Graphs

an introductory approach : a first course in discrete mathematics

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Author: Robin J. Wilson,John J. Watkins

Publisher: John Wiley & Sons Inc

ISBN: N.A

Category: Mathematics

Page: 340

View: 1892

The only text available on graph theory at the freshman/sophomore level, it covers properties of graphs, presents numerous algorithms, and describes actual applications to chemistry, genetics, music, linguistics, control theory and the social sciences. Illustrated.

Graphs & Digraphs, Fourth Edition

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Author: Gary Chartrand,Linda Lesniak,Ping Zhang

Publisher: CRC Press

ISBN: 9781584883906

Category: Mathematics

Page: 386

View: 6948

With a growing range of applications in fields from computer science to chemistry and communications networks, graph theory has enjoyed a rapid increase of interest and widespread recognition as an important area of mathematics. Through more than 20 years of publication, Graphs & Digraphs has remained a popular point of entry to the field, and through its various editions, has evolved with the field from a purely mathematical treatment to one that also addresses the mathematical needs of computer scientists. Carefully updated, streamlined, and enhanced with new features, Graphs & Digraphs, Fourth Edition reflects many of the developments in graph theory that have emerged in recent years. The authors have added discussions on topics of increasing interest, deleted outdated material, and judiciously augmented the Exercises sections to cover a range of problems that reach beyond the construction of proofs. New in the Fourth Edition: Expanded treatment of Ramsey theory Major revisions to the material on domination and distance New material on list colorings that includes interesting recent results A solutions manual covering many of the exercises available to instructors with qualifying course adoptions A comprehensive bibliography including an updated list of graph theory books Every edition of Graphs & Digraphs has been unique in its reflection the subject as one that is important, intriguing, and most of all beautiful. The fourth edition continues that tradition, offering a comprehensive, tightly integrated, and up-to-date introduction that imparts an appreciation as well as a solid understanding of the material.

Discrete Mathematics

for New Technology

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Author: Rowan Garnier

Publisher: CRC Press

ISBN: 9780750301350

Category: Technology & Engineering

Page: 696

View: 2853

In a comprehensive yet easy-to-follow manner, Discrete Mathematics for New Technology follows the progression from the basic mathematical concepts covered by the GCSE in the UK and by high-school algebra in the USA to the more sophisticated mathematical concepts examined in the latter stages of the book. The book punctuates the rigorous treatment of theory with frequent uses of pertinent examples and exercises, enabling readers to achieve a feel for the subject at hand. The exercise hints and solutions are provided at the end of the book. Topics covered include logic and the nature of mathematical proof, set theory, relations and functions, matrices and systems of linear equations, algebraic structures, Boolean algebras, and a thorough treatise on graph theory. Although aimed primarily at computer science students, the structured development of the mathematics enables this text to be used by undergraduate mathematicians, scientists, and others who require an understanding of discrete mathematics.

Algebraic Graph Theory

Morphisms, Monoids and Matrices

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Author: Ulrich Knauer

Publisher: Walter de Gruyter

ISBN: 311025509X

Category: Mathematics

Page: 324

View: 3429

This is a highly self-contained book about algebraic graph theory which is written with a view to keep the lively and unconventional atmosphere of a spoken text to communicate the enthusiasm the author feels about this subject. The focus is on homomorphisms and endomorphisms, matrices and eigenvalues. Graph models are extremely useful for almost all applications and applicators as they play an important role as structuring tools. They allow to model net structures - like roads, computers, telephones - instances of abstract data structures - like lists, stacks, trees - and functional or object oriented programming.

A First Course in Graph Theory

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Author: Gary Chartrand,Ping Zhang

Publisher: Courier Corporation

ISBN: 0486297306

Category: Mathematics

Page: 464

View: 7472

Written by two prominent figures in the field, this comprehensive text provides a remarkably student-friendly approach. Its sound yet accessible treatment emphasizes the history of graph theory and offers unique examples and lucid proofs. 2004 edition.

