"离散数学及应用电子版：Kenneth H. Rosen著作第八版全新展望"

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The electronic version of the eighth edition of "Discrete Mathematics and Its Applications" by Kenneth H. Rosen is a comprehensive and informative resource for students and professionals interested in the field of discrete mathematics. This edition covers a wide range of topics in discrete mathematics, including sets, relations, functions, graphs, trees, and combinatorics. One of the key strengths of this book is its clear and concise explanations of complex mathematical concepts. The author, Kenneth H. Rosen, has a talent for breaking down abstract ideas into understandable terms, making this book accessible to readers with varying levels of mathematical background. The book also includes numerous examples, exercises, and practice problems to help readers reinforce their understanding of the material. In addition to its thorough coverage of theoretical concepts, "Discrete Mathematics and Its Applications" also includes practical applications of discrete mathematics in various fields, such as computer science, cryptography, and operations research. By demonstrating how theoretical concepts can be applied in real-world scenarios, this book helps readers appreciate the relevance and importance of discrete mathematics in modern society. Overall, the electronic version of "Discrete Mathematics and Its Applications" is a valuable resource for anyone looking to deepen their understanding of discrete mathematics. Whether you are a student studying mathematics or a professional working in a related field, this book will provide you with the knowledge and skills needed to succeed in the field of discrete mathematics. With its comprehensive coverage, clear explanations, and practical examples, this book is sure to be a valuable addition to any mathematical library.

Preface xv

I also want to thank the many students who have provided suggestions and reported errata.

The students in the discrete mathematics courses I have taught at Monmouth University, both

undergraduate and graduate computer science students, have helped me improve the book in

many ways.

There are many people to thank at McGraw-Hill Higher Education, the publisher of this

book, as well as people who did the production work at Aptara. I would also like to thank the

original editor at Random House, Wayne Yuhasz, 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 deep appreciation to Nora Devlin, the Product Developer who has

gone far beyond her assigned duties to support the author. She has displayed many skills and

virtues working to solve a wide variety of problems that have arisen in the development of this

new edition.

I am also grateful to Peggy Selle, the Content Product Manager, who managed the produc-

tion process. She has kept the production on track and has helped resolve many issues that have

arise during this process. Thanks go to Sarita Yadav, Senior Product Manager of Aptara, and

her colleagues at Aptara, who worked diligently to ensure the production quality of this edition.

I also want to express my appreciation to the many others in 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, including

▶

Mike Ryan, VP, Portfolio & Learning Content, Higher Education

▶

Kathleen McMahon, Managing Director, Mathematics & Physical Sciences

▶

Caroline Celano, Director, Mathematics

▶

Alison Frederick, Marketing Manager

▶

Robin Reed, Lead Product Developer

▶

Sandy Ludovissey, Buyer

▶

Egzon Shaqiri, Designer

▶

Tammy Juran, Assessment Content Project Manager

▶

Cynthia Northrup, Director of Digital Content

▶

Ruth Czarnecki-Lichstein, Business Product Manager

▶

Megan Platt, Editorial Coordinator

▶

Lora Neyens and Jolynn Kilburg, Program Managers

▶

Lorraine Buczek, Content Licensing Specialist

Kenneth H. Rosen

Online Resources

xtensive eﬀort has been devoted to producing valuable web resources for this book. In-

structors should make a special eﬀort to explore these resources to identify those they feel

will help their students learn and explore discrete mathematics. These resources are available in

the Online Learning Center, which is available to all students and instructors, and the Connect

Site, designed for interactive instruction, which instructors can choose to use. To use Connect,

students purchase online access for a speciﬁc time period.

0.1

The Online Learning Center

The Online Learning Center (OLC), accessible at www.mhhe.com/rosen, includes an Informa-

tion Center,aStudent Site, and an Instructor Site. Key features of each area are described here.

0.1.1 The Information Center

The Information Center contains basic information about the book including the expanded table

of contents (including subsection heads), the preface, descriptions of the ancillaries, and a sam-

ple chapter. It also provides a link that can be used to submit errata reports and other feedback

about the book.

