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| discrete math strong induction: Discrete Mathematics Oscar Levin, 2016-08-16 This gentle introduction to discrete mathematics is written for first and second year math majors, especially those who intend to teach. The text began as a set of lecture notes for the discrete mathematics course at the University of Northern Colorado. This course serves both as an introduction to topics in discrete math and as the introduction to proof course for math majors. The course is usually taught with a large amount of student inquiry, and this text is written to help facilitate this. Four main topics are covered: counting, sequences, logic, and graph theory. Along the way proofs are introduced, including proofs by contradiction, proofs by induction, and combinatorial proofs. The book contains over 360 exercises, including 230 with solutions and 130 more involved problems suitable for homework. There are also Investigate! activities throughout the text to support active, inquiry based learning. While there are many fine discrete math textbooks available, this text has the following advantages: It is written to be used in an inquiry rich course. It is written to be used in a course for future math teachers. It is open source, with low cost print editions and free electronic editions. |
| discrete math strong induction: Handbook of Mathematical Induction David S. Gunderson, 2016-11-16 Handbook of Mathematical Induction: Theory and Applications shows how to find and write proofs via mathematical induction. This comprehensive book covers the theory, the structure of the written proof, all standard exercises, and hundreds of application examples from nearly every area of mathematics. In the first part of the book, the author discusses different inductive techniques, including well-ordered sets, basic mathematical induction, strong induction, double induction, infinite descent, downward induction, and several variants. He then introduces ordinals and cardinals, transfinite induction, the axiom of choice, Zorn's lemma, empirical induction, and fallacies and induction. He also explains how to write inductive proofs. The next part contains more than 750 exercises that highlight the levels of difficulty of an inductive proof, the variety of inductive techniques available, and the scope of results provable by mathematical induction. Each self-contained chapter in this section includes the necessary definitions, theory, and notation and covers a range of theorems and problems, from fundamental to very specialized. The final part presents either solutions or hints to the exercises. Slightly longer than what is found in most texts, these solutions provide complete details for every step of the problem-solving process. |
| discrete math strong induction: A Spiral Workbook for Discrete Mathematics Harris Kwong, 2015-11-06 A Spiral Workbook for Discrete Mathematics covers the standard topics in a sophomore-level course in discrete mathematics: logic, sets, proof techniques, basic number theory, functions,relations, and elementary combinatorics, with an emphasis on motivation. The text explains and claries the unwritten conventions in mathematics, and guides the students through a detailed discussion on how a proof is revised from its draft to a nal polished form. Hands-on exercises help students understand a concept soon after learning it. The text adopts a spiral approach: many topics are revisited multiple times, sometimes from a dierent perspective or at a higher level of complexity, in order to slowly develop the student's problem-solving and writing skills. |
| discrete math strong induction: Discrete Mathematics R. C. Penner, 1999 This book offers an introduction to mathematical proofs and to the fundamentals of modern mathematics. No real prerequisites are needed other than a suitable level of mathematical maturity. The text is divided into two parts, the first of which constitutes the core of a one-semester course covering proofs, predicate calculus, set theory, elementary number theory, relations, and functions, and the second of which applies this material to a more advanced study of selected topics in pure mathematics, applied mathematics, and computer science, specifically cardinality, combinatorics, finite-state automata, and graphs. In both parts, deeper and more interesting material is treated in optional sections, and the text has been kept flexible by allowing many different possible courses or emphases based upon different paths through the volume. |
| discrete math strong induction: A Logical Approach to Discrete Math David Gries, Fred B. Schneider, 2013-03-14 Here, the authors strive to change the way logic and discrete math are taught in computer science and mathematics: while many books treat logic simply as another topic of study, this one is unique in its willingness to go one step further. The book traets logic as a basic tool which may be applied in essentially every other area. |
| discrete math strong induction: Applied Discrete Structures Ken Levasseur, Al Doerr, 2012-02-25 ''In writing this book, care was taken to use language and examples that gradually wean students from a simpleminded mechanical approach and move them toward mathematical maturity. We also recognize that many students who hesitate to ask for help from an instructor need a readable text, and we have tried to anticipate the questions that go unasked. The wide range of examples in the text are meant to augment the favorite examples that most instructors have for teaching the topcs in discrete mathematics. To provide diagnostic help and encouragement, we have included solutions and/or hints to the odd-numbered exercises. These solutions include detailed answers whenever warranted and complete proofs, not just terse outlines of proofs. Our use of standard terminology and notation makes Applied Discrete Structures a valuable reference book for future courses. Although many advanced books have a short review of elementary topics, they cannot be complete. The text is divided into lecture-length sections, facilitating the organization of an instructor's presentation.Topics are presented in such a way that students' understanding can be monitored through thought-provoking exercises. The exercises require an understanding of the topics and how they are interrelated, not just a familiarity with the key words. An Instructor's Guide is available to any instructor who uses the text. It includes: Chapter-by-chapter comments on subtopics that emphasize the pitfalls to avoid; Suggested coverage times; Detailed solutions to most even-numbered exercises; Sample quizzes, exams, and final exams. This textbook has been used in classes at Casper College (WY), Grinnell College (IA), Luzurne Community College (PA), University of the Puget Sound (WA).''-- |