A Beginner's Guide to Graph Theory

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Author: W.D. Wallis

Publisher: Springer Science & Business Media

ISBN: 9780817645809

Category: Mathematics

Page: 260

View: 9085

Concisely written, gentle introduction to graph theory suitable as a textbook or for self-study Graph-theoretic applications from diverse fields (computer science, engineering, chemistry, management science) 2nd ed. includes new chapters on labeling and communications networks and small worlds, as well as expanded beginner's material Many additional changes, improvements, and corrections resulting from classroom use

A Transition to Mathematics with Proofs

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Author: Michael J Cullinane

Publisher: Jones & Bartlett Publishers

ISBN: 1449627781

Category: Mathematics

Page: 354

View: 9167

Developed for the "transition" course for mathematics majors moving beyond the primarily procedural methods of their calculus courses toward a more abstract and conceptual environment found in more advanced courses, A Transition to Mathematics with Proofs emphasizes mathematical rigor and helps students learn how to develop and write mathematical proofs. The author takes great care to develop a text that is accessible and readable for students at all levels. It addresses standard topics such as set theory, number system, logic, relations, functions, and induction in at a pace appropriate for a wide range of readers. Throughout early chapters students gradually become aware of the need for rigor, proof, and precision, and mathematical ideas are motivated through examples.

Combinatorics

A Guided Tour

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Author: David R. Mazur

Publisher: MAA

ISBN: 9780883857625

Category: Mathematics

Page: 391

View: 2279

Combinatorics is mathematics of enumeration, existence, construction, and optimization questions concerning finite sets. This text focuses on the first three types of questions and covers basic counting and existence principles, distributions, generating functions, recurrence relations, Polya theory, combinatorial designs, error correcting codes, partially ordered sets, and selected applications to graph theory including the enumeration of trees, the chromatic polynomial, and introductory Ramsey theory. The only prerequisites are single-variable calculus and familiarity with sets and basic proof techniques. The text emphasizes the brands of thinking that are characteristic of combinatorics: bijective and combinatorial proofs, recursive analysis, and counting problem classification. It is flexible enough to be used for undergraduate courses in combinatorics, second courses in discrete mathematics, introductory graduate courses in applied mathematics programs, as well as for independent study or reading courses. What makes this text a guided tour are the approximately 350 reading questions spread throughout its eight chapters. These questions provide checkpoints for learning and prepare the reader for the end-of-section exercises of which there are over 470. Most sections conclude with Travel Notes that add color to the material of the section via anecdotes, open problems, suggestions for further reading, and biographical information about mathematicians involved in the discoveries.

Discrete Mathematics

Numbers and Beyond

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Author: Stephen Barnett

Publisher: Addison-Wesley

ISBN: N.A

Category: Mathematics

Page: 441

View: 2000

For the increasing number of students who need an understanding of the subject, Discrete Mathematics: Numbers and Beyond provides the perfect introduction. Aimed particularly at non-specialists, its attractive style and practical approach offer easy access to this important subject. With an emphasis on methods and applications rather than rigorous proofs, the book's coverage is based an the essential topics of numbers, counting and numerical processes. Discrete Mathematics: Numbers and Beyond supplies the reader with a thorough grounding in number systems, modular arithmetic, combinatorics, networks and graphs, coding theory and recurrence relations. Throughout the book, learning is aided and reinforced by the following features: a wealth of exercises and problems of varying difficulty a wide range of illustrative applications of general interest numerous worked examples and diagrams team-based student projects in every chapter concise, informal explanations tips for further reading Discrete Mathematics: Numbers and Beyond is an ideal textbook for an introductory discrete mathematics course taken by students of economics, computer science, mathematics, business, finance, engineering and the sciences. 0201342928B04062001

Computing the Continuous Discretely

Integer-point Enumeration in Polyhedra

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Author: Matthias Beck,Sinai Robins

