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| discrete math direct proof: 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 direct proof: 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 direct proof: 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 direct proof: 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 direct proof: Discrete Mathematics with Proof Eric Gossett, 2009-06-22 A Trusted Guide to Discrete Mathematics with Proof?Now in a Newly Revised Edition Discrete mathematics has become increasingly popular in recent years due to its growing applications in the field of computer science. Discrete Mathematics with Proof, Second Edition continues to facilitate an up-to-date understanding of this important topic, exposing readers to a wide range of modern and technological applications. The book begins with an introductory chapter that provides an accessible explanation of discrete mathematics. Subsequent chapters explore additional related topics including counting, finite probability theory, recursion, formal models in computer science, graph theory, trees, the concepts of functions, and relations. Additional features of the Second Edition include: An intense focus on the formal settings of proofs and their techniques, such as constructive proofs, proof by contradiction, and combinatorial proofs New sections on applications of elementary number theory, multidimensional induction, counting tulips, and the binomial distribution Important examples from the field of computer science presented as applications including the Halting problem, Shannon's mathematical model of information, regular expressions, XML, and Normal Forms in relational databases Numerous examples that are not often found in books on discrete mathematics including the deferred acceptance algorithm, the Boyer-Moore algorithm for pattern matching, Sierpinski curves, adaptive quadrature, the Josephus problem, and the five-color theorem Extensive appendices that outline supplemental material on analyzing claims and writing mathematics, along with solutions to selected chapter exercises Combinatorics receives a full chapter treatment that extends beyond the combinations and permutations material by delving into non-standard topics such as Latin squares, finite projective planes, balanced incomplete block designs, coding theory, partitions, occupancy problems, Stirling numbers, Ramsey numbers, and systems of distinct representatives. A related Web site features animations and visualizations of combinatorial proofs that assist readers with comprehension. In addition, approximately 500 examples and over 2,800 exercises are presented throughout the book to motivate ideas and illustrate the proofs and conclusions of theorems. Assuming only a basic background in calculus, Discrete Mathematics with Proof, Second Edition is an excellent book for mathematics and computer science courses at the undergraduate level. It is also a valuable resource for professionals in various technical fields who would like an introduction to discrete mathematics. |
| discrete math direct proof: 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 direct proof: A Transition to Advanced Mathematics Douglas Smith, Maurice Eggen, Richard St. Andre, 2010-06-01 A TRANSITION TO ADVANCED MATHEMATICS helps students make the transition from calculus to more proofs-oriented mathematical study. The most successful text of its kind, the 7th edition continues to provide a firm foundation in major concepts needed for continued study and guides students to think and express themselves mathematically to analyze a situation, extract pertinent facts, and draw appropriate conclusions. The authors place continuous emphasis throughout on improving students' ability to read and write proofs, and on developing their critical awareness for spotting common errors in proofs. Concepts are clearly explained and supported with detailed examples, while abundant and diverse exercises provide thorough practice on both routine and more challenging problems. Students will come away with a solid intuition for the types of mathematical reasoning they'll need to apply in later courses and a better understanding of how mathematicians of all kinds approach and solve problems. Important Notice: Media content referenced within the product description or the product text may not be available in the ebook version. |
| discrete math direct proof: Axiomatic Geometry John M. Lee, 2013-04-10 The story of geometry is the story of mathematics itself: Euclidean geometry was the first branch of mathematics to be systematically studied and placed on a firm logical foundation, and it is the prototype for the axiomatic method that lies at the foundation of modern mathematics. It has been taught to students for more than two millennia as a mode of logical thought. This book tells the story of how the axiomatic method has progressed from Euclid's time to ours, as a way of understanding what mathematics is, how we read and evaluate mathematical arguments, and why mathematics has achieved the level of certainty it has. It is designed primarily for advanced undergraduates who plan to teach secondary school geometry, but it should also provide something of interest to anyone who wishes to understand geometry and the axiomatic method better. It introduces a modern, rigorous, axiomatic treatment of Euclidean and (to a lesser extent) non-Euclidean geometries, offering students ample opportunities to practice reading and writing proofs while at the same time developing most of the concrete geometric relationships that secondary teachers will need to know in the classroom. -- P. [4] of cover. |
