Discrete Mathematics and Its Applications 8th Edition by Kenneth Rosen, ISBN-13: 978-1259676512

Description

Discrete Mathematics and Its Applications 8th Edition by Kenneth Rosen, ISBN-13: 978-1259676512

[PDF eBook eTextbook]

• Publisher: ‎ McGraw Hill; 8th edition (July 9, 2018)
• Language: ‎ English
• 1120 pages
• ISBN-10: ‎ 125967651X
• ISBN-13: ‎ 978-1259676512

Rosen’s Discrete Mathematics and its Applications presents a precise, relevant, comprehensive approach to mathematical concepts. This world-renowned best-selling text was written to accommodate the needs across a variety of majors and departments, including mathematics, computer science, and engineering. As the market leader, the book is highly flexible, comprehensive and a proven pedagogical teaching tool for instructors. Digital is becoming increasingly important and gaining popularity, crowning Connect as the digital leader for this discipline.

Table of Contents:

About the Author vi

Preface vii

Online Resources xvi

To the Student xix

1 The Foundations: Logic and Proofs . . . . . 1

1.1 Propositional Logic . . . . . . . . . . . . . . . . . . . 1

1.2 Applications of Propositional Logic. . . . . . . . . . . . . .17

1.3 Propositional Equivalences . . . . . . . . . . . . . . . . . . . . . 26

1.4 Predicates and Quantifiers . . . . . . . . . . . . . . . . . . . . . . 40

1.5 Nested Quantifiers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

1.6 Rules of Inference. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .73

1.7 Introduction to Proofs . . . . . . . . . . . . . . . . . . . . . . . . . . 84

1.8 Proof Methods and Strategy . . . . . . . . . . . . . . . . . . . . . 96

End-of-Chapter Material . . . . . . . . . . . . . . . . . . . . . . 115

2 Basic Structures: Sets, Functions, Sequences, Sums,

and Matrices . . . . . . . . . . . . . . . . . . . . . . . . 121

2.1 Sets . . . . . . . . . . 121

2.2 Set Operations .. . . . . . . . . .133

2.3 Functions . . . . . 147

2.4 Sequences and Summations . . . . . . . . . . . . . . . . . . . . 165

2.5 Cardinality of Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . 179

2.6 Matrices . . . . . . 188

End-of-Chapter Material . . . . . . . . . . . . . . . . . . . . . . 195

3 Algorithms . . . . . . . . . . . . . . . . . . . . . . . . . . 201

3.1 Algorithms. . . .201

3.2 The Growth of Functions . . . . . . . . . . . . . . . . . . . . . . 216

3.3 Complexity of Algorithms . . . . . . . . . . . . . . . . . . . . . 231

End-of-Chapter Material . . . . . . . . . . . . . . . . . . . . . . 244

4 Number Theory and Cryptography . . .251

4.1 Divisibility and Modular Arithmetic . . . . . . . . . . . . 251

4.2 Integer Representations and Algorithms . . . . . . . . . 260

4.3 Primes and Greatest Common Divisors . . . . . . . . . 271

4.4 Solving Congruences. . . . . . . . . . . . . . . . . . . . . . . . . .290

4.5 Applications of Congruences . . . . . . . . . . . . . . . . . . 303

4.6 Cryptography . . . . . . . . . . .310

End-of-Chapter Material . . . . . . . . . . . . . . . . . . . . . . 324

5 Induction and Recursion . . . . . . . . . . . . . 331

5.1 Mathematical Induction . . . . . . . . . . . . . . . . . . . . . . . 331

5.2 Strong Induction and Well-Ordering . . . . . . . . . . . . 354

5.3 Recursive Definitions and Structural Induction . . . 365

5.4 Recursive Algorithms . . . . . . . . . . . . . . . . . . . . . . . . . 381

5.5 Program Correctness . . . . . . . . . . . . . . . . . . . . . . . . . . 393

End-of-Chapter Material . . . . . . . . . . . . . . . . . . . . . . 398

6 Counting . . . . . . . . . . . . . . . . . . . . . . . . . . . . 405

6.1 The Basics of Counting. . . . . . . . . . . . . . . . . . . . . . . .405

6.2 The Pigeonhole Principle . . . . . . . . . . . . . . . . . . . . . . 420

6.3 Permutations and Combinations . . . . . . . . . . . . . . . . 428

6.4 Binomial Coefficients and Identities . . . . . . . . . . . . 437

6.5 Generalized Permutations and Combinations . . . . 445

