Exercise

Abstract: Foreword Preface Acknowledgments I How: Methods 1 Basic Methods 1.1 When We Add and When We Subtract 1.1.1 When We Add 1.1.2 When We Subtract 1.2 When We Multiply 1.2.1 The Product Principle 1.2.2 Using Several Counting Principles 1.2.3 When Repetitions Are Not Allowed 1.3 When We Divide 1.3.1 The Division Principle 1.3.2 Subsets 1.4 Applications of Basic Counting Principles 1.4.1 Bijective Proofs 1.4.2 Properties of Binomial Coefficients 1.4.3 Permutations With Repetition 1.5 The Pigeonhole Principle 1.6 Notes 1.7 Chapter Review 1.8 Exercises 1.9 Solutions to Exercises 1.10 Supplementary Exercises 2 Direct Applications of Basic Methods 2.1 Multisets and Compositions 2.1.1 Weak Compositions 2.1.2 Compositions 2.2 Set Partitions 2.2.1 Stirling Numbers of the Second Kind 2.2.2 Recurrence Relations for Stirling Numbers of the Second Kind 2.2.3 When the Number of Blocks Is Not Fixed 2.3 Partitions of Integers 2.3.1 Nonincreasing Finite Sequences of Integers 2.3.2 Ferrers Shapes and Their Applications 2.3.3 Excursion: Euler's Pentagonal Number Theorem 2.4 The Inclusion-Exclusion Principle 2.4.1 Two Intersecting Sets 2.4.2 Three Intersecting Sets 2.4.3 Any Number of Intersecting Sets 2.5 The Twelvefold Way 2.6 Notes 2.7 Chapter Review 2.8 Exercises 2.9 Solutions to Exercises 2.10 Supplementary Exercises 3 Generating Functions 3.1 Power Series 3.1.1 Generalized Binomial Coefficients 3.1.2 Formal Power Series 3.2 Warming Up: Solving Recursions 3.2.1 Ordinary Generating Functions 3.2.2 Exponential Generating Functions 3.3 Products of Generating Functions 3.3.1 Ordinary Generating Functions 3.3.2 Exponential Generating Functions 3.4 Excursion: Composition of Two Generating Functions 3.4.1 Ordinary Generating Functions 3.4.2 Exponential Generating Functions 3.5 Excursion: A Different Type of Generating Function 3.6 Notes 3.7 Chapter Review 3.8 Exercises 3.9 Solutions to Exercises 3.10 Supplementary Exercises II What: Topics 4 Counting Permutations 4.1 Eulerian Numbers 4.2 The Cycle Structure of Permutations 4.2.1 Stirling Numbers of the First Kind 4.2.2 Permutations of a Given Type 4.3 Cycle Structure and Exponential Generating Functions 4.4 Inversions 4.4.1 Counting Permutations with Respect to Inversions 4.5 Notes 4.6 Chapter Review 4.7 Exercises 4.8 Solutions to Exercises 4.9 Supplementary Exercises 5 Counting Graphs 5.1 Counting Trees and Forests 5.1.1 Counting Trees 5.2 The Notion of Graph Isomorphisms 5.3 Counting Trees on Labeled Vertices 5.3.1 Counting Forests 5.4 Graphs and Functions 5.4.1 Acyclic Functions 5.4.2 Parking Functions 5.5 When the Vertices Are Not Freely Labeled 5.5.1 Rooted Plane Trees 5.5.2 Binary Plane Trees 5.6 Excursion: Graphs on Colored Vertices 5.6.1 Chromatic Polynomials 5.6.2 Counting k-colored Graphs 5.7 Graphs and Generating Functions 5.7.1 Generating Functions of Trees 5.7.2 Counting Connected Graphs 5.7.3 Counting Eulerian Graphs 5.8 Notes 5.9 Chapter Review 5.10 Exercises 5.11 Solutions to Exercises 5.12 Supplementary Exercises 6 Extremal Combinatorics 6.1 Extremal Graph Theory 6.1.1 Bipartite Graphs 6.1.2 Tur'an's Theorem 6.1.3 Graphs Excluding Cycles 6.1.4 Graphs Excluding Complete Bipartite Graphs 6.2 Hypergraphs 6.2.1 Hypergraphs with Pairwise Intersecting Edges 6.2.2 Hypergraphs with Pairwise Incomparable Edges 6.3 Something Is More Than Nothing: Existence Proofs 6.3.1 Property B 6.3.2 Excluding Monochromatic Arithmetic Progressions 6.3.3 Codes Over Finite Alphabets 6.4 Notes 6.5 Chapter Review 6.6 Exercises 6.7 Solutions to Exercises 6.8 Supplementary