| discrete math direct proof: 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 direct proof: Understanding Mathematical Proof John Taylor, Rowan Garnier, 2016-04-19 The notion of proof is central to mathematics yet it is one of the most difficult aspects of the subject to teach and master. In particular, undergraduate mathematics students often experience difficulties in understanding and constructing proofs.Understanding Mathematical Proof describes the nature of mathematical proof, explores the various techn |
| discrete math direct proof: 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 direct proof: Discrete and Combinatorial Mathematics: An applied Introduction ( For VTU) Grimaldi Ralph P., 2013 |
| discrete math direct proof: 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 direct proof: Discrete Mathematics with Ducks Sarah-marie Belcastro, 2018-11-15 Discrete Mathematics with Ducks, Second Edition is a gentle introduction for students who find the proofs and abstractions of mathematics challenging. At the same time, it provides stimulating material that instructors can use for more advanced students. The first edition was widely well received, with its whimsical writing style and numerous exercises and materials that engaged students at all levels. The new, expanded edition continues to facilitate effective and active learning. It is designed to help students learn about discrete mathematics through problem-based activities. These are created to inspire students to understand mathematics by actively practicing and doing, which helps students better retain what they’ve learned. As such, each chapter contains a mixture of discovery-based activities, projects, expository text, in-class exercises, and homework problems. The author’s lively and friendly writing style is appealing to both instructors and students alike and encourages readers to learn. The book’s light-hearted approach to the subject is a guiding principle and helps students learn mathematical abstraction. Features: The book’s Try This! sections encourage students to construct components of discussed concepts, theorems, and proofs Provided sets of discovery problems and illustrative examples reinforce learning Bonus sections can be used by instructors as part of their regular curriculum, for projects, or for further study |
| discrete math direct proof: Write Your Own Proofs Amy Babich, Laura Person, 2019-08-14 Written by a pair of math teachers and based on their classroom notes and experiences, this introductory treatment of theory, proof techniques, and related concepts is designed for undergraduate courses. No knowledge of calculus is assumed, making it a useful text for students at many levels. The focus is on teaching students to prove theorems and write mathematical proofs so that others can read them. Since proving theorems takes lots of practice, this text is designed to provide plenty of exercises. The authors break the theorems into pieces and walk readers through examples, encouraging them to use mathematical notation and write proofs themselves. Topics include propositional logic, set notation, basic set theory proofs, relations, functions, induction, countability, and some combinatorics, including a small amount of probability. The text is ideal for courses in discrete mathematics or logic and set theory, and its accessibility makes the book equally suitable for classes in mathematics for liberal arts students or courses geared toward proof writing in mathematics. |
| discrete math direct proof: Mathematical Reasoning Theodore A. Sundstrom, 2007 Focusing on the formal development of mathematics, this book shows readers how to read, understand, write, and construct mathematical proofs.Uses elementary number theory and congruence arithmetic throughout. Focuses on writing in mathematics. Reviews prior mathematical work with “Preview Activities” at the start of each section. Includes “Activities” throughout that relate to the material contained in each section. Focuses on Congruence Notation and Elementary Number Theorythroughout.For professionals in the sciences or engineering who need to brush up on their advanced mathematics skills. Mathematical Reasoning: Writing and Proof, 2/E Theodore Sundstrom |
| discrete math direct proof: Discrete Mathematics Douglas E. Ensley, J. Winston Crawley, 2005-10-07 These active and well-known authors have come together to create a fresh, innovative, and timely approach to Discrete Math. One innovation uses several major threads to help weave core topics into a cohesive whole. Throughout the book the application of mathematical reasoning is emphasized to solve problems while the authors guide the student in thinking about, reading, and writing proofs in a wide variety of contexts. Another important content thread, as the sub-title implies, is the focus on mathematical puzzles, games and magic tricks to engage students. |