Publisher: Springer Science & Business Media

ISBN: 0387461124

Category: Mathematics

Page: 227

View: 8790

This textbook illuminates the field of discrete mathematics with examples, theory, and applications of the discrete volume of a polytope. The authors have weaved a unifying thread through basic yet deep ideas in discrete geometry, combinatorics, and number theory. We encounter here a friendly invitation to the field of "counting integer points in polytopes", and its various connections to elementary finite Fourier analysis, generating functions, the Frobenius coin-exchange problem, solid angles, magic squares, Dedekind sums, computational geometry, and more. With 250 exercises and open problems, the reader feels like an active participant.

Introduction to Calculus and Classical Analysis

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Author: Omar Hijab

Publisher: Springer Science & Business Media

ISBN: 0387693165

Category: Mathematics

Page: 342

View: 1494

Intended for an honors calculus course or for an introduction to analysis, this is an ideal text for undergraduate majors since it covers rigorous analysis, computational dexterity, and a breadth of applications. The book contains many remarkable features: * complete avoidance of /epsilon-/delta arguments by using sequences instead * definition of the integral as the area under the graph, while area is defined for every subset of the plane * complete avoidance of complex numbers * heavy emphasis on computational problems * applications from many parts of analysis, e.g. convex conjugates, Cantor set, continued fractions, Bessel functions, the zeta functions, and many more * 344 problems with solutions in the back of the book.

Discrete Mathematics

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Author: Richard Johnsonbaugh

Publisher: Prentice Hall

ISBN: 0131593188

Category: Mathematics

Page: 766

View: 7530

Focused on helping readers understand and construct proofs – and, generally, expanding their mathematical maturity – this best-seller is an accessible introduction to discrete mathematics. Takes an algorithmic approach that emphasizes problem-solving techniques. Expands discussion on how to construct proofs and treatment of problem solving. Increases number of examples and exercises throughout.

Discrete Mathematics

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Author: Norman L. Biggs

Publisher: OUP Oxford

ISBN: 9780198507178

Category: Computers

Page: 440

View: 5664

This much-awaited new edition of Biggs' best-selling text includes new chapters on statements and proof, logical framework, and natural numbers and the integers, in addition to updated chapters, over 1000 tailored exercises and an accompanying website containing hints and solutions to all exercises. The text is designed explicitly for mathematicians and computer scientists seeking a first approach to this important topic.

Reshaping college mathematics

a project of the Committee on the Undergraduate Program in Mathematics

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Author: Lynn Arthur Steen,Mathematical Association of America. Committee on the Undergraduate Program in Mathematics

Publisher: Mathematical Assn of Amer

ISBN: N.A

Category: Mathematics

Page: 125

View: 7619

Calculus of Several Variables

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Author: Serge Lang

Publisher: Springer Science & Business Media

ISBN: 1461210682

Category: Mathematics

Page: 619

View: 7626

This new, revised edition covers all of the basic topics in calculus of several variables, including vectors, curves, functions of several variables, gradient, tangent plane, maxima and minima, potential functions, curve integrals, Green’s theorem, multiple integrals, surface integrals, Stokes’ theorem, and the inverse mapping theorem and its consequences. It includes many completely worked-out problems.

Graph Theory and Its Applications, Second Edition

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Author: Jonathan L. Gross,Jay Yellen