0.1.2 Student Site

The Student Site contains a wealth of resources available for student use, including the follow-

ing, tied into the text wherever the special icons displayed below are found in the text:

▶

Extra Examples You can ﬁnd a large number of additional examples on the site, cov-

Extra

Examples

ering all chapters of the book. These examples are concentrated in areas where students

often ask for additional material. Although most of these examples amplify the basic

concepts, more-challenging examples can also be found here. Many new extra examples

have been recently added for the eighth edition. Each icon in the book corresponds to

one or more extra examples on the website.

▶

Interactive Demonstration Applets These applets enable you to interactively explore

Demo

how important algorithms work, and are tied directly to material in the text with linkages

to examples and exercises. Additional resources are provided on how to use and apply

these applets.

▶

Self Assessments These interactive guides help you assess your understanding of 14

ssessment

key concepts, providing a question bank where each question includes a brief tutorial

followed by a multiple-choice question. If you select an incorrect answer, advice is pro-

vided to help you understand your error. Using these Self Assessments, you should be

able to diagnose your problems and ﬁnd appropriate help.

▶

Web Resources Guide This guide provides annotated links to hundreds of external

Links

websites containing relevant material such as historical and biographical information,

puzzles and problems, discussions, applets, programs, and more. These links are keyed

to the text by page number.

xvi

Online Resources xvii

Additional resources in the Student Site include:

▶

Exploring Discrete Mathematics This ancillary provides help for using a computer alge-

bra system to do a wide range of computations in discrete mathematics. Each chapter provides

a description of relevant functions in the computer algebra system and how they are used, pro-

grams to carry out computations in discrete mathematics, examples, and exercises that can be

worked using this computer algebra system. Two versions, Exploring Discrete Mathematics

with Maple

and Exploring Discrete Mathematics with Mathematica

, are available.

▶

Applications of Discrete Mathematics This ancillary contains 24 chapters—each with

its own set of exercises—presenting a wide variety of interesting and important applications

covering three general areas in discrete mathematics: discrete structures, combinatorics, and

graph theory. These applications are ideal for supplementing the text or for independent study.

▶

A Guide to Proof-Writing This guide provides additional help for writing proofs, a skill

that many students ﬁnd diﬃcult to master. By reading this guide at the beginning of the

course and periodically thereafter when proof writing is required, you will be rewarded as

your proof-writing ability grows. (Also available in the Student’s Solutions Guide.)

▶

Common Mistakes in Discrete Mathematics This guide includes a detailed list of com-

mon misconceptions that students of discrete mathematics often have and the kinds of errors

they tend to make. You are encouraged to review this list from time to time to help avoid

these common traps. (Also available in the Student’s Solutions Guide.)

▶

Advice on Writing Projects This guide oﬀers helpful hints and suggestions for the Writing

Projects in the text, including an extensive bibliography of helpful books and articles for

research, discussion of various resources available in print and online, tips on doing library

research, and suggestions on how to write well. (Also available in the Student’s Solutions

Guide.)

0.1.3 Instructor Site

This part of the website provides access to all of the resources on the Student Site, as well as

these resources for instructors:

▶

Suggested Syllabi Detailed course outlines are shown, oﬀering suggestions for

courses with diﬀerent emphases and diﬀerent student backgrounds and ability levels.

▶

Teaching Suggestions This guide contains detailed teaching suggestions for instruc-

tors, including chapter overviews for the entire text, detailed remarks on each section,

and comments on the exercise sets.

▶

Printable Tests Printable tests are oﬀered in TeX and Word format for every chapter,

and can be customized by instructors.

▶

PowerPoint Lecture Slides and PowerPoint Figures and Tables An extensive col-

lection of PowerPoint lecture notes for all chapters of the text are provided for instructor

use. In addition, images of all ﬁgures and tables from the text are provided as PowerPoint

slides.

0.1.4 Connect

A comprehensive online learning package has been developed in conjunction with the text. A

high-level description of this site will be provided here. Interested instructors and students can

ﬁnd out more about Connect from McGraw-Hill Higher Education. When instructors choose to

use this option, students in their classes must obtain access to Connect for this text, either by

purchasing a copy of the textbook that also includes access privileges or by purchasing access

only with the option of buying a loose-leaf version of the textbook.

xviii Online Resources

Instructors who adopt Connect have access to a full-featured course management system.