| discrete math direct proof: Understanding Analysis Stephen Abbott, 2012-12-06 This elementary presentation exposes readers to both the process of rigor and the rewards inherent in taking an axiomatic approach to the study of functions of a real variable. The aim is to challenge and improve mathematical intuition rather than to verify it. The philosophy of this book is to focus attention on questions which give analysis its inherent fascination. Each chapter begins with the discussion of some motivating examples and concludes with a series of questions. |
| discrete math direct proof: Handbook of Discrete and Combinatorial Mathematics Kenneth H. Rosen, 1999-09-28 The importance of discrete and combinatorial mathematics continues to increase as the range of applications to computer science, electrical engineering, and the biological sciences grows dramatically. Providing a ready reference for practitioners in the field, the Handbook of Discrete and Combinatorial Mathematics, Second Edition presents additional material on Google's matrix, random graphs, geometric graphs, computational topology, and other key topics. New chapters highlight essential background information on bioinformatics and computational geometry. Each chapter includes a glossary, definitions, facts, examples, algorithms, major applications, and references. |
| discrete math direct proof: 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 direct proof: Building Proofs: A Practical Guide David Stewart, Suely Oliveira, 2015-06-10 This book introduces students to the art and craft of writing proofs, beginning with the basics of writing proofs and logic, and continuing on with more in-depth issues and examples of creating proofs in different parts of mathematics, as well as introducing proofs-of-correctness for algorithms. The creation of proofs is covered for theorems in both discrete and continuous mathematics, and in difficulty ranging from elementary to beginning graduate level.Just beyond the standard introductory courses on calculus, theorems and proofs become central to mathematics. Students often find this emphasis difficult and new. This book is a guide to understanding and creating proofs. It explains the standard “moves” in mathematical proofs: direct computation, expanding definitions, proof by contradiction, proof by induction, as well as choosing notation and strategies. |
| discrete math direct proof: Discrete Mathematics and Its Applications Kenneth H. Rosen, 2018-05 A precise, relevant, comprehensive approach to mathematical concepts... |
| discrete math direct proof: Logic and Discrete Mathematics Willem Conradie, Valentin Goranko, Claudette Robinson, 2015-05-08 Solutions manual to accompany Logic and Discrete Mathematics: A Concise Introduction This book features a unique combination of comprehensive coverage of logic with a solid exposition of the most important fields of discrete mathematics, presenting material that has been tested and refined by the authors in university courses taught over more than a decade. Written in a clear and reader-friendly style, each section ends with an extensive set of exercises, most of them provided with complete solutions which are available in this accompanying solutions manual. |
| discrete math direct proof: 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 direct proof: Ask a Manager Alison Green, 2018-05-01 'I'm a HUGE fan of Alison Green's Ask a Manager column. This book is even better' Robert Sutton, author of The No Asshole Rule and The Asshole Survival Guide 'Ask A Manager is the book I wish I'd had in my desk drawer when I was starting out (or even, let's be honest, fifteen years in)' - Sarah Knight, New York Times bestselling author of The Life-Changing Magic of Not Giving a F*ck A witty, practical guide to navigating 200 difficult professional conversations Ten years as a workplace advice columnist has taught Alison Green that people avoid awkward conversations in the office because they don't know what to say. Thankfully, Alison does. In this incredibly helpful book, she takes on the tough discussions you may need to have during your career. You'll learn what to say when: · colleagues push their work on you - then take credit for it · you accidentally trash-talk someone in an email and hit 'reply all' · you're being micromanaged - or not being managed at all · your boss seems unhappy with your work · you got too drunk at the Christmas party With sharp, sage advice and candid letters from real-life readers, Ask a Manager will help you successfully navigate the stormy seas of office life. |
| discrete math direct proof: 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 direct proof: 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 direct proof: Proof in Mathematics James Franklin, Albert Daoud, 2010 |