| discrete math strong induction: Fundamentals of Discrete Math for Computer Science Tom Jenkyns, Ben Stephenson, 2012-10-16 This textbook provides an engaging and motivational introduction to traditional topics in discrete mathematics, in a manner specifically designed to appeal to computer science students. The text empowers students to think critically, to be effective problem solvers, to integrate theory and practice, and to recognize the importance of abstraction. Clearly structured and interactive in nature, the book presents detailed walkthroughs of several algorithms, stimulating a conversation with the reader through informal commentary and provocative questions. Features: no university-level background in mathematics required; ideally structured for classroom-use and self-study, with modular chapters following ACM curriculum recommendations; describes mathematical processes in an algorithmic manner; contains examples and exercises throughout the text, and highlights the most important concepts in each section; selects examples that demonstrate a practical use for the concept in question. |
| discrete math strong induction: Proofs from THE BOOK Martin Aigner, Günter M. Ziegler, 2013-06-29 According to the great mathematician Paul Erdös, God maintains perfect mathematical proofs in The Book. This book presents the authors candidates for such perfect proofs, those which contain brilliant ideas, clever connections, and wonderful observations, bringing new insight and surprising perspectives to problems from number theory, geometry, analysis, combinatorics, and graph theory. As a result, this book will be fun reading for anyone with an interest in mathematics. |
| discrete math strong induction: Mathematics for Computer Science Eric Lehman, F. Thomson Leighton, Albert R. Meyer, 2017-03-08 This book covers elementary discrete mathematics for computer science and engineering. It emphasizes mathematical definitions and proofs as well as applicable methods. Topics include formal logic notation, proof methods; induction, well-ordering; sets, relations; elementary graph theory; integer congruences; asymptotic notation and growth of functions; permutations and combinations, counting principles; discrete probability. Further selected topics may also be covered, such as recursive definition and structural induction; state machines and invariants; recurrences; generating functions. |
| discrete math strong induction: The Tools of Mathematical Reasoning Tamara J. Lakins, 2016-09-08 This accessible textbook gives beginning undergraduate mathematics students a first exposure to introductory logic, proofs, sets, functions, number theory, relations, finite and infinite sets, and the foundations of analysis. The book provides students with a quick path to writing proofs and a practical collection of tools that they can use in later mathematics courses such as abstract algebra and analysis. The importance of the logical structure of a mathematical statement as a framework for finding a proof of that statement, and the proper use of variables, is an early and consistent theme used throughout the book. |
| discrete math strong induction: How to Prove It Daniel J. Velleman, 2006-01-16 Many students have trouble the first time they take a mathematics course in which proofs play a significant role. This new edition of Velleman's successful text will prepare students to make the transition from solving problems to proving theorems by teaching them the techniques needed to read and write proofs. The book begins with the basic concepts of logic and set theory, to familiarize students with the language of mathematics and how it is interpreted. These concepts are used as the basis for a step-by-step breakdown of the most important techniques used in constructing proofs. The author shows how complex proofs are built up from these smaller steps, using detailed 'scratch work' sections to expose the machinery of proofs about the natural numbers, relations, functions, and infinite sets. To give students the opportunity to construct their own proofs, this new edition contains over 200 new exercises, selected solutions, and an introduction to Proof Designer software. No background beyond standard high school mathematics is assumed. This book will be useful to anyone interested in logic and proofs: computer scientists, philosophers, linguists, and of course mathematicians. |
| discrete math strong induction: Resources for Teaching Discrete Mathematics Brian Hopkins, 2009 Hopkins collects the work of 35 instructors who share their innovations and insights about teaching discrete mathematics at the high school and college level. The book's 9 classroom-tested projects, including building a geodesic dome, come with student handouts, solutions, and notes for the instructor. The 11 history modules presented draw on original sources, such as Pascal's Treatise on the Arithmetical Triangle, allowing students to explore topics in their original contexts. Three articles address extensions of standard discrete mathematics content. Two other articles explore pedagogy specifically related to discrete mathematics courses: adapting a group discovery method to larger classes, and using logic in encouraging students to construct proofs. |
| discrete math strong induction: Journey into Discrete Mathematics Owen D. Byer, Deirdre L. Smeltzer, Kenneth L. Wantz, 2018-11-13 Journey into Discrete Mathematics is designed for use in a first course in mathematical abstraction for early-career undergraduate mathematics majors. The important ideas of discrete mathematics are included—logic, sets, proof writing, relations, counting, number theory, and graph theory—in a manner that promotes development of a mathematical mindset and prepares students for further study. While the treatment is designed to prepare the student reader for the mathematics major, the book remains attractive and appealing to students of computer science and other problem-solving disciplines. The exposition is exquisite and engaging and features detailed descriptions of the thought processes that one might follow to attack the problems of mathematics. The problems are appealing and vary widely in depth and difficulty. Careful design of the book helps the student reader learn to think like a mathematician through the exposition and the problems provided. Several of the core topics, including counting, number theory, and graph theory, are visited twice: once in an introductory manner and then again in a later chapter with more advanced concepts and with a deeper perspective. Owen D. Byer and Deirdre L. Smeltzer are both Professors of Mathematics at Eastern Mennonite University. Kenneth L. Wantz is Professor of Mathematics at Regent University. Collectively the authors have specialized expertise and research publications ranging widely over discrete mathematics and have over fifty semesters of combined experience in teaching this subject. |