Publisher: CRC Press

ISBN: 158488505X

Category: Mathematics

Page: 800

View: 9541

Already an international bestseller, with the release of this greatly enhanced second edition, Graph Theory and Its Applications is now an even better choice as a textbook for a variety of courses -- a textbook that will continue to serve your students as a reference for years to come. The superior explanations, broad coverage, and abundance of illustrations and exercises that positioned this as the premier graph theory text remain, but are now augmented by a broad range of improvements. Nearly 200 pages have been added for this edition, including nine new sections and hundreds of new exercises, mostly non-routine. What else is new? New chapters on measurement and analytic graph theory Supplementary exercises in each chapter - ideal for reinforcing, reviewing, and testing. Solutions and hints, often illustrated with figures, to selected exercises - nearly 50 pages worth Reorganization and extensive revisions in more than half of the existing chapters for smoother flow of the exposition Foreshadowing - the first three chapters now preview a number of concepts, mostly via the exercises, to pique the interest of reader Gross and Yellen take a comprehensive approach to graph theory that integrates careful exposition of classical developments with emerging methods, models, and practical needs. Their unparalleled treatment provides a text ideal for a two-semester course and a variety of one-semester classes, from an introductory one-semester course to courses slanted toward classical graph theory, operations research, data structures and algorithms, or algebra and topology.

Real Analysis

A Historical Approach

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Author: Saul Stahl

Publisher: John Wiley & Sons

ISBN: 1118096851

Category: Mathematics

Page: 316

View: 1816

A provocative look at the tools and history of realanalysis This new edition of Real Analysis: A Historical Approachcontinues to serve as an interesting read for students of analysis.Combining historical coverage with a superb introductory treatment,this book helps readers easily make the transition from concrete toabstract ideas. The book begins with an exciting sampling of classic and famousproblems first posed by some of the greatest mathematicians of alltime. Archimedes, Fermat, Newton, and Euler are each summoned inturn, illuminating the utility of infinite, power, andtrigonometric series in both pure and applied mathematics. Next,Dr. Stahl develops the basic tools of advanced calculus, whichintroduce the various aspects of the completeness of the realnumber system as well as sequential continuity anddifferentiability and lead to the Intermediate and Mean ValueTheorems. The Second Edition features: A chapter on the Riemann integral, including the subject ofuniform continuity Explicit coverage of the epsilon-delta convergence A discussion of the modern preference for the viewpoint ofsequences over that of series Throughout the book, numerous applications and examplesreinforce concepts and demonstrate the validity of historicalmethods and results, while appended excerpts from originalhistorical works shed light on the concerns of influentialmathematicians in addition to the difficulties encountered in theirwork. Each chapter concludes with exercises ranging in level ofcomplexity, and partial solutions are provided at the end of thebook. Real Analysis: A Historical Approach, Second Edition isan ideal book for courses on real analysis and mathematicalanalysis at the undergraduate level. The book is also a valuableresource for secondary mathematics teachers and mathematicians.

Representation Theory of Finite Groups

An Introductory Approach

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Author: Benjamin Steinberg

Publisher: Springer Science & Business Media

ISBN: 9781461407768

Category: Mathematics

Page: 157

View: 1156

This book is intended to present group representation theory at a level accessible to mature undergraduate students and beginning graduate students. This is achieved by mainly keeping the required background to the level of undergraduate linear algebra, group theory and very basic ring theory. Module theory and Wedderburn theory, as well as tensor products, are deliberately avoided. Instead, we take an approach based on discrete Fourier Analysis. Applications to the spectral theory of graphs are given to help the student appreciate the usefulness of the subject. A number of exercises are included. This book is intended for a 3rd/4th undergraduate course or an introductory graduate course on group representation theory. However, it can also be used as a reference for workers in all areas of mathematics and statistics.

Discrete Mathematics and Its Applications, 7thEd, Kenneth H. Rosen, 2012

Discrete Mathematics and Its Applications,

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Author: The McGraw-Hill Companies, Inc

