Jan 19, 2004 This approachable text studies discrete objects and the relationsips that bind them. It helps students understand and apply the power of discrete math to digital computer systems and other modern applications. It provides excellent preparation for courses in linear algebra, number theory, and modern/abstract algebra and for computer science courses in data structures, algorithms. It helps students understand and apply the power of discrete math to digital computer systems and other modern applications. It provides excellent preparation for courses in linear algebra, number theory, and modern/abstract algebra and for computer science courses in data structures, algorithms, programming languages, compilers, databases,.
TRANSCRIPT
PrefaceA journey of a thousand miles begins with a single step.- - A Chinese proverb
eople often ask: What is discrete mathematics? It's the mathematics of discrete (distinct and disconnected) objects. In other words, it is the study of discrete objects and relationships that bind them. The geometric representations of discrete objects have gaps in them. For example, integers are discrete objects, therefore (elementary) number theory, for instance, is part of discrete mathematics; so are linear algebra and abstract algebra. On the other hand, calculus deals with sets of connected (without any gaps) objects. The set of real numbers and the set of points on a plane are two such sets; they have continuous pictorial representations. Therefore, calculus does not belong to discrete mathematics, but to continuous mathematics. However, calculus is relevant in the study of discrete mathematics. The sets in discrete mathematics are often finite or countable, whereas those in continuous mathematics are often uncountable. Interestingly, an analogous situation exists in the field of computers. Just as mathematics can be divided into discrete and continuous mathematics, computers can be divided into digital and analog. Digital computers process the discrete objects 0 and 1, whereas analog computers process continuous d a t a ~ t h a t is, data obtained through measurement. Thus the terms discrete and continuous are analogous to the terms digital and analog, respectively. The advent of modern digital computers has increased the need for understanding discrete mathematics. The tools and techniques of discrete mathematics enable us to appreciate the power and beauty of mathematics in designing problem-solving strategies in everyday life, especially in computer science, and to communicate with ease in the language of discrete mathematics.
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The Realization of a Dream
This book is the fruit of many years of many dreams; it is the end-product of my fascination for the myriad applications of discrete mathematics to a variety of courses, such as Data Structures, Analysis of Algorithms, Programming Languages, Theory of Compilers, and Databases. Data structures and Discrete Mathematics compliment each other. The information in this book is applicable to quite a few areas in mathematics; discretexiii
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Prefacemathematics is also an excellent preparation for number theory and abstract
algebra.A logically conceived, self-contained, well-organized, and a user-friendly book, it is suitable for students and amateurs as well; so the language employed is, hopefully, fairly simple and accessible. Although the book features a well-balanced mix of conversational and formal writing style, mathematical rigor has not been sacrificed. Also great care has been taken to be attentive to even minute details.
AudienceThe book has been designed for students in computer science, electrical engineering, and mathematics as a one- or two-semester course in discrete mathematics at the sophomore/junior level. Several earlier versions of the text were class-tested at two different institutions, with positive responses from students.
PrerequisitesNo formal prerequisites are needed to enjoy the material or to employ its power, except a very strong background in college algebra. A good background in pre-calculus mathematics is desirable, but not essential. Perhaps the most important requirement is a bit of sophisticated mathematical maturity: a combination of patience, logical and analytical thinking, motivation, systematism, decision-making, and the willingness to persevere through failure until success is achieved. Although no programming background is required to enjoy the discrete mathematics, knowledge of a structured programming language, such as Java or C + +, can make the study of discrete mathematics more rewarding.
CoverageThe text contains in-depth coverage of all major topics proposed by professional associations for a discrete mathematics course. It emphasizes problem-solving techniques, pattern recognition, conjecturing, induction, applications of varying nature, proof techniques, algorithm development, algorithm correctness, and numeric computations. Recursion, a powerful problem-solving strategy, is used heavily in both mathematics and computer science. Initially, for some students, it can be a bitter-sweet and demanding experience, so the strategy is presented with great care to help amateurs feel at home with this fascinating and frequently used technique for program development. This book also includes discussions on Fibonacci and Lucas numbers, Fermat numbers, and figurate numbers and their geometric representations, all excellent tools for exploring and understanding recursion.
Preface
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A sufficient amount of theory is included for those who enjoy the beauty in the development of the subject, and a wealth of applications as well for those who enjoy the power of problem-solving techniques. Hopefully, the student will benefit from the nice balance between theory and applications. Optional sections in the book are identified with an asterisk (.) in the left margin. Most of these sections deal with interesting applications or discussions. They can be omitted without negatively affecting the logical development of the topic. However, students are strongly encouraged to pursue the optional sections to maximize their learning.
