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Published by: McGraw-Hill EducationPublished by: McGraw-Hill EducationPublished by: McGraw-Hill EducationPublished by: McGraw-Hill EducationPublished by: McGraw-Hill EducationPublished by: McGraw-Hill Education. Evidence: Published by: McGraw-Hill EducationPublished by: McGraw-Hill Education. The solutions are fully answered and mostly in English. The presentation of mathematical induction has been enhanced with further explanations and interesting examples. Probability: Coverage of discrete probability and aspects of probability theory have been expanded and placed in their own chapter (Chapter 5). This allows for more thorough coverage of probability while increasing the accessibility to counting techniques. Web site: This text continues its efforts to support and expand upon the textual material with enhanced coverage of topics on its web site. Features include a detailed Web Resources Guide, Additional Exercises and Answers, a Guide to Writing Proofs, and many other elements. The site will continue its expansion throughout the life of the edition, providing tools that students and instructors can use to assess student's understanding of key topics--including implications, quantifiers, proofs, mathematical introduction, and counting problems. Extensive additional examples in key areas and more detailed explanations of key examples in the text will also be provided. Features: Accessibility: This text has proven to be easily read and understood by beginning students with no mathematical background beyond college algebra. Once basic mathematical concepts have been carefully developed, more difficult material and applications to other areas of study are presented. There are a few places in the text where calculus is referred to and these places are carefully noted. Flexibility: This text has been carefully designed for flexible use. Each chapter is divided into sections of approximately the same length, and each section is divided into subsections that form natural blocks of material teaching. The dependence of chapters on previous material has been minimized. Many new examples have been added, as well as a focused expansion of key examples. These examples are used to illustrate concepts, relate different topics, and introduce applications. In the 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. Exercises: There are over 3000 exercises in the text. 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. Exercises sets contain special discussions, with exercises, that develop new concepts not covered in the text, permitting students to discover new ideas through their own work. Supplementary exercises follow each chapter and provide a rich and varied set of additional exercises. These exercises are generally more difficult than those in the section exercise sets and integrate different topics more effectively. Computer Projects: Each chapter is followed by a set of computer projects. These 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 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. 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. Historical Information: The background of many topics is succinctly described in the text. Brief biographies of more than 55 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. In addition, numerous historical footnotes are included that supplement the historical information in the main body of the text. These themes include mathematical reasoning, combinatorial analysis, discrete structures, algorithmic thinking, and enhanced problem-solving skills through modeling. Its intent is to demonstrate the relevance and practicality of discrete mathematics to all students. The Fifth Edition includes a more thorough and linear presentation of logic, proof types and proof writing, and mathematical reasoning. This enhanced coverage will provide students with a solid understanding of the material as it relates to their immediate field of study and other relevant subjects. The inclusion of applications and examples to key topics has been significantly addressed to add clarity to every subject. True to the Fourth Edition, the text-specific web site supplements the subject matter in meaningful ways, offering additional material for students and instructors. Discrete math is an active subject with new discoveries made every year. The continual growth and updates to the web site reflect the active nature of the topics being discussed.