Exercises III What Else: Special Topics 7 Symmetric Structures 7.1 Hypergraphs with Symmetries 7.2 Finite Projective Planes 7.2.1 Excursion: Finite Projective Planes of Prime Power Order 7.3 Error-Correcting Codes 7.3.1 Words Far Apart 7.3.2 Codes from Hypergraphs 7.3.3 Perfect Codes 7.4 Counting Symmetric Structures 7.5 Notes 7.6 Chapter Review 7.7 Exercises 7.8 Solutions to Exercises 7.9 Supplementary Exercises 8 Sequences in Combinatorics 8.1 Unimodality 8.2 Log-Concavity 8.2.1 Log-Concavity Implies Unimodality 8.2.2 The Product Property 8.2.3 Injective Proofs 8.3 The Real Zeros Property 8.4 Notes 8.5 Chapter Review 8.6 Exercises 8.7 Solutions to Exercises 8.8 Supplementary Exercises 9 Counting Magic Squares and Magic Cubes 9.1 An Interesting Distribution Problem 9.2 Magic Squares of Fixed Size 9.2.1 The Case of n = 3 9.2.2 The Function Hn(r) for Fixed n 9.3 Magic Squares of Fixed Line Sum 9.4 Why Magic Cubes Are Different 9.5 Notes 9.6 Chapter Review 9.7 Exercises 9.8 Supplementary Exercises A The Method of Mathematical Induction A.1 Weak Induction A.2 Strong Induction Bibliography Index Frequently Used Notation

Abstract: 1. SETS, PROOF TEMPLATES, AND INDUCTION. Basic Definitions. Exercises. Operations on Sets. Exercises. The Principle of Inclusion-Exclusion. Exercises. Mathematical Induction. Program Correctness. Exercises. Strong Form of Mathematical Induction. Exercises. Chapter Review. 2. FORMAL LOGIC. Introduction to Propositional Logic. Exercises. Truth and Logical Truth. Exercises. Normal Forms. Exercises. Predicates and Quantification. Exercises. Chapter Review. 3. RELATIONS. Binary Relations. Operations on Binary Relations. Exercises. Special Types of Relations. Exercises. Equivalence Relations. Exercises. Ordering Relations. Exercises. Relational Databases: An Introduction. Exercises. Chapter Review. 4. FUNCTIONS. Basic Definitions. Exercises. Operations on Functions. Sequences and Subsequences. Exercises. The Pigeon-Hole Principle. Exercises. Countable and Uncountable Sets. Exercises. Chapter Review. 5. ANALYSIS OF ALGORITHMS. Comparing Growth Rates of Functions. Exercises. Complexity of Programs. Exercises. Uncomputability. Chapter Review. 6. GRAPH THEORY. Introduction to Graph Theory. The Handshaking Problem. Paths and Cycles. Graph Isomorphism. Representation of Graphs. Exercises. Connected Graphs. The Konigsberg Bridge Problem. Exercises. Trees. Spanning Trees. Rooted Trees. Exercises. Directed Graphs. Applications: Scheduling a Meeting Facility. Finding a Cycle in a Directed Graph. Priority in Scheduling. Connectivity in Directed Graphs. Eulerian Circuits in Directed Graphs. Exercises. Chapter Review. 7. COUNTING AND COMBINATORICS. Traveling Salesperson. Counting Principles. Set Decomposition Principle. Exercises. Permutations and Combinations. Constructing the kth Permutation. Exercises. Counting with Repeated Objects. Combinatorial Identities. Pascals Triangle. Exercises. Chapter Review. 8. DISCRETE PROBABILITY. Ideas of Chance in Computer Science. Exercises. Cross Product Sample Spaces. Exercises. Independent Events and Conditional Probability. Exercises. Discrete Random Variables. Exercises. Variance, Standard Deviation, and the Law of Averages. Exercises. Chapter Review. 9. RECURRENCE RELATIONS. The Tower of Hanoi Problem. Solving First-Order Recurrence Relations. Exercises. Second-Order Recurrence Relations. Exercises. Divide-and-Conquer Paradigm. Binary Search. Merge Sort. Multiplication of n-Bit Numbers. Divide-and-Conquer Recurrence Relations. Exercises. Chapter Review.