| discrete math direct proof: An Introduction to Discrete Mathematics Steven Roman, 1989 Intended for a one-term course in discrete mathematics, to prepare freshmen and sophomores for further work in computer science as well as mathematics. Sets, proof techniques, logic, combinatorics, and graph theory are covered in concise form. All topics are motivated by concrete examples, often emphasizing the interplay between computer science and mathematics. Examples also illustrate all definitions. Applications and references cover a wide variety of realistic situations. Coverage of mathematical induction includes the stroung form of induction, and new sections have been added on nonhomogeneous recurrence relations and the essentials of probability. |
| discrete math direct proof: Introduction to Discrete Mathematics with ISETL William E. Fenton, Ed Dubinsky, 1996-09-19 Intended for first- or second-year undergraduates, this introduction to discrete mathematics covers the usual topics of such a course, but applies constructivist principles that promote - indeed, require - active participation by the student. Working with the programming language ISETL, whose syntax is close to that of standard mathematical language, the student constructs the concepts in her or his mind as a result of constructing them on the computer in the syntax of ISETL. This dramatically different approach allows students to attempt to discover concepts in a Socratic dialog with the computer. The discussion avoids the formal definition-theorem approach and promotes active involvement by the reader by its questioning style. An instructor using this text can expect a lively class whose students develop a deep conceptual understanding rather than simply manipulative skills. Topics covered in this book include: the propositional calculus, operations on sets, basic counting methods, predicate calculus, relations, graphs, functions, and mathematical induction. |
| discrete math direct proof: 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 direct proof: 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 direct proof: Discrete Mathematics and Graph Theory K. Erciyes, 2021-01-28 This textbook can serve as a comprehensive manual of discrete mathematics and graph theory for non-Computer Science majors; as a reference and study aid for professionals and researchers who have not taken any discrete math course before. It can also be used as a reference book for a course on Discrete Mathematics in Computer Science or Mathematics curricula. The study of discrete mathematics is one of the first courses on curricula in various disciplines such as Computer Science, Mathematics and Engineering education practices. Graphs are key data structures used to represent networks, chemical structures, games etc. and are increasingly used more in various applications such as bioinformatics and the Internet. Graph theory has gone through an unprecedented growth in the last few decades both in terms of theory and implementations; hence it deserves a thorough treatment which is not adequately found in any other contemporary books on discrete mathematics, whereas about 40% of this textbook is devoted to graph theory. The text follows an algorithmic approach for discrete mathematics and graph problems where applicable, to reinforce learning and to show how to implement the concepts in real-world applications. |
| discrete math direct proof: Practical Discrete Mathematics Ryan T. White, Archana Tikayat Ray, 2021-02-22 A practical guide simplifying discrete math for curious minds and demonstrating its application in solving problems related to software development, computer algorithms, and data science Key FeaturesApply the math of countable objects to practical problems in computer scienceExplore modern Python libraries such as scikit-learn, NumPy, and SciPy for performing mathematicsLearn complex statistical and mathematical concepts with the help of hands-on examples and expert guidanceBook Description Discrete mathematics deals with studying countable, distinct elements, and its principles are widely used in building algorithms for computer science and data science. The knowledge of discrete math concepts will help you understand the algorithms, binary, and general mathematics that sit at the core of data-driven tasks. Practical Discrete Mathematics is a comprehensive introduction for those who are new to the mathematics of countable objects. This book will help you get up to speed with using discrete math principles to take your computer science skills to a more advanced level. As you learn the language of discrete mathematics, you'll also cover methods crucial to studying and describing computer science and machine learning objects and algorithms. The chapters that follow will guide you through how memory and CPUs work. In addition to this, you'll understand how to analyze data for useful patterns, before finally exploring how to apply math concepts in network routing, web searching, and data science. By the end of this book, you'll have a deeper understanding of discrete math and its applications in computer science, and be ready to work on real-world algorithm development