| discrete math strong induction: Discrete Mathematics for Computer Science Gary Haggard, John Schlipf, Sue Whitesides, 2006 Master the fundamentals of discrete mathematics with DISCRETE MATHEMATICS FOR COMPUTER SCIENCE with Student Solutions Manual CD-ROM! An increasing number of computer scientists from diverse areas are using discrete mathematical structures to explain concepts and problems and this mathematics text shows you how to express precise ideas in clear mathematical language. Through a wealth of exercises and examples, you will learn how mastering discrete mathematics will help you develop important reasoning skills that will continue to be useful throughout your career. |
| discrete math strong induction: Book of Proof Richard H. Hammack, 2016-01-01 This book is an introduction to the language and standard proof methods of mathematics. It is a bridge from the computational courses (such as calculus or differential equations) that students typically encounter in their first year of college to a more abstract outlook. It lays a foundation for more theoretical courses such as topology, analysis and abstract algebra. Although it may be more meaningful to the student who has had some calculus, there is really no prerequisite other than a measure of mathematical maturity. |
| discrete math strong induction: Discrete Mathematics with Applications, Metric Edition Susanna Epp, 2019 DISCRETE MATHEMATICS WITH APPLICATIONS, 5th Edition, Metric Edition explains complex, abstract concepts with clarity and precision and provides a strong foundation for computer science and upper-level mathematics courses of the computer age. Author Susanna Epp presents not only the major themes of discrete mathematics, but also the reasoning that underlies mathematical thought. Students develop the ability to think abstractly as they study the ideas of logic and proof. While learning about such concepts as logic circuits and computer addition, algorithm analysis, recursive thinking, computability, automata, cryptography and combinatorics, students discover that the ideas of discrete mathematics underlie and are essential to today's science and technology. |
| discrete math strong induction: A Short Course in Discrete Mathematics Edward A. Bender, S. Gill Williamson, 2005-01-01 What sort of mathematics do I need for computer science? In response to this frequently asked question, a pair of professors at the University of California at San Diego created this text. Its sources are two of the university's most basic courses: Discrete Mathematics, and Mathematics for Algorithm and System Analysis. Intended for use by sophomores in the first of a two-quarter sequence, the text assumes some familiarity with calculus. Topics include Boolean functions and computer arithmetic; logic; number theory and cryptography; sets and functions; equivalence and order; and induction, sequences, and series. Multiple choice questions for review appear throughout the text. Original 2005 edition. Notation Index. Subject Index. |
| discrete math strong induction: Discrete Mathematics and Computing Malik Magdon-Ismail, 2019-12-14 This text is a semester course in the basic mathematical and theoretical foundations of computer science. Students who make heavy use of computing should learn these foundations well, setting a base for a follow-on course in algorithms. A solid theoretical and algorithmic foundation in computer science sets the stage for developing good programs, programs that work, always and efficiently.Each chapter is a lecture that has been taught as such. Part I starts with basic logic, proofs and discrete mathematics, including: induction, recursion, summation, asymptotics and number theory. We then continue with graphs, counting and combinatorics, and wrap up the coverage of discrete mathematics with discrete probability. Part II presents the blockbuster application of discrete mathematics: the digital computer and a theory of computing. The goal is to understand what a computer can and cannot do. We start small, with automata, and end big with Turing Machines.Our approach is Socratic. The reader is encouraged to participate actively in the learning process by doing the quizzes and exercises that are liberally sprinkled through the text. The pace and level is appropriate for readers with one year of training in programming and calculus (college sophomores). |
| discrete math strong induction: Discrete Mathematics Using a Computer John O'Donnell, Cordelia Hall, Rex Page, 2007-01-04 Computer science abounds with applications of discrete mathematics, yet s- dents of computer science often study discrete mathematics in the context of purely mathematical applications. They have to ?gure out for themselves how to apply the ideas of discrete mathematics to computing problems. It is not easy. Most students fail to experience broad success in this enterprise, which is not surprising, since many of the most important advances in science and engineeringhavebeen, precisely, applicationsofmathematicstospeci?cscience and engineering problems. Tobesure,mostdiscretemathtextbooksincorporatesomeaspectsapplying discrete math to computing, but it usually takes the form of asking students to write programs to compute the number of three-ball combinations there are in a set of ten balls or, at best, to implement a graph algorithm. Few texts ask students to use mathematical logic to analyze properties of digital circuits or computer programs or to apply the set theoretic model of functions to understand higher-order operations. A major aim of this text is to integrate, tightly, the study of discrete mathematics with the study of central problems of computer science. |
