基本信息
- 原书名：Discrete Mathematics and Its Applications
- 原出版社： McGraw-Hill Higher Education
- 作者： Kenneth H. Rosen
- 丛书名： 经典原版书库
- 出版社：机械工业出版社
- ISBN：9787111313298
- 上架时间：2010-11-3
- 出版日期：2010 年10月
- 开本：16开
- 页码：441
- 版次：6-1
- 所属分类：数学 > 代数，数论及组合理论 > 离散数学
教材
【插图】
编辑推荐
该书是介绍离散数学理论和方法的经典教材，已经成为采用率最高的离散数学教材，仅在美国就被600多所高校用作教材，并获得了极大的成功。第6版在前5版的基础上做了大量的改进，使其成为更有效的教学工具。
内容简介
目录
Adapter\'s Forword<BR>Preface<BR>To the Student<BR>LIST OF SYMBOLS<BR>Chapter 1 The Foundations: Logic and Proofs<BR>1.1 Propositional Logic<BR>1.2 Propositional Equivalences<BR>1.3 Predicates and Quantifiers<BR>1.4 Nested Quantifiers<BR>1.5 Rules of Inference<BR>1.6 Introduction to Proofs<BR>1.7 Proof Methods and Strategy<BR>End-of-Chapter Material<BR>Chapter 2 Basic Structures: Sets, Functions, Sequences, and Sums<BR>2.1 Sets<BR>2.2 Set Operations<BR>2.3 Functions<BR>2.4 Sequences and Summations<BR>End-of-Chapter Material<BR>Chapter3 Counting<BR>3.1 The Basics of Counting<BR>3.2 The Pigeonhole Principle<BR>3.3 Permutations and Combinations<BR>3.4 Binomial Coefficients<BR>3.5 Generalized Permutations and Combinations<BR>3.6 Generating Permutations and Combinations<BR>End-of-Chapter Material<BR>Chapter 4 Advanced Counting Techniques<BR>4.1 Recurrence Relations<BR>4.2 Solving Linear Recurrence Relations<BR>4.3 Divide-and-Conquer Algorithms and Recurrence Relations<BR>4.4 Generating Functions<BR>4.5 Inclusion-Exclusion<BR>4.6 Applications of Inclusion-Exclusion<BR>End-of-Chapter Material<BR>Chapter 5 Relations<BR>5.1 Relations and Their Properties<BR>5.2 n-ary Relations and Their Applications<BR>5.3 Representing Relations<BR>5.4 Closures of Relations<BR>5.5 Equivalence Relations<BR>5.6 Partial Orderings<BR>End-of-Chapter Material<BR>Chapter 6 Graphs<BR>6.1 Graphs and Graph Models<BR>6.2 Graph Terminology and Special Types of Graphs<BR>6.3 Representing Graphs and Graph Isomorphism<BR>6.4 Connectivity<BR>6.5 Euler and Hamilton Paths<BR>6.6 Shortest-Path Problems<BR>6.7 Planar Graphs<BR>6.8 Graph Coloring<BR>End-of-Chapter Material<BR>Chapter 7 Trees<BR>7.1 Introduction to Trees<BR>7.2 Applications of Trees<BR>7.3 Tree Traversal<BR>7.4 Spanning Trees<BR>7.5 Minimum Spanning Trees<BR>End-of-Chapter Material<BR>Answers to Exercises
前言
In writing this book, I was guided by my long-standing experience and interest in teaching discrete mathematics. For the student, my purpose was to present material in a precise, readable manner, with the concepts and techniques of discrete mathematics clearly presented and demonstrated. My goal was to show the relevance and practicality of discrete mathematics to students, who are often skeptical. I wanted to give students studying computer science all of the mathematical foundations they need for their future studies. I wanted to give mathematics students an understanding of important mathematical concepts together with a sense of why these concepts are important for applications. And most importantly, I wanted to accomplish these goals without watering down the material.
For the instructor, my purpose was to design a .exible, comprehensive teaching tool using proven pedagogical techniques in mathematics. I wanted to provide instructors with a package of materials that they could use to teach discrete mathematics effectively and ef.ciently in the most appropriate manner for their particular set of students. I hope that I have achieved these goals.
I have been extremely grati.ed by the tremendous success of this text. The many improve-ments in the sixth edition have been made possible by the feedback and suggestions of a large number of instructors and students at many of the more than 600 schools where this book has been successfully used. There are many enhancements in this edition. The companion web-site has been substantially enhanced and more closely integrated with the text, providing helpful material to make it easier for students and instructors to achieve their goals.