and machine learning. What you will learnUnderstand the terminology and methods in discrete math and their usage in algorithms and data problemsUse Boolean algebra in formal logic and elementary control structuresImplement combinatorics to measure computational complexity and manage memory allocationUse random variables, calculate descriptive statistics, and find average-case computational complexitySolve graph problems involved in routing, pathfinding, and graph searches, such as depth-first searchPerform ML tasks such as data visualization, regression, and dimensionality reductionWho this book is for This book is for computer scientists looking to expand their knowledge of discrete math, the core topic of their field. University students looking to get hands-on with computer science, mathematics, statistics, engineering, or related disciplines will also find this book useful. Basic Python programming skills and knowledge of elementary real-number algebra are required to get started with this book. |
| discrete math direct proof: 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 direct proof: Discrete Mathematics Rajendra Akerkar, Rupali Akerkar, 2004-09 |
| discrete math direct proof: 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 direct proof: Fundamental Proof Methods in Computer Science Konstantine Arkoudas, David Musser, 2017-05-05 A textbook that teaches students to read and write proofs using Athena. Proof is the primary vehicle for knowledge generation in mathematics. In computer science, proof has found an additional use: verifying that a particular system (or component, or algorithm) has certain desirable properties. This book teaches students how to read and write proofs using Athena, a freely downloadable computer language. Athena proofs are machine-checkable and written in an intuitive natural-deduction style. The book contains more than 300 exercises, most with full solutions. By putting proofs into practice, it demonstrates the fundamental role of logic and proof in computer science as no other existing text does. Guided by examples and exercises, students are quickly immersed in the most useful high-level proof methods, including equational reasoning, several forms of induction, case analysis, proof by contradiction, and abstraction/specialization. The book includes auxiliary material on SAT and SMT solving, automated theorem proving, and logic programming. The book can be used by upper undergraduate or graduate computer science students with a basic level of programming and mathematical experience. Professional programmers, practitioners of formal methods, and researchers in logic-related branches of computer science will find it a valuable reference. |
| discrete math direct proof: Discrete Mathematical Structures for Computer Science Bernard Kolman, Robert C. Busby, 1987 This text has been designed as a complete introduction to discrete mathematics, primarily for computer science majors in either a one or two semester course. The topics addressed are of genuine use in computer science, and are presented in a logically coherent fashion. The material has been organized and interrelated to minimize the mass of definitions and the abstraction of some of the theory. For example, relations and directed graphs are treated as two aspects of the same mathematical idea. Whenever possible each new idea uses previously encountered material, and then developed in such a way that it simplifies the more complex ideas that follow. |
| discrete math direct proof: Discrete Mathematics with Applications Thomas Koshy, 2004-01-19 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, programming languages, compilers, databases, and computation.* Covers all recommended topics in a self-contained, comprehensive, and understandable format for students and new professionals * Emphasizes problem-solving techniques, pattern recognition, conjecturing, induction, applications of varying nature, proof techniques, algorithm development and correctness, and numeric computations* Weaves numerous applications into the text* Helps students learn by doing with a wealth of examples and exercises: - 560 examples worked out in detail - More than 3,700 exercises - More than 150 computer assignments - More than 600 writing projects* Includes chapter summaries of important vocabulary, formulas, and properties, plus the chapter review exercises* Features interesting anecdotes and biographies of 60 mathematicians and computer scientists* Instructor's Manual available for adopters* Student Solutions Manual available separately for purchase (ISBN: 0124211828) |
Why is My Discrete GPU Idle? Expert Answers and Solutions
NVIDIA Control Panel/AMD Radeon Settings: Depending on your GPU, you can manually set the discrete GPU for specific applications. For NVIDIA: Right-click on the desktop and select …
Expert Solutions for Discrete GPU Idle Issues in NitroSense
Use the arrow keys to select the Advanced tab. Use the arrow keys to select the Display mode and change it from Optimus to Discrete GPU only. Select the Exit tab. Select Exit Saving …
What does mild coarsening of the liver echo texture mean?