| discrete math strong induction: Concrete Mathematics Ronald L. Graham, Donald E. Knuth, Oren Patashnik, 1994-02-28 This book introduces the mathematics that supports advanced computer programming and the analysis of algorithms. The primary aim of its well-known authors is to provide a solid and relevant base of mathematical skills - the skills needed to solve complex problems, to evaluate horrendous sums, and to discover subtle patterns in data. It is an indispensable text and reference not only for computer scientists - the authors themselves rely heavily on it! - but for serious users of mathematics in virtually every discipline. Concrete Mathematics is a blending of CONtinuous and disCRETE mathematics. More concretely, the authors explain, it is the controlled manipulation of mathematical formulas, using a collection of techniques for solving problems. The subject matter is primarily an expansion of the Mathematical Preliminaries section in Knuth's classic Art of Computer Programming, but the style of presentation is more leisurely, and individual topics are covered more deeply. Several new topics have been added, and the most significant ideas have been traced to their historical roots. The book includes more than 500 exercises, divided into six categories. Complete answers are provided for all exercises, except research problems, making the book particularly valuable for self-study. Major topics include: Sums Recurrences Integer functions Elementary number theory Binomial coefficients Generating functions Discrete probability Asymptotic methods This second edition includes important new material about mechanical summation. In response to the widespread use of the first edition as a reference book, the bibliography and index have also been expanded, and additional nontrivial improvements can be found on almost every page. Readers will appreciate the informal style of Concrete Mathematics. Particularly enjoyable are the marginal graffiti contributed by students who have taken courses based on this material. The authors want to convey not only the importance of the techniques presented, but some of the fun in learning and using them. |
| discrete math strong induction: Mathematics Higher Level for the IB Diploma Option Topic 10 Discrete Mathematics Paul Fannon, Vesna Kadelburg, Ben Woolley, Stephen Ward, 2013-04-25 This title forms part of the completely new Mathematics for the IB Diploma series. This highly illustrated book covers topic 10 of the IB Diploma Higher Level Mathematics syllabus, the optional topic Discrete Mathematics. It is also for use with the further mathematics course. Based on the new group 5 aims, the progressive approach encourages cumulative learning. Features include: a dedicated chapter exclusively for mixed examination practice; plenty of worked examples; questions colour-coded according to grade; exam-style questions; feature boxes throughout of exam hints and tips. |
| discrete math strong induction: Discrete Mathematics Susanna S. Epp, 2011 |
| discrete math strong induction: Discrete Mathematics for Computer Scientists Clifford Stein, Robert L. Drysdale, Kenneth P. Bogart, 2011 Stein/Drysdale/Bogart's Discrete Mathematics for Computer Scientists is ideal for computer science students taking the discrete math course. Written specifically for computer science students, this unique textbook directly addresses their needs by providing a foundation in discrete math while using motivating, relevant CS applications. This text takes an active-learning approach where activities are presented as exercises and the material is then fleshed out through explanations and extensions of the exercises. |
| discrete math strong induction: Discrete Mathematics Gary Chartrand, Ping Zhang, 2011-03-31 Chartrand and Zhangs Discrete Mathematics presents a clearly written, student-friendly introduction to discrete mathematics. The authors draw from their background as researchers and educators to offer lucid discussions and descriptions fundamental to the subject of discrete mathematics. Unique among discrete mathematics textbooks for its treatment of proof techniques and graph theory, topics discussed also include logic, relations and functions (especially equivalence relations and bijective functions), algorithms and analysis of algorithms, introduction to number theory, combinatorics (counting, the Pascal triangle, and the binomial theorem), discrete probability, partially ordered sets, lattices and Boolean algebras, cryptography, and finite-state machines. This highly versatile text provides mathematical background used in a wide variety of disciplines, including mathematics and mathematics education, computer science, biology, chemistry, engineering, communications, and business. Some of the major features and strengths of this textbook Numerous, carefully explained examples and applications facilitate learning. More than 1,600 exercises, ranging from elementary to challenging, are included with hints/answers to all odd-numbered exercises. Descriptions of proof techniques are accessible and lively. Students benefit from the historical discussions throughout the textbook. |
| discrete math strong induction: Fundamentals of Discrete Math for Computer Science Tom Jenkyns, Ben Stephenson, 2012-08-28 This textbook provides an engaging and motivational introduction to traditional topics in discrete mathematics, in a manner specifically designed to appeal to computer science students. The text empowers students to think critically, to be effective problem solvers, to integrate theory and practice, and to recognize the importance of abstraction. Clearly structured and interactive in nature, the book presents detailed walkthroughs of several algorithms, stimulating a conversation with the reader through informal commentary and provocative questions. Features: no university-level background in mathematics required; ideally structured for classroom-use and self-study, with modular chapters following ACM curriculum recommendations; describes mathematical processes in an algorithmic manner; contains examples and exercises throughout the text, and highlights the most important concepts in each section; selects examples that demonstrate a practical use for the concept in question. |
| discrete math strong induction: Essential Discrete Mathematics for Computer Science Harry Lewis, Rachel Zax, 2019-03-19 A more intuitive approach to the mathematical foundation of computer science Discrete mathematics is the basis of much of computer science, from algorithms and automata theory to combinatorics and graph theory. This textbook covers the discrete mathematics that every computer science student needs to learn. Guiding students quickly through thirty-one short chapters that discuss one major topic each, this flexible book can be tailored to fit the syllabi for a variety of courses. Proven in the classroom, Essential Discrete Mathematics for Computer Science aims to teach mathematical reasoning as well as concepts and skills by stressing the art of proof. It is fully illustrated in color, and each chapter includes a concise summary as well as a set of exercises. The text requires only precalculus, and where calculus is needed, a quick summary of the basic facts is provided. Essential Discrete Mathematics for Computer Science is the ideal introductory textbook for standard undergraduate courses, and is also suitable for high school courses, distance education for adult learners, and self-study. The essential introduction to discrete mathematics Features thirty-one short chapters, each suitable for a single class lesson Includes more than 300 exercises Almost every formula and theorem proved in full Breadth of content makes the book adaptable to a variety of courses Each chapter includes a concise summary Solutions manual available to instructors |