This text is designed for a one-or two-term introductory discrete mathematics course taken by students in a wide variety of majors, including mathematics, computer science, and engineer-ing. College algebra is the only explicit prerequisite, although a certain degree of mathematical maturity is needed to study discrete mathematics in a meaningful way.
Goals of a Discrete Mathematics Course
A discrete mathematics course has more than one purpose. Students should learn a particular set of mathematical facts and how to apply them; more importantly, such a course should teach students how to think logically and mathematically. To achieve these goals, this text stresses mathematical reasoning and the different ways problems are solved. Five important themes are interwoven in this text: mathematical reasoning, combinatorial analysis, discrete structures, al-gorithmic thinking, and applications and modeling. A successful discrete mathematics course should carefully blend and balance all .ve themes.
1. Mathematical Reasoning: Students must understand mathematical reasoning in order to read, comprehend, and construct mathematical arguments. This text starts with a discussion of mathematical logic, which serves as the foundation for the subsequent discussions of methods of proof. Both the science and the art of constructing proofs are addressed. The technique of mathematical induction is stressed through many different types of examples of such proofs and a careful explanation of why mathematical induction is a valid proof technique.
2. Combinatorial Analysis: An important problem-solving skill is the ability to count or enumerate objects. The discussion of enumeration in this book begins with the basic techniques of counting. The stress is on performing combinatorial analysis to solve counting problems and analyze algorithms, not on applying formulae.
3. Discrete Structures: A course in discrete mathematics should teach students how to work with discrete structures, which are the abstract mathematical structures used to represent discrete objects and relationships between these objects. These discrete structures include sets, permuta-tions, relations, graphs, trees, and .nite-state machines.
4. Algorithmic Thinking: Certain classes of problems are solved by the speci.cation of an algorithm. After an algorithm has been described, a computer program can be constructed im-plementing it. The mathematical portions of this activity, which include the speci.cation of the algorithm, the veri.cation that it works properly, and the analysis of the computer memory and time required to perform it, are all covered in this text. Algorithms are described using both English and an easily understood form of pseudocode.
5. Applications and Modeling: Discrete mathematics has applications to almost every con-ceivable area of study. There are many applications to computer science and data networking in this text, as well as applications to such diverse areas as chemistry, botany, zoology, linguistics, geography, business, and the Internet. These applications are natural and important uses of dis-crete mathematics and are not contrived. Modeling with discrete mathematics is an extremely important problem-solving skill, which students have the opportunity to develop by constructing their own models in some of the exercises.
Special Features
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 of this text. Stu-dents needing extra help will .nd tools on the MathZone 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 dif.cult material and applications to other areas of study are presented. Flexibility This text has been carefully designed for .exible 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 Rigor and Precision All de.nitions and theorems in this text are stated ex-tremely 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 justi.ed. 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 de.ni-tions are explained and used extensively. Worked Examples Examples are used to illustrate concepts, relate different topics, and intro-duce applications. In most examples, a question is .rst posed, then its solution is presented with the appropriate amount of detail. Applications The applications included in this text demonstrate the utility of discrete mathe-matics 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, linguis-tics, 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. The computational complexity of the algorithms in the text is also analyzed at an elementary level. 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, not every term de.ned in the chapter. Exercises There are 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 interme-diate exercises, and many challenging exercises. Exercises are stated clearly and unambiguously, and all are carefully graded for level of dif.culty. 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 dif.cult 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 identi.ed with the symbol . Answers or outlined solutions to 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. Computer Projects Each chapter is followed by a set of computer projects. The computer projects tie together what students may have learned in computing and in discrete mathematics. Computer projects that are more dif.cult than average, from both a mathematical and a program-ming 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 are designed to be completed using existing software tools, such as programs that students or instructors have written or mathematical computation packages such as Maple or Mathematica. 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 with Maple companion workbook 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.) Suggested Readings A list of suggested readings for each chapter is provided in a section at the end of the text. These suggested readings include books at or below the level of this text, more dif.cult 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 within the last few years.
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 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: 0-07-310779-4) (ISBN-13: 978-0-07-310779-0)
Instructor's Resource Guide This manual, available 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 1300 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 emphasis and student ability levels, and a complete section and exercise migration guide is included to help users of the .fth edition update their course materials to match the sixth edition.
For the instructor, my purpose was to design a .exible, comprehensive teaching tool using proven pedagogical techniques in mathematics. I wanted to provide instructors with a package of materials that they could use to teach discrete mathematics effectively and ef.ciently in the most appropriate manner for their particular set of students. I hope that I have achieved these goals.