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Understanding Blunting and Fraying of the Labrum: Expert Answers
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Understanding Pyriform Sinus CT Scans: Expert Q&A - JustAnswer
1. New mild asymmetric fullness of the left piriform sinus without discrete mass. This is likely due to underdistention, but correlation with direct visualization is recommended. You can have …
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Understanding ANA Titer 1:1280 and Its Patterns - Expert Q&A
Hello, this is Dr. David. I have read your question and am ready to help. the ANA speckled patter titer of 1:1280 means you are positive for antineuclear antibodies which means your body is …
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CSE 215 Practice Questions - cs.stonybrook.edu
Direct Proofs Prove each of the following using a direct proof method: 1.The sum of any two odd integers is even. 2.If nand mare odd, then nmis also odd. 3.The product of any two …
Practice with Proofs - University of California, Berkeley
Here’s another proof, which I was thinking of when I wrote the hint. Proof. Given x, we need to nd ysuch that y2 >x. If x 1, then x 1 <232; so we can take y= 23. Otherwise x>1. Multiplying both …
Methods of Proofs - University of Houston–Downtown
A direct proof begins by assuming p is true. : : until we conclude q. An indirect proof begins by assuming ~q is true. : : until we conclude ~p . An example of a proof by contradiction. Example …
CS103 Handout 07 Summer 2019 June 28, 2019 Guide to …
tackle writing a proof. It might seem weird to approach writing a proof of a result when you still haven’t figured out how everything fits together. And that’s a good intuition to have. However, …
Guide to Proofs on Discrete Structures - Stanford University
Statement Form Proof Approach ∀x. P Direct proof: Pick an arbitrary x, then prove P is true for that choice of x. By contradiction: Suppose for the sake of contradiction that there is some x …
Direct Proof - Gordon College
MAT231 (Transition to Higher Math) Direct Proof Fall 2014 12 / 24. GCD and LCM De nition The greatest common divisor (GCD) of integers a and b, denoted gcd(a;b), is the largest integer …
CS/Math 240: Introduction to Discrete Mathematics
CS/Math 240: Introduction to Discrete Mathematics Reading 5 : Induction Author: Dieter van Melkebeek (updates by Beck Hasti and Gautam Prakriya) In the last reading we began …
Proofs Involving Sets - Gordon College
Proof. Suppose k 2Z and let K = fn 2Z : njkgand S = fn 2Z : njk2g. Let x 2K so that xjk. We can write k = ax for some a 2Z. Then k2 = (ax)2 = x(a2x) so xjk2. Thus, x 2S. Since any element x …
Section 3.1: Direct Proof and Counterexample 1 - University …
(and in math, we are nearly always considering statements about infinite sized sets). Therefore, in general, we would try a method more like the following: Result 4.2. (Method of Direct Proof) …
Informal proofs. Types of proofs. - University of Pittsburgh
Direct proof • Direct proof may not be the best option. It may become hard to prove the conclusion follows from the premises. Example: Prove If 3n + 2 is odd then n is odd. Proof: • Assume that …
Notes on Discrete Mathematics - Yale University
May 9, 2025 · Contents Tableofcontentsii Listoffiguresxvii Listoftablesxix Listofalgorithmsxx Prefacexxi Resourcesxxii 1 Introduction1 1.1 ...