| discrete math strong induction: Discrete Mathematics László Lovász, József Pelikán, Katalin Vesztergombi, 2006-05-10 Aimed at undergraduate mathematics and computer science students, this book is an excellent introduction to a lot of problems of discrete mathematics. It discusses a number of selected results and methods, mostly from areas of combinatorics and graph theory, and it uses proofs and problem solving to help students understand the solutions to problems. Numerous examples, figures, and exercises are spread throughout the book. |
| discrete math strong induction: Discrete Mathematics and Its Applications Kenneth H. Rosen, 2018-05 A precise, relevant, comprehensive approach to mathematical concepts... |
| discrete math strong induction: Mathematics Edward R. Scheinerman, 2006 Master the fundamentals of discrete mathematics and proof-writing with MATHEMATICS: A DISCRETE INTRODUCTION! With a wealth of learning aids and a clear presentation, the mathematics text teaches you not only how to write proofs, but how to think clearly and present cases logically beyond this course. Though it is presented from a mathematician's perspective, you will learn the importance of discrete mathematics in the fields of computer science, engineering, probability, statistics, operations research, and other areas of applied mathematics. Tools such as Mathspeak, hints, and proof templates prepare you to succeed in this course. |
| discrete math strong induction: How to Think Like a Mathematician Kevin Houston, 2009-02-12 Looking for a head start in your undergraduate degree in mathematics? Maybe you've already started your degree and feel bewildered by the subject you previously loved? Don't panic! This friendly companion will ease your transition to real mathematical thinking. Working through the book you will develop an arsenal of techniques to help you unlock the meaning of definitions, theorems and proofs, solve problems, and write mathematics effectively. All the major methods of proof - direct method, cases, induction, contradiction and contrapositive - are featured. Concrete examples are used throughout, and you'll get plenty of practice on topics common to many courses such as divisors, Euclidean algorithms, modular arithmetic, equivalence relations, and injectivity and surjectivity of functions. The material has been tested by real students over many years so all the essentials are covered. With over 300 exercises to help you test your progress, you'll soon learn how to think like a mathematician. |
| discrete math strong induction: Discrete and Combinatorial Mathematics: An applied Introduction ( For VTU) Grimaldi Ralph P., 2013 |
| discrete math strong induction: Foundations of Discrete Mathematics K. D. Joshi, 1989 This Book Is Meant To Be More Than Just A Text In Discrete Mathematics. It Is A Forerunner Of Another Book Applied Discrete Structures By The Same Author. The Ultimate Goal Of The Two Books Are To Make A Strong Case For The Inclusion Of Discrete Mathematics In The Undergraduate Curricula Of Mathematics By Creating A Sequence Of Courses In Discrete Mathematics Parallel To The Traditional Sequence Of Calculus-Based Courses.The Present Book Covers The Foundations Of Discrete Mathematics In Seven Chapters. It Lays A Heavy Emphasis On Motivation And Attempts Clarity Without Sacrificing Rigour. A List Of Typical Problems Is Given In The First Chapter. These Problems Are Used Throughout The Book To Motivate Various Concepts. A Review Of Logic Is Included To Gear The Reader Into A Proper Frame Of Mind. The Basic Counting Techniques Are Covered In Chapters 2 And 7. Those In Chapter 2 Are Elementary. But They Are Intentionally Covered In A Formal Manner So As To Acquaint The Reader With The Traditional Definition-Theorem-Proof Pattern Of Mathematics. Chapters 3 Introduces Abstraction And Shows How The Focal Point Of Todays Mathematics Is Not Numbers But Sets Carrying Suitable Structures. Chapter 4 Deals With Boolean Algebras And Their Applications. Chapters 5 And 6 Deal With More Traditional Topics In Algebra, Viz., Groups, Rings, Fields, Vector Spaces And Matrices.The Presentation Is Elementary And Presupposes No Mathematical Maturity On The Part Of The Reader. Instead, Comments Are Inserted Liberally To Increase His Maturity. Each Chapter Has Four Sections. Each Section Is Followed By Exercises (Of Various Degrees Of Difficulty) And By Notes And Guide To Literature. Answers To The Exercises Are Provided At The End Of The Book. |
| discrete math strong induction: Proof Patterns Mark Joshi, 2015-03-17 This innovative textbook introduces a new pattern-based approach to learning proof methods in the mathematical sciences. Readers will discover techniques that will enable them to learn new proofs across different areas of pure mathematics with ease. The patterns in proofs from diverse fields such as algebra, analysis, topology and number theory are explored. Specific topics examined include game theory, combinatorics and Euclidean geometry, enabling a broad familiarity. The author, an experienced lecturer and researcher renowned for his innovative view and intuitive style, illuminates a wide range of techniques and examples from duplicating the cube to triangulating polygons to the infinitude of primes to the fundamental theorem of algebra. Intended as a companion for undergraduate students, this text is an essential addition to every aspiring mathematician’s toolkit. |
| discrete math strong induction: Essential Discrete Mathematics for Computer Science Todd Feil, Joan Krone, 2003 This book introduces readers to the mathematics of computer science and prepares them for the math they will encounter in other college courses. It includes applications that are specific to computer science, helps learners to develop reasoning skills, and provides the fundamental mathematics necessary for computer scientists. Chapter topics include sets, functions and relations, Boolean algebra, natural numbers and induction, number theory, recursion, solving recurrences, counting, matrices, and graphs. For computer scientists and the enhancement of programming skills. |