Historical Anecdotes and BiographiesBiographical sketches of about 60 mathematicians and computer scientists who have played a significant role in the development of the field are threaded into the text. Hopefully, they provide a h u m a n dimension and attach a h u m a n face to major discoveries. A biographical index, keyed to page, appears on the inside of the back cover for easy access.
Examples and ExercisesEach section in the book contains a generous selection of carefully tailored examples to clarify and illuminate various concepts and facts. The backbone of the book is the 560 examples worked out in detail for easy understanding. Every section ends with a large collection of carefully prepared and wellgraded exercises (more than 3700 in total), including thought-provoking true-false questions. Some exercises enhance routine computational skills; some reinforce facts, formulas, and techniques; and some require mastery of various proof techniques coupled with algebraic manipulation. Often exercises of the latter category require a mathematically sophisticated mind and hence are meant to challenge the mathematically curious. Most of the exercise sets contain optional exercises, identified by the letter o in the left margin. These are intended for more mathematically sophisticated students. Exercises marked with one asterisk (.) are slightly more advanced than the ones that precede them. Double-starred (**) exercises are more challenging than the single-starred; they require a higher level of mathematical maturity. Exercises identified with the letter c in the left margin require a calculus background; they can be omitted by those with no or minimal calculus. Answers or partial solutions to all odd-numbered exercises are given at the end of the book.
FoundationTheorems are the backbones of mathematics. Consequently, this book contains the various proof techniques, explained and illustrated in detail.
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PrefaceThey provide a strong foundation in problem-solving techniques, algorithmic approach, verification and analysis of algorithms, as well as in every discrete mathematics topic needed to pursue computer science courses such as Data Structures, Analysis of Algorithms, Programming Languages, Theory of Compilers, Databases, and Theory of Computation.
ProofsMost of the concepts, definitions, and theorems in the book are illustrated with appropriate examples. Proofs shed additional light on the topic and enable students to sharpen their problem-solving skills. The various proof techniques appear throughout the text.
ApplicationsNumerous current and relevant applications are woven into the text, taken from computer science, chemistry, genetics, sports, coding theory, banking, casino games, electronics, decision-making, and gambling. They enhance understanding and show the relevance of discrete mathematics to everyday life. A detailed index of applications, keyed to pages, is given at the end of the book. Algorithms Clearly written algorithms are presented throughout the text as problemsolving tools. Some standard algorithms used in computer science are developed in a straightforward fashion; they are analyzed and proved to enhance problem-solving techniques. The computational complexities of a number of standard algorithms are investigated for comparison. Algorithms are written in a simple-to-understand pseudocode that can easily be translated into any programming language. In this pseudocode: 9 Explanatory comments are enclosed within the delimeters (* and *). 9 The body of the algorithm begins with a B e g i n and ends in an E n d ; they serve as the outermost parentheses. 9 Every compound statement begins with a b e g i n and ends in an end; again, they serve as parentheses. In particular, for easy readability, a while (for) loop with a compound statement ends in e n d w h i l e (endfor).
Chapter SummariesEach chapter ends with a summary of important vocabulary, formulas, and properties developed in the chapter. All the terms are keyed to the text pages for easy reference and a quick review.
Preface Review and Supplementary Exercises
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Each chapter summary is followed by an extensive set of well-constructed review exercises. Used along with the summary, these provide a comprehensive review of the chapter. Chapter-end supplementary exercises provide additional challenging opportunities for the mathematically sophisticated and curious-minded for further experimentation and exploration. The book contains about 950 review and supplementary exercises.
Computer AssignmentsOver 150 relevant computer assignments are given at the end of chapters. They provide hands-on experience with concepts and an opportunity to enhance programming skills. A computer algebra system, such as Maple, Derive, or Mathematica, or a programming language of choice can be used.
Exploratory Writing ProjectsEach chapter contains a set of well-conceived writing projects, for a total of about 600. These expository projects allow students to explore areas not pursued in the book, as well as to enhance research techniques and to practice writing skills. They can lead to original research, and can be assigned as group projects in a real world environment. For convenience, a comprehensive list of references for the writing projects, compiled from various sources, is provided in the S t u d e n t ' sSolutions Manual.
Enrichment ReadingsEach chapter ends with a list of selected references for further exploration and enrichment. Most expand the themes studied in this book.