Publisher: Bukupedia

ISBN: N.A

Category: Mathematics

Page: 1071

View: 565

[1] The satisfiability problem is addressed in greater depth, with Sudoku modeled in terms of satisfiability. [1] Hilbert’s Grand Hotel is used to help explain uncountability. [1] Proofs throughout the book have been made more accessible by adding steps and reasons behind these steps. [1] A template for proofs by mathematical induction has been added. [1] The step that applies the inductive hypothesis in mathematical induction proof is now explicitly noted. Algorithms [1] The pseudocode used in the book has been updated. [1] Explicit coverage of algorithmic paradigms, including brute force, greedy algorithms, and dynamic programing, is now provided. [1] Useful rules for big-O estimates of logarithms, powers, and exponential functions have been added. Number Theory and Cryptography [1] Expanded coverage allows instructors to include just a little or a lot of number theory in their courses. [1] The relationship between the mod function and congruences has been explained more fully. [1] The sieve of Eratosthenes is now introduced earlier in the book. [1] Linear congruences and modular inverses are now covered in more detail. [1] Applications of number theory, including check digits and hash functions, are covered in great depth. [1] A new section on cryptography integrates previous coverage, and the notion of a cryptosystem has been introduced. [1] Cryptographic protocols, including digital signatures and key sharing, are now covered. x Preface Graph Theory [1] A structured introduction to graph theory applications has been added. [1] More coverage has been devoted to the notion of social networks. [1] Applications to the biological sciences and motivating applications for graph isomorphism and planarity have been added. [1] Matchings in bipartite graphs are now covered, including Hall’s theorem and its proof. [1] Coverage of vertex connectivity, edge connectivity, and n-connectedness has been added, providing more insight into the connectedness of graphs. Enrichment Material [1] Many biographies have been expanded and updated, and new biographies of Bellman, Bézout Bienyamé, Cardano, Catalan, Cocks, Cook, Dirac, Hall, Hilbert, Ore, and Tao have been added. [1] Historical information has been added throughout the text. [1] Numerous updates for latest discoveries have been made. Expanded Media [1] Extensive effort has been devoted to producing valuable web resources for this book. [1] Extra examples in key parts of the text have been provided on companion website. [1] Interactive algorithms have been developed, with tools for using them to explore topics and for classroom use. [1] A new online ancillary, The Virtual Discrete Mathematics Tutor, available in fall 2012, will help students overcome problems learning discrete mathematics. [1] A new homework delivery system, available in fall 2012, will provide automated homework for both numerical and conceptual exercises. [1] Student assessment modules are available for key concepts. [1] Powerpoint transparencies for instructor use have been developed. [1] Asupplement Exploring Discrete Mathematics has been developed, providing extensive support for using MapleTM or MathematicaTM in conjunction with the book. [1] An extensive collection of external web links is provided. Features of the Book ACCESSIBILITY This text has proved to be easily read and understood by beginning students. There are no mathematical prerequisites beyond college algebra for almost all the content of the text. Students needing extra help will find tools on the companion website for bringing their mathematical maturity up to the level of the text. The few places in the book where calculus is referred to are explicitly noted. Most students should easily understand the pseudocode used in the text to express algorithms, regardless of whether they have formally studied programming languages. There is no formal computer science prerequisite. Each chapter begins at an easily understood and accessible level. Once basic mathematical concepts have been carefully developed, more difficult material and applications to other areas of study are presented. Preface xi FLEXIBILITY This text has been carefully designed for flexible use. The dependence of chapters on previous material has been minimized. Each chapter is divided into sections of approximately the same length, and each section is divided into subsections that form natural blocks of material for teaching. Instructors can easily pace their lectures using these blocks. WRITING STYLE The writing style in this book is direct and pragmatic. Precise mathematical language is used without excessive formalism and abstraction. Care has been taken to balance the mix of notation and words in mathematical statements. MATHEMATICAL RIGORAND PRECISION All