Course management capabilities are provided that allow instructors to create assignments, au-

tomatically assign and grade homework, quiz, and test questions from a bank of questions tied

directly to the text, create and edit their own questions, manage course announcements and due

dates, and track student progress.

Instructors can create their own assignments using Connect. They select the particular ex-

ercises from each section of the book that they want to assign. They can also assign chapters

from the SmartBook version of the text, which provides an adaptive learning environment. Stu-

dents have access to the interactive version of the textbook, the online homework exercises, and

SmartBook.

Interactive Textbook Students have access to an easy-to-use interactive version of the textbook

when they use Connect. The interactive site provides the full content of the text, as well as the

many extra resources that enrich the book. The resources include extra examples, interactive

demonstrations, and self-assessments.

Homework and Learning Management Solution An extensive learning management solution

has been developed that instructors can use to construct homework assignments. Approximately

800 online questions are available, including questions from every section of the text. These

questions are tied to the most commonly assigned exercises in the book.

These online questions have been constructed to support the same objectives as the corre-

sponding written homework questions. This challenge has been met by stretching the capabili-

ties of diﬀerent question types supported by the Connect platform.

SmartBook Connect also provides another extended online version of the text in the McGraw-

Hill SmartBook platform. The SmartBook version of the text includes a set of objectives for each

chapter of the text. A collection of questions, called probes, is provided to assess student un-

derstanding of each objective. Students are directed to the appropriate part of the text to review

the material needed for each of these objectives. SmartBook provides an adaptive learning en-

vironment; it selects probes for students based on their performance answering earlier probes.

Instructors can assign SmartBook as assignments or can have their students use SmartBook as

a learning tool.

To the Student

hat is discrete mathematics? Discrete mathematics is the part of mathematics devoted

to the study of discrete objects. (Here discrete means consisting of distinct or uncon-

nected elements.) The kinds of problems solved using discrete mathematics include:

▶

How many ways are there to choose a valid password on a computer system?

▶

What is the probability of winning a lottery?

▶

Is there a link between two computers in a network?

▶

How can I identify spam e-mail messages?

▶

How can I encrypt a message so that no unintended recipient can read it?

▶

What is the shortest path between two cities using a transportation system?

▶

How can a list of integers be sorted so that the integers are in increasing order?

▶

How many steps are required to do such a sorting?

▶

How can it be proved that a sorting algorithm correctly sorts a list?

▶

How can a circuit that adds two integers be designed?

▶

How many valid Internet addresses are there?

You will learn the discrete structures and techniques needed to solve problems such as these.

More generally, discrete mathematics is used whenever objects are counted, when relation-

ships between ﬁnite (or countable) sets are studied, and when processes involving a ﬁnite num-

ber of steps are analyzed. A key reason for the growth in the importance of discrete mathematics

is that information is stored and manipulated by computing machines in a discrete fashion.

WHY STUDY DISCRETE MATHEMATICS? There are several important reasons for

studying discrete mathematics. First, through this course you can develop your mathematical

maturity: that is, your ability to understand and create mathematical arguments. You will not

get very far in your studies in the mathematical sciences without these skills.

Second, discrete mathematics is the gateway to more advanced courses in all parts of the

mathematical sciences. Discrete mathematics provides the mathematical foundations for many

computer science courses, including data structures, algorithms, database theory, automata the-

ory, formal languages, compiler theory, computer security, and operating systems. Students ﬁnd

these courses much more diﬃcult when they have not had the appropriate mathematical foun-

dations from discrete mathematics. One student sent me an e-mail message saying that she used

the contents of this book in every computer science course she took!

Math courses based on the material studied in discrete mathematics include logic, set theory,

number theory, linear algebra, abstract algebra, combinatorics, graph theory, and probability

theory (the discrete part of the subject).

Also, discrete mathematics contains the necessary mathematical background for solv-

ing problems in operations research (including discrete optimization), chemistry, engineering,

biology, and so on. In the text, we will study applications to some of these areas.

Many students ﬁnd their introductory discrete mathematics course to be signiﬁcantly more

challenging than courses they have previously taken. One reason for this is that one of the pri-

mary goals of this course is to teach mathematical reasoning and problem solving, rather than a

discrete set of skills. The exercises in this book are designed to reﬂect this goal. Although there

are plenty of exercises in this text similar to those addressed in the examples, a large percentage

xix

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