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All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.

PREFACE My purpose in writing this book was to provide a clear, accessible treatment of discrete mathematics for students majoring or minoring in computer science, mathematics, mathematics education, and engineering.

The goal of the book is to lay the mathematical foundation for computer science courses such as data structures, algorithms, relational database theory, automata theory and formal languages, compiler design, and cryptography, and for mathematics courses such as linear and abstract algebra, combinatorics, probability, logic and set theory, and number theory.

By combining discussion of theory and practice, I have tried to show that mathematics has engaging and important applications as well as being interesting and beautiful in its own right. A good background in algebra is the only prerequisite; the course may be taken by students either before or after a course in calculus.

Previous editions of the book have been used successfully by students at hundreds of institutions in North and South America, Europe, the Middle East, Asia, and Australia. This book includes the topics recommended by those organizations and can be used effectively for either a one-semester or a two-semester course.

At one time, most of the topics in discrete mathematics were taught only to upperlevel undergraduates. The presentation was developed over a long period of experimentation during which my students were in many ways my teachers.

Many of the changes in this edition have resulted from continuing interaction with students. Themes of a Discrete Mathematics Course Discrete mathematics describes processes that consist of a sequence of individual steps. This contrasts with calculus, which describes processes that change in a continuous fashion. Whereas the ideas of calculus were fundamental to the science and technology of the industrial revolution, the ideas of discrete mathematics underlie the science and technology of the computer age.

Preface xv argument to derive new results from those already known to be true, and being able to work with symbolic representations as if they were concrete objects. Discrete Structures Discrete mathematical structures are the abstract structures that describe, categorize, and reveal the underlying relationships among discrete mathematical objects. Combinatorics and Discrete Probability Combinatorics is the mathematics of counting and arranging objects, and probability is the study of laws concerning the measurement of random or chance events.

Skill in using combinatorics and probability is needed in almost every discipline where mathematics is applied, from economics to biology, to computer science, to chemistry and physics, to business management. Designing an algorithm requires an understanding of the mathematics underlying the problem to be solved. Determining whether or not an algorithm is correct requires a sophisticated use of mathematical induction. Applications and Modeling Mathematical topics are best understood when they are seen in a variety of contexts and used to solve problems in a broad range of applied situations.

A goal of this book is to show students the extraordinary practical utility of some very abstract mathematical ideas. For many years I taught an intensively interactive transition-to-abstract-mathematics course to mathematics and computer science majors.

This experience showed me that while it is possible to teach the majority of students to Copyright Cengage Learning. It must also include enough concrete examples and exercises to enable students to develop the mental models needed to conceptualize more abstract problems.

The treatment of logic and proof in this book blends common sense and rigor in a way that explains the essentials, yet avoids overloading students with formal detail. Spiral Approach to Concept Development A number of concepts in this book appear in increasingly more sophisticated forms in successive chapters to help students develop the ability to deal effectively with increasing levels of abstraction. This approach builds in useful review and develops mathematical maturity in natural stages.

Support for the Student Students at colleges and universities inevitably have to learn a great deal on their own. Though it is often frustrating, learning to learn through selfstudy is a crucial step toward eventual success in a professional career. This format allows students to read the problem and skip immediately to the summary, if they wish, only going back to the discussion if they have trouble understanding the summary.

The format also saves time for students who are rereading the text in preparation for an examination. Marginal Notes and Test Yourself Questions Notes about issues of particular importance and cautionary comments to help students avoid common mistakes are included in the margins throughout the book. Questions designed to focus attention on the main ideas of each section are located between the text and the exercises. Exercises The book contains almost exercises. Solutions for Exercises To provide adequate feedback for students between class sessions, Appendix B contains a large number of complete solutions to exercises.

Students are strongly urged not to consult solutions until they have tried their best to answer the questions on their own. In addition, many problems, including some of the most challenging, have partial solutions or hints so that students can determine whether they are on the right track and make adjustments if necessary.

Copyright Cengage Learning. Preface xvii There are also plenty of exercises without solutions to help students learn to grapple with mathematical problems in a realistic environment. Reference Features Many students have written me to say that the book helped them succeed in their advanced courses.

Figures and tables are included where doing so would help readers to a better understanding. In most, a second color is used to highlight meaning. Support for the Instructor I have received a great deal of valuable feedback from instructors who have used previous editions of this book. Many aspects of the book have been improved through their suggestions. Exercises with solutions in the back of the book have numbers in blue, and those whose solutions are given in a separate Student Solutions Manual and Study Guide have numbers that are a multiple of three.

There are exercises of every type that are represented in this book that have no answer in either location to enable instructors to assign whatever mixture they prefer of exercises with and without answers. Flexible Sections Most sections are divided into subsections so that an instructor who is pressed for time can choose to cover certain subsections only and either omit the rest or leave them for the students to study on their own. The division into subsections also makes it easier for instructors to break up sections if they wish to spend more then one day on them.

Presentation of Proof Methods It is inevitable that the proofs and disproofs in this book will seem easy to instructors. In showing students how to discover and construct proofs and disproofs, I have tried to describe the kinds of approaches that mathematicians use when confronting challenging problems in their own research.

Instructors can sign up for access at www. In response to requests from some instructors, core material is now placed together in Chapter 1—8, with the chapter on recursion now joined to the chapter on induction. Chapters 9—12 were placed together at the end because, although many instructors cover one or more of them, there is considerable diversity in their choices, with some of the topics from these chapters being included in other courses.

Approximately new exercises have been added. Discussion of historical background and recent results has been expanded and the number of photographs of mathematicians and computer scientists whose contributions are discussed in the book has been increased. The descriptions of methods of proof have been made clearer. The subsections in the section on sequences have been reorganized. Increased attention has been given to structural induction.

The presentation in the section on modular arithmetic and cryptography has been streamlined.

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