definitions and theorems in this text are stated extremely carefully so that students will appreciate the precision of language and rigor needed in mathematics. Proofs are motivated and developed slowly; their steps are all carefully justified. The axioms used in proofs and the basic properties that follow from them are explicitly described in an appendix, giving students a clear idea of what they can assume in a proof. Recursive definitions are explained and used extensively. WORKEDEXAMPLES Over 800 examples are used to illustrate concepts, relate different topics, and introduce applications. In most examples, a question is first posed, then its solution is presented with the appropriate amount of detail. APPLICATIONS The applications included in this text demonstrate the utility of discrete mathematics in the solution of real-world problems. This text includes applications to a wide variety of areas, including computer science, data networking, psychology, chemistry, engineering, linguistics, biology, business, and the Internet. ALGORITHMS Results in discrete mathematics are often expressed in terms of algorithms; hence, key algorithms are introduced in each chapter of the book. These algorithms are expressed in words and in an easily understood form of structured pseudocode, which is described and specified in Appendix 3. The computational complexity of the algorithms in the text is also analyzed at an elementary level. HISTORICAL INFORMATION The background of many topics is succinctly described in the text. Brief biographies of 83 mathematicians and computer scientists are included as footnotes. These biographies include information about the lives, careers, and accomplishments of these important contributors to discrete mathematics and images, when available, are displayed. In addition, numerous historical footnotes are included that supplement the historical information in the main body of the text. Efforts have been made to keep the book up-to-date by reflecting the latest discoveries. KEY TERMS AND RESULTS A list of key terms and results follows each chapter. The key terms include only the most important that students should learn, and not every term defined in the chapter. EXERCISES There are over 4000 exercises in the text, with many different types of questions posed. There is an ample supply of straightforward exercises that develop basic skills, a large number of intermediate exercises, and many challenging exercises. Exercises are stated clearly and unambiguously, and all are carefully graded for level of difficulty. Exercise sets contain special discussions that develop new concepts not covered in the text, enabling students to discover new ideas through their own work. Exercises that are somewhat more difficult than average are marked with a single star ∗; those that are much more challenging are marked with two stars ∗∗. Exercises whose solutions require calculus are explicitly noted. Exercises that develop results used in the text are clearly identified with the right pointing hand symbol . Answers or outlined solutions to all oddxii Preface numbered exercises are provided at the back of the text. The solutions include proofs in which most of the steps are clearly spelled out. REVIEW QUESTIONS A set of review questions is provided at the end of each chapter. These questions are designed to help students focus their study on the most important concepts and techniques of that chapter. To answer these questions students need to write long answers, rather than just perform calculations or give short replies. SUPPLEMENTARY EXERCISE SETS Each chapter is followed by a rich and varied set of supplementary exercises. These exercises are generally more difficult than those in the exercise sets following the sections. The supplementary exercises reinforce the concepts of the chapter and integrate different topics more effectively. COMPUTER PROJECTS Each chapter is followed by a set of computer projects. The approximately 150 computer projects tie together what students may have learned in computing and in discrete mathematics. Computer projects that are more difficult than average, from both a mathematical and a programming point of view, are marked with a star, and those that are extremely challenging are marked with two stars. COMPUTATIONS AND EXPLORATIONS A set of computations and explorations is included at the conclusion of each chapter. These exercises (approximately 120 in total) are designed to be completed using existing software tools, such as programs that students or instructors have written or mathematical computation packages such as MapleTM or MathematicaTM. Many of these exercises give students the opportunity to uncover new facts and ideas through computation. (Some of these exercises are discussed in the Exploring Discrete Mathematics companion workbooks