| discrete math strong induction: Handbook of Mathematical Induction David S. Gunderson, 2010-09-14 Handbook of Mathematical Induction: Theory and Applications shows how to find and write proofs via mathematical induction. This comprehensive book covers the theory, the structure of the written proof, all standard exercises, and hundreds of application examples from nearly every area of mathematics. In the first part of the book, the author discusses different inductive techniques, including well-ordered sets, basic mathematical induction, strong induction, double induction, infinite descent, downward induction, and several variants. He then introduces ordinals and cardinals, transfinite induction, the axiom of choice, Zorn’s lemma, empirical induction, and fallacies and induction. He also explains how to write inductive proofs. The next part contains more than 750 exercises that highlight the levels of difficulty of an inductive proof, the variety of inductive techniques available, and the scope of results provable by mathematical induction. Each self-contained chapter in this section includes the necessary definitions, theory, and notation and covers a range of theorems and problems, from fundamental to very specialized. The final part presents either solutions or hints to the exercises. Slightly longer than what is found in most texts, these solutions provide complete details for every step of the problem-solving process. |
| discrete math strong induction: Combinatorics and Graph Theory John Harris, Jeffry L. Hirst, Michael Mossinghoff, 2009-04-03 These notes were first used in an introductory course team taught by the authors at Appalachian State University to advanced undergraduates and beginning graduates. The text was written with four pedagogical goals in mind: offer a variety of topics in one course, get to the main themes and tools as efficiently as possible, show the relationships between the different topics, and include recent results to convince students that mathematics is a living discipline. |
| discrete math strong induction: Essentials of Discrete Mathematics David Hunter, 2010-12-29 This is the ideal text for a one-term discrete mathematics course to serve computer scientists as well as other students. It introduces students to the mathematical way of thinking, and also to many important modern applications. |
| discrete math strong induction: An Introduction to Abstract Mathematics Robert J. Bond, William J. Keane, 2007-08-24 Bond and Keane explicate the elements of logical, mathematical argument to elucidate the meaning and importance of mathematical rigor. With definitions of concepts at their disposal, students learn the rules of logical inference, read and understand proofs of theorems, and write their own proofs all while becoming familiar with the grammar of mathematics and its style. In addition, they will develop an appreciation of the different methods of proof (contradiction, induction), the value of a proof, and the beauty of an elegant argument. The authors emphasize that mathematics is an ongoing, vibrant disciplineits long, fascinating history continually intersects with territory still uncharted and questions still in need of answers. The authors extensive background in teaching mathematics shines through in this balanced, explicit, and engaging text, designed as a primer for higher- level mathematics courses. They elegantly demonstrate process and application and recognize the byproducts of both the achievements and the missteps of past thinkers. Chapters 1-5 introduce the fundamentals of abstract mathematics and chapters 6-8 apply the ideas and techniques, placing the earlier material in a real context. Readers interest is continually piqued by the use of clear explanations, practical examples, discussion and discovery exercises, and historical comments. |
| discrete math strong induction: Discrete Mathematics with Combinatorics James Andrew Anderson, Jerome L. Lewis, 2004 For one-/two- semester, freshman courses in Discrete Mathematics. This carefully organized, very readable text covers every essential topic in discrete mathematics in a logical fashion. Placing each topic in context, it covers concepts associated with discrete mathematical systems that have applications in computer science, engineering, and mathematics. The author introduces more basic concepts at the freshman level than are found in other texts, in a simple, accessible form. Introductory material is balanced with extensive coverage of graphs, trees, recursion, algebra, theory of computing, and combinatorics. Extensive examples throughout the text reinforce concepts. |
| discrete math strong induction: Discrete Mathematics by Example Andrew Simpson, 2002 Designed for an undergraduate course, this book provides exercises and examples, enabling the development of students' understanding of the principles of discrete mathematics through appropriate pedagogical methods. It includes information such as on propositional and predicate logic, typed set theory, boolean algebra, induction and graph theory. |
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CS 70 Discrete Mathematics for CS Spring 2005 …
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Strong InductionI Another form of induction is called the \strong form". Despite the name, it is not a stronger proof technique. In fact, we have the following. Lemma The following are …
Mathematical Induction - Colorado State University
How does discrete math help us ! Helps create a solution (program) ! Helps analyze a program How does discrete math help (CS160)? ! Helps create a solution (program) " Logic helps you …
Discrete Structures Final exam sample questions| Solutions …
Use strong induction on nto prove that for all n 2, f(n) 2nlog 2 n. You may write log to indicate log 2. Here is a reminder of some facts about bxcand logx: bxc x log1 = 0, log2 = 1 log(x=2) = logx …
induction and resursion - cs.tulane.edu
CMPS/MATH 2170: Discrete Mathematics. Outline •Mathematical induction (5.1) •Sequences and Summations (2.4) •Strong induction (5.2) •Recursive definitions (5.3) •Recurrence Relations …
Discrete Mathematics (Math 151)
Strong Induction (King Saud University) Discrete Mathematics (Math 151) 3/56. Chapter 2: Proof Methods 1 Introduction to Proofs Direct Proofs ... Discrete Mathematics (Math 151) 11/56. …
Mathematical Induction - University of Texas at Austin
Instructor: Is l Dillig, CS311H: Discrete Mathematics Mathematical Induction 16/26 Strong Induction I Slight variation on the inductive proof technique isstrong induction I Regular and …
MATH 3336 Discrete Mathematics Mathematical Induction …
MATH 3336 – Discrete Mathematics Mathematical Induction (5.1) (from Discrete Mathematics and Its Applications by K. Rosen) Suppose we have an infinite ladder: ... Mathematical induction …
MAT 2534 Discrete Math - Virginia Tech
Mar 31, 2023 · MAT 2534 Discrete Math Joe Wells Virginia Tech Spring 2023 Last Updated: March 31, 2023 Contents ... 5 Sequences, Mathematical Induction, and Recursion 50 ... 5.4 …
CS 30: Discrete Math in CS (Winter 2019): Lecture 15
CS 30: Discrete Math in CS (Winter 2019): Lecture 15 Date: 30th January, 2019 (Wednesday) Topic: Strong Induction Disclaimer: These notes have not gone through scrutiny and in all …
Discrete Mathematics (Math 151) - هيئة التدريس جامعة ...
