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This text includes applications to a wide va- riety of areas, including computer science, data networking, psychology, chemistry, engineering, linguistics, biology, business, and the Internet. These algorithms are expressed in words and in an easily understood form of structured pseudocode, which is described and specified in Appendix A.

The computational complexity of the algorithms in the text is also analyzed at an elementary level. Briefbiographies of more than 65 mathematicians and computer scientists, accom- panied by photos or images, are included as footnotes. These biographies include information about the lives, careers, and accomplishments of these important contributors to discrete mathe- matics and images ofthese contributors 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. The key terms include only the most important that students should learn, not every term defined in the chapter. 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 whose solutions require calculus are explicitly noted. Answers or outlined solutions to all odd-numbered exercises are provided at the back of the text. The solutions include proofs in which most of the steps are clearly spelled out. These questions are designed to help students focus their study on the most important concepts IiI Preface and techniques of that chapter.

To answer these questions students need to write long answers, rather than just perfonn calculations or give short replies. 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. The approximately 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.

These exercises approximately 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 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.

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.

The first introduces axioms for real numbers and the integers, and illustrates how facts 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. 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 within 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-tenn 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 instructor.

A two-tenn 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. The dependence of chapters on earlier chapters is shown in the following chart.

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. Express each of these quantifi- cations in English. Express each of these quantifications in English. Let W x , y mean that student x has visited website y , where the domain for x consists of all students in your school and the domain for y consists of all websites.

Ex- press each of these statements by a simple English sen- tence. Veja mais.

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