available online.) WRITING PROJECTS Each chapter is followed by a set of writing projects. To do these projects students need to consult the mathematical literature. Some of these projects are historical in nature and may involve looking up original sources. Others are designed to serve as gateways to new topics and ideas. All are designed to expose students to ideas not covered in depth in the text. These projects tie mathematical concepts together with the writing process and help expose students to possible areas for future study. (Suggested references for these projects can be found online or in the printed Student’s Solutions Guide.) APPENDIXES There are three appendixes to the text. The first introduces axioms for real numbers and the positive integers, and illustrates howfacts are proved directly from these axioms. The second covers exponential and logarithmic functions, reviewing some basic material used heavily in the course. The third specifies the pseudocode used to describe algorithms in this text. SUGGESTED READINGS A list of suggested readings for the overall book and for each chapter is provided after the appendices. These suggested readings include books at or below the level of this text, more difficult books, expository articles, and articles in which discoveries in discrete mathematics were originally published. Some of these publications are classics, published many years ago, while others have been published in the last few years. How to Use This Book This text has been carefully written and constructed to support discrete mathematics courses at several levels and with differing foci. The following table identifies the core and optional sections. An introductory one-term course in discrete mathematics at the sophomore level can be based on the core sections of the text, with other sections covered at the discretion of the Preface xiii instructor. A two-term introductory course can include all the optional mathematics sections in addition to the core sections. A course with a strong computer science emphasis can be taught by covering some or all of the optional computer science sections. Instructors can find sample syllabi for a wide range of discrete mathematics courses and teaching suggestions for using each section of the text can be found in the Instructor’s Resource Guide available on the website for this book. Chapter Core Optional CS Optional Math 1 1.1–1.8 (as needed) 2 2.1–2.4, 2.6 (as needed) 2.5 3 3.1–3.3 (as needed) 4 4.1–4.4 (as needed) 4.5, 4.6 5 5.1–5.3 5.4, 5.5 6 6.1–6.3 6.6 6.4, 6.5 7 7.1 7.4 7.2, 7.3 8 8.1, 8.5 8.3 8.2, 8.4, 8.6 9 9.1, 9.3, 9.5 9.2 9.4, 9.6 10 10.1–10.5 10.6–10.8 11 11.1 11.2, 11.3 11.4, 11.5 12 12.1–12.4 13 13.1–13.5 Instructors using this book can adjust the level of difficulty of their course by choosing either to cover or to omit the more challenging examples at the end of sections, as well as the more challenging exercises. The chapter dependency chart shown here displays the strong dependencies.A star indicates that only relevant sections of the chapter are needed for study of a later chapter.Weak dependencies have been ignored. More details can be found in the Instructor Resource Guide. Chapter 9* Chapter 10* Chapter 11 Chapter 13 Chapter 12 Chapter 2* Chapter 7 Chapter 8 Chapter 6* Chapter 3* Chapter 1 Chapter 4* Chapter 5* Ancillaries STUDENT’S SOLUTIONS GUIDE This student manual, available separately, contains full solutions to all odd-numbered problems in the exercise sets. These solutions explain why a particular method is used and why it works. For some exercises, one or two other possible approaches are described to show that a problem can be solved in several different ways. Suggested references for the writing projects found at the end of each chapter are also included in this volume. Also included are a guide to writing proofs and an extensive description of common xiv Preface mistakes students make in discrete mathematics, plus sample tests and a sample crib sheet for each chapter designed to help students prepare for exams. (ISBN-10: 0-07-735350-1) (ISBN-13: 978-0-07-735350-6) INSTRUCTOR’S RESOURCE GUIDE This manual, available on the website and in printed form by request for instructors, contains full solutions to even-numbered exercises in the text. Suggestions on how to teach the material in each chapter of the book are provided, including the points to stress in each section and how to put the material into perspective. It also offers sample tests for each chapter and a test bank containing over 1500 exam questions to choose from. Answers to all sample tests and test bank questions are included. Finally, several sample syllabi are presented for courses with differing emphases and student ability levels. (ISBN-10: 0-07-735349-8) (ISBN-13: 978-0-07-735349-0) Acknowledgments I would like to thank the many