Strong Induction (King Saud University) Discrete Mathematics (Math 151) 3/56. Chapter 2: Proof Methods 1 Introduction to Proofs Direct Proofs ... Discrete Mathematics (Math 151) 11/56. …
Complete Induction - Stanford University
Complete Induction If the following are true: P(0) is true, and If P(0), P(1), P(2), …, P(k) are true, then P(k+1) is true as well. then P(n) is true for all n ∈ ℕ. This is called the principle of …
Prime factorization - Department of Mathematics
CU, Discrete Math, March 20 & 30, 2020. Finally we’ll realize our longterm goal of proving: ... We prove this conjecture by strong induction using the base cases above. I Assumption for strong …
Induction Problem Set - gotohaggstrom.com
p 0 = a 0, p 1= a 0 a 1 + 1, p n= a n p n-1 + p n-2 for 2 § n § N and symmetrically, q 0 = 1, q 1= a 1, q n= a n q n-1 + q n-2 for 2 § n § N Then [a0, a 1, ..., a n-1, a n] = p n q n. Prove this …
Discrete Mathematics, Chapter 5: Induction and Recursion
Note: Compared to mathematical induction, strong induction has a stronger induction hypothesis. You assume not only P(k) but even [P(0) ^P(1) ^P(2) ^^ P(k)] to then prove P(k + 1). Again the …
Induction and Recursion - uOttawa
Induction Strong Induction Recursive Defs and Structural Induction Program Correctness Mathematical Induction Mathematical Induction Principle (of Mathematical Induction) Suppose …
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proofs logic proof techniques so far direct proofs proof of the contrapositive proof by example / disproof by counterexample using cases induction today: more induction, including strong …
151 Math Exercises (3,3) Methods of Proof
Math 151 Discrete Mathematics [Mathematical induction, 2nd principle] By: Malek Zein AL-Abidin Strong Induction STRONG INDUCTION To prove that P(n) is true for all positive integers n, …
10 Mathematical Induction - Clemson University
10.11. Prove by induction that for every nonnegative integer n, 42n+1+11n is divisible by 5. 10.12. Prove by induction that for every nonnegative integer n, 62n+1+43n is divisible by 7. 10.13. …
CSE 20 DISCRETE MATH - University of California, San Diego
Fulfilling promises • We now have all the tools we need to rigorously prove • Correctness of greedy change-making algorithm with quarters, dimes, nickels, and pennies Proof by …
7.1 Induction Continued - University of Wisconsin–Madison
Lecture 7: Strong Induction 7.2. Strong Induction Consider the predicate Q(n): For all natural numbers m ≤ n, P(m) holds. In the case of Theorem 7.2, Q(n) says that for all m ≤ n, m can be …
CSE 20 Discrete math - cseweb.ucsd.edu
• The division algorithm is correct Strong induction, Rosen p. 341 • Russian peasant multiplication is correct Induction • Largest n-bit binary number is 2n-1 Induction, Rosen p. 318 • …
csci54 – discrete math & functional programming recurrence …
proofs logic proof techniques so far direct proofs proof of the contrapositive proof by example / disproof by counterexample using cases induction today: more induction, including strong …
CSE 311 Lecture 17: Strong Induction - University of …
We use strong induction to prove that a factorization into primes exists (but not that it is unique). 15. Prove that every integer ≥ 2 is a product of primes 16. Prove that every integer is a product …
Announcements CompSci 230 Discrete Math for Computer …
Strong Induction is sometimes called the second principle of mathematical induction or complete induction. 3 Strong Induction and the Infinite Ladder Strong induction tells us that we can …
strong) - Department of Mathematics
1 The induction format (\strong") Katherine Stange, Math 2001, CU Boulder For each theorem, envision it as falling into cases that are parametrized by some integer. Imagine how larger …
MATH 3336 Discrete Mathematics Strong Induction (5.2)
MATH 3336 – Discrete Mathematics Strong Induction (5.2) Picile f S g Mahema ical I d c i: To prove that 2 : J ; is true for all positive ... Conclusion: By the principle of strong mathematical …
Recitation 5: Weak and Strong Induction - Duke University
COMPSCI 230: Discrete Mathematics for Computer Science Recitation 5: Weak and Strong Induction Spring 2020 Created By: David Fischer Recall the boiler plate for weak induction: …
Induction, Sequences and Series - University of California, …
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Principle of Mathematical Induction - ualberta.ca
n 2 S; then the second property of S implies that n+1 2 S also. By the principle of strong mathematical induction we must have S = fx 2 Zjx ag: Therefore the principle of mathematical …
Claim: Every positive integer greater than - Duke University
Proof by Strong Induction: Variable: " is an element of positive natural numbers greater than 1 i.e., {2,3,4…}, so " Î ++ - {1} Prop: ,("):" is either a prime number or the product of prime numbers. …
MATH 224: Discrete Mathematics - Southern Illinois …
MATH 224: Discrete Mathematics (Adopted - Spring 2012; Committee: M. Agustin, V. Kieftenbeld, S. Rigdon, G.S. Staples) ... 5.2 Strong Induction and Well-Ordering 5.3 Recursive Definitions …
Review CS243: Discrete Structures - University of Texas at …
Is l Dillig, CS243: Discrete Structures Structural Induction 21/30 Mathematical vs. Strong vs. Structural Induction I So far, saw three types of induction: 1.Mathematical (standard) induction …
Discrete Mathematics (MATH 151) - هيئة التدريس جامعة ...