Abstract: Preface 1. Numbers 2. Sequences 3. Continuous functions 4. The differential calculus 5. Infinite series 6. The special functions of analysis 7. The integral calculus 8. Functions of several variables Notes on the exercises Index.

Abstract: 0 Communicating Mathematics Learning Mathematics What Others Have Said About Writing Mathematical Writing Using Symbols Writing Mathematical Expressions Common Words and Phrases in Mathematics Some Closing Comments about Writing 1 Sets 11 Describing a Set 12 Subsets 13 Set Operations 14 Indexed Collections of Sets 15 Partitions of Sets 16 Cartesian Products of Sets Exercises for Chapter 1 2 Logic 21 Statements 22 The Negation of a Statement 23 The Disjunction and Conjunction of Statements 24 The Implication 25 More on Implications 26 The Biconditional 27 Tautologies and Contradictions 28 Logical Equivalence 29 Some Fundamental Properties of Logical Equivalence 210 Quantified Statements 211 Characterizations of Statements Exercises for Chapter 2 3 Direct Proof and Proof by Contrapositive 31 Trivial and Vacuous Proofs 32 Direct Proofs 33 Proof by Contrapositive 34 Proof by Cases 35 Proof Evaluations Exercises for Chapter 3 4 More on Direct Proof and Proof by Contrapositive 41 Proofs Involving Divisibility of Integers 42 Proofs Involving Congruence of Integers 43 Proofs Involving Real Numbers 44 Proofs Involving Sets 45 Fundamental Properties of Set Operations 46 Proofs Involving Cartesian Products of Sets Exercises for Chapter 4 5 Existence and Proof by Contradiction 51 Counterexamples 52 Proof by Contradiction 53 A Review of Three Proof Techniques 54 Existence Proofs 55 Disproving Existence Statements Exercises for Chapter 5 6 Mathematical Induction 61 The Principle of Mathematical Induction 62 A More General Principle of Mathematical Induction 63 Proof by Minimum Counterexample 64 The Strong Principle of Mathematical Induction Exercises for Chapter 6 7 Prove or Disprove 71 Conjectures in Mathematics 72 Revisiting Quantified Statements 73 Testing Statements 74 A Quiz of "Prove or Disprove" Problems Exercises for Chapter 7 8 Equivalence Relations 81 Relations 82 Properties of Relations 83 Equivalence Relations 84 Properties of Equivalence Classes 85 Congruence Modulo n 86 The Integers Modulo n Exercises for Chapter 8 9 Functions 91 The Definition of Function 92 The Set of All Functions from A to B 93 One-to-one and Onto Functions 94 Bijective Functions 95 Composition of Functions 96 Inverse Functions 97 Permutations Exercises for Chapter 9 10 Cardinalities of Sets 101 Numerically Equivalent Sets 102 Denumerable Sets 103 Uncountable Sets 104 Comparing Cardinalities of Sets 105 The Schroder-Bernstein Theorem Exercises for Chapter 10 11 Proofs in Number Theory 111 Divisibility Properties of Integers 112 The Division Algorithm 113 Greatest Common Divisors 114 The Euclidean Algorithm 115 Relatively Prime Integers 116 The Fundamental Theorem of Arithmetic 117 Concepts Involving Sums of Divisors Exercises for Chapter 11 12 Proofs in Calculus 121 Limits of Sequences 122 Infinite Series 123 Limits of Functions 124 Fundamental Properties of Limits of Functions 125 Continuity 126 Differentiability Exercises for Chapter 12 13 Proofs in Group Theory 131 Binary Operations 132 Groups 133 Permutation Groups 134 Fundamental Properties of Groups 135 Subgroups 136 Isomorphic Groups Exercises for Chapter 13 Answers and Hints to Selected Odd-Numbered Exercises References Index of Symbols Index of Mathematical Terms