I have been extremely grati.ed by the tremendous success of this text. The many improve-ments in the sixth edition have been made possible by the feedback and suggestions of a large number of instructors and students at many of the more than 600 schools where this book has been successfully used. There are many enhancements in this edition. The companion web-site has been substantially enhanced and more closely integrated with the text, providing helpful material to make it easier for students and instructors to achieve their goals.
This text is designed for a one-or two-term introductory discrete mathematics course taken by students in a wide variety of majors, including mathematics, computer science, and engineer-ing. College algebra is the only explicit prerequisite, although a certain degree of mathematical maturity is needed to study discrete mathematics in a meaningful way.
Goals of a Discrete Mathematics Course
A discrete mathematics course has more than one purpose. Students should learn a particular set of mathematical facts and how to apply them; more importantly, such a course should teach students how to think logically and mathematically. To achieve these goals, this text stresses mathematical reasoning and the different ways problems are solved. Five important themes are interwoven in this text: mathematical reasoning, combinatorial analysis, discrete structures, al-gorithmic thinking, and applications and modeling. A successful discrete mathematics course should carefully blend and balance all .ve themes.
1. Mathematical Reasoning: Students must understand mathematical reasoning in order to read, comprehend, and construct mathematical arguments. This text starts with a discussion of mathematical logic, which serves as the foundation for the subsequent discussions of methods of proof. Both the science and the art of constructing proofs are addressed. The technique of mathematical induction is stressed through many different types of examples of such proofs and a careful explanation of why mathematical induction is a valid proof technique.
2. Combinatorial Analysis: An important problem-solving skill is the ability to count or enumerate objects. The discussion of enumeration in this book begins with the basic techniques of counting. The stress is on performing combinatorial analysis to solve counting problems and analyze algorithms, not on applying formulae.
3. Discrete Structures: A course in discrete mathematics should teach students how to work with discrete structures, which are the abstract mathematical structures used to represent discrete objects and relationships between these objects. These discrete structures include sets, permuta-tions, relations, graphs, trees, and .nite-state machines.
4. Algorithmic Thinking: Certain classes of problems are solved by the speci.cation of an algorithm. After an algorithm has been described, a computer program can be constructed im-plementing it. The mathematical portions of this activity, which include the speci.cation of the algorithm, the veri.cation that it works properly, and the analysis of the computer memory and time required to perform it, are all covered in this text. Algorithms are described using both English and an easily understood form of pseudocode.
5. Applications and Modeling: Discrete mathematics has applications to almost every con-ceivable area of study. There are many applications to computer science and data networking in this text, as well as applications to such diverse areas as chemistry, botany, zoology, linguistics, geography, business, and the Internet. These applications are natural and important uses of dis-crete mathematics and are not contrived. Modeling with discrete mathematics is an extremely important problem-solving skill, which students have the opportunity to develop by constructing their own models in some of the exercises.
Special Features
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 of this text. Stu-dents needing extra help will .nd tools on the MathZone 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 dif.cult material and applications to other areas of study are presented. Flexibility This text has been carefully designed for .exible 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 Rigor and Precision All de.nitions and theorems in this text are stated ex-tremely 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 justi.ed. 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 de.ni-tions are explained and used extensively. Worked Examples Examples are used to illustrate concepts, relate different topics, and intro-duce applications. In most examples, a question is .rst posed, then its solution is presented with the appropriate amount of detail. Applications The applications included in this text demonstrate the utility of discrete mathe-matics 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, linguis-tics, 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. The computational complexity of the algorithms in the text is also analyzed at an elementary level. 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, not every term de.ned in the chapter. Exercises There are 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 interme-diate exercises, and many challenging exercises. Exercises are stated clearly and unambiguously, and all are carefully graded for level of dif.culty. 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 dif.cult 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 identi.ed with the symbol . Answers or outlined solutions to 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. Computer Projects Each chapter is followed by a set of computer projects. The computer projects tie together what students may have learned in computing and in discrete mathematics. Computer projects that are more dif.cult than average, from both a mathematical and a program-ming 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 are designed to be completed using existing software tools, such as programs that students or instructors have written or mathematical computation packages such as Maple or Mathematica. 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 with Maple companion workbook 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.) Suggested Readings A list of suggested readings for each chapter is provided in a section at the end of the text. These suggested readings include books at or below the level of this text, more dif.cult 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 within the last few years.
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 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: 0-07-310779-4) (ISBN-13: 978-0-07-310779-0)
Instructor's Resource Guide This manual, available 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 1300 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 emphasis and student ability levels, and a complete section and exercise migration guide is included to help users of the .fth edition update their course materials to match the sixth edition.