Mathematical Induction Discrete Mathematics (MATH 151) Dr. BorhenHalouani King Saud University February9,2020 Dr. Borhen Halouani Discrete Mathematics (MATH 151) Outline ...
Math 310 Class Notes 4: The Well-ordering Principle
out to be logically equivalent to the mathematical induction, the fth axiom of Peano, which is quite surprising. Theorem 1. The mathematical induction is logically equivalent to the well-ordering …
MATH 215 Discrete Mathematics Worksheets Induction and …
MATH 215 Discrete Mathematics Worksheets Induction and Recursion Let P(n) be the statement that a postage of ncents can be formed using just 3-cent stamps and 5-cent stamps. The parts …
CMSC 27130 Honors Discrete Mathematics
3 Strong Induction and the Well-Ordering Principle 3.1 Strong Induction Remark 3.1. The principle of mathematical induction is an incredibly powerful tool, but we need not \restrict" ourselves by …
MAT 2534 Discrete Math - Virginia Tech
Jul 25, 2023 · 1 Logic 1.1 Propositions and logical operations Definition 1.1.1 A proposition is a sentence which is either true or false, but not both. Example 1.1.2
Mathematical Induction - University of Utah
A stronger statement (sometimes called “strong induction”) that is sometimes easier to work with is this: Let S(n) be any statement about a natural number n. To show using strong induction …
CSE 20 DISCRETE MATH - University of California, San Diego
Structural Induction, example Rosen Sec 5.3 Define the subset S of binary strings {0,1}* by Basis step: where is the empty string. Recursive step: If , then each of Claim: Every element in S has …
Math 270 Basic Discrete Math Practice Test 3 Sections 5.1, …
Math 270 Basic Discrete Math Practice Test 3 ... but no other electronic devices. Show all work. 1. Prove, using mathematical induction, that for all integers n 1, 3 + 7 + 11 + + (4n 1) = 2n(n+ 1) …
Resources for Teaching Discrete Mathematics
War Stories from Applied Math: Undergraduate Consultancy Projects, Robert Fraga, Editor. 72. Hands On History: A Resource for Teaching Mathematics, Amy Shell-Gellasch, Editor. ...
Discrete Mathematics Induction - Lahore University of …
Discrete Mathematics Induction Principle of Mathematical Induction Proofs by Induction Strong Induction Well Ordering Principle Imdad ullah Khan Imdad ullah Khan (LUMS) Induction 1/11. …
MAT 2534 Discrete Math - Virginia Tech
MAT 2534 Discrete Math Joe Wells Virginia Tech Spring 20241 Last Updated: May 4, 2024 1This courses is using Susanna Epp’s Discrete Mathematics with Applications, 4th Edition. The ...
Recursive De nitions CS311H: Discrete Mathematics I
Inductive Proofs for Recursively De ned Structures I Recursive de nitions and inductive proofs are very similar I Natural to use induction to prove properties about recursively de ned structures …
CS103X: Discrete Structures Homework Assignment 1: Solutions
CS103X: Discrete Structures Homework Assignment 1: Solutions Due January 25, 2008 Exercise 1 (10 Points). Prove or give a counterexample for each of the following: ... Consider the …
CHAPTER 5 SEQUENCES, MATHEMATICAL INDUCTION, …
228 Chapter 5 Sequences, Mathematical Induction, and Recursion numbers are 8, 16, 32, 64, and 128, which equal 23,24,25,26, and 27, respectively.For a general value of k,letA k be the …
Induction Proofs, IV: Fallacies and pitfalls - Department of …
induction hypothesis to i and j, we get 5i = 0 and 5j = 0. Then 5(k + 1) = 5(i+ j) = 5i+ 5j = 0 + 0 = 0; proving ( ). Hence the induction step is complete. Conclusion: By the principle of strong …
