 12 Order in the Reals 13 Consequences of the Completeness of R Tips: You Solved ItNow What? 14 Functions, Domain, and Range Spotlight: The Definition of Function 15 Functions, One-to-One, and Onto 16 Inverses 17 Images and Inverse Images Spotlight: Minimum or Infimum? 18 Mathematical Induction 19 Sequences 20 Convergence of Sequences of Real Numbers 21 Equivalent Sets 22 Finite Sets and an Infinite Set 23 Countable and Uncountable Sets 24 The Cantor–Schroder–Bernstein Theorem Spotlight: The Continuum Hypothesis ……

1 Preliminaries The Integers 3 Groups 4 Cyclic Groups 5 Permutation Groups 6.1 Cosets 7 Introduction to Cryptography 8 Algebraic Coding Theory 9 Isomorphisms 10 Normal Subgroups and Factor Groups 11 Homomorphisms 12 Matrix Groups and Symmetry 13 The Structure of Groups 14 Group Actions 15 The Sylow Theorems 16 Rings 17 Polynomials 18 Integral Domains 19 Lattices and Boolean Algebras 20 Vector Spaces 21 Fields 22 Finite Fields 23 Galois Theory

The first three chapters cover the standard material on sets, relations, and functions and algorithms. Next come chapters on logic, counting, and probability. We then have three chapters on graph theory: graphs, directed graphs, and binary trees. Finally there are individual chapters on properties of the integers, languages, machines, ordered sets and lattices, and Boolean algebra, and appendices on vectors and matrices, and algebraic systems. The chapter on functions and algorithms includes a discussion of cardinality and countable sets, and complexity. The chapters on graph theory include discussions on planarity, traversability, minimal paths, and Warshall’s and Huffman’s algorithms. We emphasize that the chapters have been written so that the order can be changed without difficulty and without loss of continuity

CHAPTER ONE:ANALYSIS OF ALGORITHMS CHAPTER TWO:RECURRENCE RELATIONS CHAPTER THREE:GENERATING FUNCTIONS CHAPTER FOUR:ASYMPTOTIC APPROXIMATIONS CHAPTER FIVE:ANALYTIC COMBINATORICS CHAPTER SIX:TREES CHAPTER SEVEN:PERMUTATIONS CHAPTER EIGHT:STRINGS AND TRIES CHAPTER NINE:WORDS AND MAPPINGS

1 Recurrent Problems 2 Sums 3 Integer Functions 4 Number Theory 5 Binomial Coefficients 6 Special Numbers 7 Generating Functions 8 Discrete Probability 9 Asymptotics

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