Proof by Contradiction - Gordon College
Use a direct proof, a contrapositive proof, or a proof by contradiction to prove each of the following propositions. Proposition Suppose a;b 2Z. If a +b 19, then a 10 or b 10. Proposition Suppose …
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 …
Direct Proof – Divisibility
Robb T. Koether (Hampden-Sydney College) Direct Proof – Divisibility Fri, Feb 7, 2014 18 / 23. Greatest Common Divisors Definition Let a;b 2Z, not both 0. Thegreatest common divisorof a …
Discrete Math Direct Proof Copy - staging …
Discrete Math Direct Proof Gary Chartrand,Ping Zhang. Discrete Math Direct Proof: Discrete Mathematics Oscar Levin,2018-07-30 Note This is a custom edition of Levin s full Discrete …
Indirect Proofs - Stanford University
Theorem: For any n ∈ ℤ, if n2 is even, then n is even. Proof: By contrapositive; we prove that if n is odd, then n2 is odd. Since n is odd, there is some integer k such that n = 2k + 1. Squaring …
Direct Proof Examples - cs.arizona.edu
Direct Proof Examples 1. Conjecture: n2 −3 is even if nis odd, n∈ Z. Discussion: The first thing to do is identify the hypothesis and the conclusion. Why? Because in a direct proof, we are …
Guide to Proofs on Discrete Structures
Statement Form Proof Approach ∀x. P Direct proof: Pick an arbitrary x, then prove P is true for that choice of x. By contradiction: Suppose for the sake of contradiction that there is some x …
Discrete Math Direct Proof Full PDF - staging …
Discrete Math Direct Proof: Discrete Mathematics Oscar Levin,2018-07-30 Note This is a custom edition of Levin s full Discrete Mathematics text arranged specifically for use in a discrete math …
(Basic Mathematical Proofs) - SNRU
relevant techniques to prove. The first is direct proof and the other is indirect proof. Direct proof or prove by construction is a basic approach to prove “p . q” ú. by assuming p is true and then …
The Foundations: Logic and Proofs - William & Mary
• q: I will study discrete math. • r: I will get an A. • If it snows, then I will study discrete math. • If I study discrete math, I will get an A. • Therefore, if it snows, I will get an A.
csci54 – discrete math & functional programming proofs: …
proof techniques direct proof: start with known facts. repeatedly infer additional new facts until can conclude what you want to show. may divide work into cases proof of the contrapositive: if …
Logical Inference and Mathematical Proof - University at Buffalo
c Xin He (University at Buffalo) CSE 191 Discrete Structures 22 / 66 Inference with quantiers Many inferences in Math and CS involve quantiers. Example 1: All computer science majors …
Section 1.6, selected answers Math 114 Discrete Mathematics
Math 114 Discrete Mathematics D Joyce, Spring 2018 6. Use a direct proof to show that the product of two odd integers is odd. Use the de nition that n is an odd when 9k;n = 2k + 1. …
DISCRETE MATHEMATICS AND APPLICATIONS Proof …
4.1 Direct Proof 4.2 Indirect Proof 4.3 Contradiction Method Aims –Apply direct method to prove a theorem ... Epp S.S, Discrete Mathematics with Applications, (Fourth Edition), Thomson …
Florida State University Course Notes MAD 2104 Discrete …
MAD 2104 Discrete Mathematics I. Florida State University Tallahassee, Florida 32306-4510 ... Methods of Proof 69 2.1. Types of Proofs 69 2.2. Trivial Proof/Vacuous Proof 69 2.3. Direct …
Quantifiers, Proofs and Sets - Department of Computer Science
Direct proof – Start with something known to be true – Repeatedly derive a statement that is implied by the previous one(s), until arriving at the conclusion – Application of modus ponens: …
Discrete Mathematics (Math 151)
A direct proof of a conditional statement p →q is constructed when the first step is the assumption thatp is true; subsequent steps are ... Discrete Mathematics (Math 151) 11/56. Introduction to …
Proof of the Fundamental Theorem of Arithmetic - potechin.org
Discrete Mathematics 27100 Winter 2022 January 31, 2022 ... This follows from a lemma which we proved last lecture as a stepping stone for the proof that there are infinitely many prime …
PROOF TEMPLATES - Ohio State University
PROOF TEMPLATES 1. Basic form of a proof of an existential statement using the method of constructive proof. 2. Proving an existential statement without a domain using the method of …
and Proof Methods - Jarrar
Keywords: Number Theory, Prove, Disapprove, Direct Proofs, divisibility, factorization Mustafa Jarrar: Lecture Notes on Number Theory and Proofs. Birzeit University, Palestine, 2015 …