Numbering SystemA concise numbering system is used to label each item, where an item can be an algorithm, figure, example, exercises, section, table, or theorem. Item m . n refers to item n in Chapter 'm'. For example, Section 3.4 is Section 4 in Chapter 3.
Special SymbolsColored boxes are used to highlight items that may need special attention. The letter o in the left margin of an exercise indicates that it is optional, whereas a c indicates that it requires the knowledge of calculus. Besides, every theorem is easily identifiable, and the end of every proof and example
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Prefaceis marked with a solid square (l l). An asterisk (.) next to an exercise indicates that it is challenging, whereas a double-star (**) indicates that it is even more challenging. While ' - ' stands for equality, the closely related symbol '~' means is approximately equal to:0 C
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optional exercises requires a knowledge of calculus end of a proof or a solution a challenging exercise a more challenging exercise is equal to is approximately equal to
AbbreviationsFor the sake of brevity, four useful abbreviations are used throughout the text: LHS, RHS, PMI, and IH: LHS RHS PMI IH Left-Hand Side Right-Hand Side Principle of Mathematical Induction Inductive Hypothesis
Symbols IndexAn index of symbols used in the text and the page numbers where they occur can be found inside the front and back covers.
Web LinksThe World Wide Web can be a useful resource for collecting information about the various topics and algorithms. Web links also provide biographies and discuss the discoveries of major mathematical contributors. Some Web sites for specific topics are listed in the Appendix.
Student's Solutions Manual The Student's Solutions Manual contains detailed solutions of all oddnumbered exercises. It also includes suggestions for studying mathematics, and for preparing to take an math exam. The Manual also contains a comprehensive list of references for the various writing projects and assignments.
Preface Instructor's Manual
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The Instructor's Manual contains detailed solutions to all even-numbered exercises, two sample tests and their keys for each chapter, and two sample final examinations and their keys.Acknowledgments
A number of people, including many students, have played a major role in substantially improving the quality of the manuscript through its development. I am truly grateful to every one of them for their unfailing encouragement, cooperation, and support. To begin with, I am sincerely indebted to the following reviewers for their unblemished enthusiasm and constructive suggestions: Gerald Alexanderson Stephen Brick Neil Calkin Andre Chapuis Luis E. Cuellar H. K. Dai Michael Daven Henry Etlinger Jerrold R. Griggs John Harding Nan Jiang Warren McGovern Tim O'Neil Michael O'Sullivan Stanley Selkow Santa Clara University University of South Alabama Clemson University Indiana University McNeese State University Oklahoma State University Mt. St. Mary College Rochester Institute of Technology University of South Carolina New Mexico State University University of South Dakota Bowling Green State University University of Notre Dame San Diego State University Worcester Polytechnic Institute
Thanks also go to Henry Etlinger of Rochester Institute of Technology and Jerrold R. Griggs of the University of South Carolina for reading the entire manuscript for accuracy; to Michael Dillencourt of the University of California at Irvine, and Thomas E. Moore of Bridgewater State College for preparing the solutions to the exercises; and to Margarite Roumas for her excellent editorial assistance. My sincere thanks also go to Senior Editor, Barbara Holland, Production Editor, Marcy Barnes-Henrie, Copy Editor, Kristin Landon, and Associate Editor, Thomas Singer for their devotion, cooperation, promptness, and patience, and for their unwavering support for the project. Finally, I must accept responsibility...
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Discrete Mathematics With Applications by Thomas Koshy
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Discrete Mathematics With Applications
Hari added it May 01, Meghal Chhabria marked it as to-read Oct 07, Sumaiya marked it as to-read Feb 21, Common terms and phrases adjacency matrix algorithm alphabet Assume P k begin bijective binary tree boolean algebra boolean expression boolean function column combinatorial circuit compute connected graph contains defined definition denote the number Determine DFSA diagram digit digraph ny elements equivalence relation Eulerian Eulerian path Evaluate exactly example illustrates Exercises Fibonacci number Find the number finite set formula function f given grammar Hamiltonian induction input integer Karnaugh map language logic loop mathematician mathematics minterms operations output pair path permutations kosh principle planar poset positive integer proof proposition Prove real number recurrence relation recursive Section sequence shows SOLUTION Solve spanning tree string sublist subsets subtree Suppose surjective symbols symmetric Theorem total number transitive true truth table truth value variables verify vertex vertices weight words wxyz yz yz.
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