instructors and students at a variety of schools who have used this book and provided me with their valuable feedback and helpful suggestions. Their input has made this a much better book than it would have been otherwise. I especially want to thank Jerrold Grossman, Jean-Claude Evard, and Georgia Mederer for their technical reviews of the seventh edition and their “eagle eyes,” which have helped ensure the accuracy of this book. I also appreciate the help provided by all those who have submitted comments via the website. I thank the reviewers of this seventh and the six previous editions. These reviewers have provided much helpful criticism and encouragement to me. I hope this edition lives up to their high expectations. Reviewers for the Seventh Edition Philip Barry University of Minnesota, Minneapolis Miklos Bona University of Florida Kirby Brown Queens College John Carter University of Toronto Narendra Chaudhari Nanyang Technological University Allan Cochran University of Arkansas Daniel Cunningham Buffalo State College George Davis Georgia State University Andrzej Derdzinski The Ohio State University Ronald Dotzel University of Missouri-St. Louis T.J. Duda Columbus State Community College Bruce Elenbogen University of Michigan, Dearborn Norma Elias Purdue University, Calumet-Hammond Herbert Enderton University of California, Los Angeles Anthony Evans Wright State University Kim Factor Marquette University Margaret Fleck University of Illinois, Champaign Peter Gillespie Fayetteville State University Johannes Hattingh Georgia State University Ken Holladay University of New Orleans Jerry Ianni LaGuardia Community College Ravi Janardan University of Minnesota, Minneapolis Norliza Katuk University of Utara Malaysia William Klostermeyer University of North Florida Przemo Kranz University of Mississippi Jaromy Kuhl University of West Florida Loredana Lanzani University of Arkansas, Fayetteville Steven Leonhardi Winona State University Xu Liutong Beijing University of Posts and Telecommunications Vladimir Logvinenko De Anza Community College Preface xv Darrell Minor Columbus State Community College Keith Olson Utah Valley University Yongyuth Permpoontanalarp King Mongkut’s University of Technology, Thonburi Galin Piatniskaia University of Missouri, St. Louis Stefan Robila Montclair State University Chris Rodger Auburn University Sukhit Singh Texas State University, San Marcos David Snyder Texas State University, San Marcos Wasin So San Jose State University Bogdan Suceava California State University, Fullerton Christopher Swanson Ashland University Bon Sy Queens College MatthewWalsh Indiana-Purdue University, Fort Wayne GideonWeinstein Western Governors University DavidWilczynski University of Southern California I would like to thank Bill Stenquist, Executive Editor, for his advocacy, enthusiasm, and support. His assistance with this edition has been essential. I would also like to thank the original editor,WayneYuhasz, whose insights and skills helped ensure the book’s success, as well as all the many other previous editors of this book. I want to express my appreciation to the staff of RPK Editorial Services for their valuable work on this edition, including Rose Kernan, who served as both the developmental editor and the production editor, and the other members of the RPK team, Fred Dahl, Martha McMaster, ErinWagner, Harlan James, and Shelly Gerger-Knecthl. I thank Paul Mailhot of PreTeX, Inc., the compositor, for the tremendous amount to work he devoted to producing this edition, and for his intimate knowledge of LaTeX. Thanks also to Danny Meldung of Photo Affairs, Inc., who was resourceful obtaining images for the new biographical footnotes. The accuracy and quality of this new edition owe much to Jerry Grossman and Jean-Claude Evard, who checked the entire manuscript for technical accuracy and Georgia Mederer, who checked the accuracy of the answers at the end of the book and the solutions in the Student’s Solutions Guide and Instructor’s Resource Guide. As usual, I cannot thank Jerry Grossman enough for all his work authoring these two essential ancillaries. I would also express my appreciation the Science, Engineering, and Mathematics (SEM) Division of McGraw-Hill Higher Education for their valuable support for this new edition and the associated media content. In particular, thanks go to Kurt Strand: President, SEM, McGraw- Hill Higher Education, Marty Lange: Editor-in-Chief, SEM, Michael Lange: Editorial Director, Raghothaman Srinivasan: Global Publisher, Bill Stenquist: Executive Editor, Curt Reynolds: Executive Marketing Manager, Robin A. Reed: Project Manager, Sandy Ludovissey: Buyer, Lorraine Buczek: In-house Developmental Editor, Brenda Rowles: Design Coordinator, Carrie K. Burger: Lead Photo Research Coordinator, and Tammy Juran: Media Project Manager. Kenneth H. Rosen.