Discrete Mathematics: Mathematical Reasoning and Proof with Puzzles, Patterns, and Games

1 Puzzles, Patterns, and Mathematical Language 1

1.1 First Examples 1

1.2 Number Puzzles and Sequences 9

1.3 Truth-tellers, Liars, and Propositional Logic 24

1.4 Predicates 40

1.5 Implications 53

1.6 Excursion Validity of Arguments 68

2 A Primer of Mathematical Writing 81

2.1 Mathematical Writing 82

2.2 Proofs about Numbers 98

2.3 Mathematical Induction 110

2.4 More about Induction 122

2.5 Contradiction and the Pigeonhole Principle 132

2.6 Excursion Representations of Numbers 150

2.7 Excursion Modular Arithmetic and Cryptography 166

3 Sets and Boolean Algebra 181

3.1 Set Definitions and Operations 181

3.2 More Operations on Sets 198

3.3 Proving Set Properties 210

3.4 Boolean Algebra 221

3.5 Excursion Logic Circuits 229

4 Functions and Relations 248

4.1 Definitions, Diagrams, and Inverses 249

4.2 The Composition Operation 268

4.3 Properties of Functions and Set Cardinality 283

4.4 Properties of Relations 301

4.5 Equivalence Relations 313

4.6 Numerical Functions in Discrete Math 324

4.7 Excursion Iterated Functions and Chaos 334

4.8 Excursion Growth of Functions 345

5 Combinatorics 368

5.1 Introduction 369

5.2 Basic Rules for Counting 386

5.3 Combinations and the Binomial Theorem 398

5.4 Binary Sequences 408

5.5 Recursive Counting 418

5.6 Excursion Solving Recurrence Relations 423

6 Probability 440

6.1 Introduction 440

6.2 Sum and Product Rules for Probability 448

6.3 Probability in Games of Chance 460

6.4 Expected Value in Games of Chance 466

6.5 Excursion Recursion Revisited 475

6.6 Excursion Matrices and Markov Chains 482

7 Graphs and Trees 505

7.1 Graph Theory 506

7.2 Proofs about Graphs and Trees 519

7.3 Isomorphism and Planarity 533

7.4 Connections to Matrices and Relations 546

7.5 Graphs in Puzzles and Games 567

7.6 Excursion Binary Trees 581

7.7 Excursion Hamiltonian Cycles and the TSP 596

A Rules of the Game 613

Cards 613

Sports 614

Miscellaneous Games 615

B Matrices and Their Operations 618

Matrix Operations 618

Matrix Arithmetic with Technology 620

Selected Answers and Hints 625

References and Further Reading 682