Practice Questions with Solutions - University of Victoria
Discrete Mathematics: Study Guide for MAT212-S - Dr. Kieka Myndardt Discrete Mathematics - Norman L. Biggs Applied Combinatorics, fourth edition - Alan Tucker Discrete Mathematics, …
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
Logic & Discrete Math in Software Engineering (CAS 701)
Proof Systems Hilbert System Natural Deduction Soundness and Completeness of Natural Deduction Proof By Resolution SAT Solving Algorithms ... Dr. Borzoo BonakdarpourLogic & …
The Foundations: Logic and Proofs - University of Richmond
Let qbe “I will study discrete math.” ... Direct Proof: Assume that pis true. Use rules of inference, axioms, and logical equivalences to show that qmust also be true. Example: Give a direct …
Math 127: Induction - CMU
Direct Proof, so we assume p(n) is true, and derive p(n + 1). This is called the \Inductive Step." The Base Case and Inductive Step are often labeled as such in a proof. The assumption that …
CSE 215 Practice Questions - cs.stonybrook.edu
Direct Proofs Prove each of the following using a direct proof method: 1.The sum of any two odd integers is even. Let two odd integers be aand b.By definition, an odd integer can be written …
Proof Techniques - Computer Science
1.1.2 Proof by contradiction In proof by contradiction, you assume your statement is not true, and then derive a con-tradiction. This is really a special case of proof by contrapositive (where your …
Module A-5: Injective, Surjective, and Bijective Functions
Nov 10, 2019 · The best way to prove that some function is injective is to use a direct proof. Let x 1 2 D,letx 2 2 D, and suppose that f(x 1)=f(x 2). Then you do some algebra, until you reach x 1 …
The Foundations: Logic and Proofs - Hand and Machine
Let qbe “I will study discrete math. ... Direct Proofs. Indirect Proofs. •Proof of the Contrapositive. ... Proofs of Mathematical Statements A proofis a valid argument that establishes the truth of a …
Discrete Mathematics - University of Cambridge
DISCRETE MATHEMATICS SUMMARY Proof patterns The standard proof techniques used in logic – you can use them systematically to prove most of the logical propositions you …
and Proof Methods - Jarrar
into a formal proof. 3.2 Direct Proof and Counterexample II: Rational Numbers 143 Note that you can use an argument similar to this one show any repeating decimal is a rational number. In …
csci54 – discrete math & functional programming induction
Notes on writing proofs by induction we prove the claim using a proof by induction <...> unless it's a direct proof should state the proof technique. base case show true on the smallest element …
Chapter III Basic Proof Techniques - mathdept.byu.edu
54 CHAPTER III. BASIC PROOF TECHNIQUES Warning 6.8. The word \trivial" should not be used in a proof to mean \this step is easy, so I will skip it." Sometimes \trivial" proofs are not …
Announcements CS243: Discrete Structures - University of …
Is l Dillig, CS243: Discrete Structures Mathematical Proof Techniques 8/38 Direct Proof I To prove p ! q in a direct proof, rst assume p is true. I Then use rules of inference, axioms, previously …
Mathematical induction & Recursion - University of …
CS 441 Discrete mathematics for CS M. Hauskrecht CS 441 Discrete Mathematics for CS Lecture 15 Milos Hauskrecht milos@cs.pitt.edu 5329 Sennott Square Mathematical induction & …
Discrete Mathematics: Practice Problems - IIT Hyderabad
natorial proof that D(n) = (n 1)(D(n 1) + D(n 2)). 15. Let A= 1 1 2 3 and let An = a n b n c n d n . Find a recurrence relation for a n. 16. For each pair f;gof functions below, decide which of the …
Discrete Math - Some Practice Problems - University of Chicago
Discrete Math - Some Practice Problems 1. Give a combinatorial proof of the following identity: Xn k=0 n k!2 = 2 n! 2. Give a closed form expression for n 0! + n 2! + n 4! + n 6! +.... 3. Count the …
Introduction I Proofs Introduction II Introduction
Introduction to Discrete Mathematics Sections 1.5, 1.6, and 1.7 of Rosen cse235@cse.unl.edu Introduction I \A proof is a proof. What kind of a proof? It's a proof. A proof ... To begin with, …
Inference Rules and Proof Methods - uOttawa
A formal proof demonstrates that if the premises are true, then the conclusion is true. Note that the word formal here is not a synomym of rigorous. A formal proof is based simply on symbol …