6.6 Generating Permutations and Combinations . . . . . 457

End-of-Chapter Material . . . . . . . . . . . . . . . . . . . . . . 461

7 Discrete Probability . . . . . . . . . . . . . . . . . .469

7.1 An Introduction to Discrete Probability . . . . . . . . . 469

7.2 Probability Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . 477

7.3 Bayes’ Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 494

7.4 Expected Value and Variance . . . . . . . . . . . . . . . . . . 503

End-of-Chapter Material . . . . . . . . . . . . . . . . . . . . . . 520

8 Advanced Counting Techniques . . . . . . 527

8.1 Applications of Recurrence Relations . . . . . . . . . . . 527

8.2 Solving Linear Recurrence Relations . . . . . . . . . . . 540

8.3 Divide-and-Conquer Algorithms and Recurrence Relations . . . . . . . . . 553

8.4 Generating Functions . . . . . . . . . . . . . . . . . . . . . . . . . 563

8.5 Inclusion–Exclusion . . . . . . . . . . . . . . . . . . . . . . . . . . 579

8.6 Applications of Inclusion–Exclusion . . . . . . . . . . . . 585

End-of-Chapter Material . . . . . . . . . . . . . . . . . . . . . . 592

9 Relations. . . . . . . . . . . . . . . . . . . . . . . . . . . . 599

9.1 Relations and Their Properties . . . . . . . . . . . . . . . . . 599

9.2 n-ary Relations and Their Applications. . . . . . . . . .611

9.3 Representing Relations . . . . . . . . . . . . . . . . . . . . . . . . 621

9.4 Closures of Relations. . . . . . . . . . . . . . . . . . . . . . . . . .628

9.5 Equivalence Relations . . . . . . . . . . . . . . . . . . . . . . . . . 638

9.6 Partial Orderings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 650

End-of-Chapter Material . . . . . . . . . . . . . . . . . . . . . . 665

10 Graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .673

10.1 Graphs and Graph Models . . . . . . . . . . . . . . . . . . . . . 673

10.2 Graph Terminology and Special Types of Graphs 685

10.3 Representing Graphs and Graph Isomorphism . . . 703

10.4 Connectivity . . 714

10.5 Euler and Hamilton Paths . . . . . . . . . . . . . . . . . . . . . .728

10.6 Shortest-Path Problems. . . . . . . . . . . . . . . . . . . . . . . .743

10.7 Planar Graphs . 753

10.8 Graph Coloring 762

End-of-Chapter Material . . . . . . . . . . . . . . . . . . . . . . 771

11 Trees .781

11.1 Introduction to Trees . . . . . . . . . . . . . . . . . . . . . . . . . . 781

11.2 Applications of Trees . . . . . . . . . . . . . . . . . . . . . . . . . 793

11.3 Tree Traversal . 808

11.4 Spanning Trees 821

11.5 Minimum Spanning Trees . . . . . . . . . . . . . . . . . . . . . 835

End-of-Chapter Material . . . . . . . . . . . . . . . . . . . . . . 841

12 Boolean Algebra . . . . . . . . . . . . . . . . . . . . . 847

12.1 Boolean Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 847

12.2 Representing Boolean Functions . . . . . . . . . . . . . . . 855

12.3 Logic Gates . . .. . . . . . . . . . 858

12.4 Minimization of Circuits . . . . . . . . . . . . . . . . . . . . . . 864

End-of-Chapter Material . . . . . . . . . . . . . . . . . . . . . . 879

13 Modeling Computation . . . . . . . . . . . . . . 885

13.1 Languages and Grammars . . . . . . . . . . . . . . . . . . . . . 885

13.2 Finite-State Machines with Output. . . . . . . . . . . . . .897

13.3 Finite-State Machines with No Output . . . . . . . . . . 904

13.4 Language Recognition . . . . . . . . . . . . . . . . . . . . . . . . 917

13.5 Turing Machines. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .927

End-of-Chapter Material . . . . . . . . . . . . . . . . . . . . . . 938

Appendices . . . . . . . . . . . . . . . . . . . . . . . . . .A-1

1 Axioms for the Real Numbers and the Positive Integers . . . . . . . . . .A-1

2 Exponential and Logarithmic Functions . . . . . . . . .A-7

3 Pseudocode . . A-11

Suggested Readings B-1

Answers to Odd-Numbered Exercises S-1

Index of Biographies I-1

Ken Rosen (Middletown, NJ) is a distinguished member of the technical staff at AT & T Labs.

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