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| discrete math proof examples: 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 proof examples: Introduction to Discrete Mathematics via Logic and Proof Calvin Jongsma, 2019-11-08 This textbook introduces discrete mathematics by emphasizing the importance of reading and writing proofs. Because it begins by carefully establishing a familiarity with mathematical logic and proof, this approach suits not only a discrete mathematics course, but can also function as a transition to proof. Its unique, deductive perspective on mathematical logic provides students with the tools to more deeply understand mathematical methodology—an approach that the author has successfully classroom tested for decades. Chapters are helpfully organized so that, as they escalate in complexity, their underlying connections are easily identifiable. Mathematical logic and proofs are first introduced before moving onto more complex topics in discrete mathematics. Some of these topics include: Mathematical and structural induction Set theory Combinatorics Functions, relations, and ordered sets Boolean algebra and Boolean functions Graph theory Introduction to Discrete Mathematics via Logic and Proof will suit intermediate undergraduates majoring in mathematics, computer science, engineering, and related subjects with no formal prerequisites beyond a background in secondary mathematics. |
| discrete math proof examples: Discrete Mathematics Rowan Garnier, John Taylor, 2009-11-09 Taking an approach to the subject that is suitable for a broad readership, Discrete Mathematics: Proofs, Structures, and Applications, Third Edition provides a rigorous yet accessible exposition of discrete mathematics, including the core mathematical foundation of computer science. The approach is comprehensive yet maintains an easy-to-follow prog |
| discrete math proof examples: 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 proof examples: 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 proof examples: 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 proof examples: Discrete Mathematics - Proof Techniques And Mathematical Structures Robert Clark Penner, 1999-10-19 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 proof examples: 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 proof examples: 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 proof examples: 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 proof examples: 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 proof examples: 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 proof examples: 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 proof examples: 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 proof examples: 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 proof examples: 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 proof examples: Proofs and Fundamentals Ethan D. Bloch, 2011-02-15 “Proofs and Fundamentals: A First Course in Abstract Mathematics” 2nd edition is designed as a transition course to introduce undergraduates to the writing of rigorous mathematical proofs, and to such fundamental mathematical ideas as sets, functions, relations, and cardinality. The text serves as a bridge between computational courses such as calculus, and more theoretical, proofs-oriented courses such as linear algebra, abstract algebra and real analysis. This 3-part work carefully balances Proofs, Fundamentals, and Extras. Part 1 presents logic and basic proof techniques; Part 2 thoroughly covers fundamental material such as sets, functions and relations; and Part 3 introduces a variety of extra topics such as groups, combinatorics and sequences. A gentle, friendly style is used, in which motivation and informal discussion play a key role, and yet high standards in rigor and in writing are never compromised. New to the second edition: 1) A new section about the foundations of set theory has been added at the end of the chapter about sets. This section includes a very informal discussion of the Zermelo– Fraenkel Axioms for set theory. We do not make use of these axioms subsequently in the text, but it is valuable for any mathematician to be aware that an axiomatic basis for set theory exists. Also included in this new section is a slightly expanded discussion of the Axiom of Choice, and new discussion of Zorn's Lemma, which is used later in the text. 2) The chapter about the cardinality of sets has been rearranged and expanded. There is a new section at the start of the chapter that summarizes various properties of the set of natural numbers; these properties play important roles subsequently in the chapter. The sections on induction and recursion have been slightly expanded, and have been relocated to an earlier place in the chapter (following the new section), both because they are more concrete than the material found in the other sections of the chapter, and because ideas from the sections on induction and recursion are used in the other sections. Next comes the section on the cardinality of sets (which was originally the first section of the chapter); this section gained proofs of the Schroeder–Bernstein theorem and the Trichotomy Law for Sets, and lost most of the material about finite and countable sets, which has now been moved to a new section devoted to those two types of sets. The chapter concludes with the section on the cardinality of the number systems. 3) The chapter on the construction of the natural numbers, integers and rational numbers from the Peano Postulates was removed entirely. That material was originally included to provide the needed background about the number systems, particularly for the discussion of the cardinality of sets, but it was always somewhat out of place given the level and scope of this text. The background material about the natural numbers needed for the cardinality of sets has now been summarized in a new section at the start of that chapter, making the chapter both self-contained and more accessible than it previously was. 4) The section on families of sets has been thoroughly revised, with the focus being on families of sets in general, not necessarily thought of as indexed. 5) A new section about the convergence of sequences has been added to the chapter on selected topics. This new section, which treats a topic from real analysis, adds some diversity to the chapter, which had hitherto contained selected topics of only an algebraic or combinatorial nature. 6) A new section called ``You Are the Professor'' has been added to the end of the last chapter. This new section, which includes a number of attempted proofs taken from actual homework exercises submitted by students, offers the reader the opportunity to solidify her facility for writing proofs by critiquing these submissions as if she were the instructor for the course. 7) All known errors have been corrected. 8) Many minor adjustments of wording have been made throughout the text, with the hope of improving the exposition. |
| discrete math proof examples: Discrete Mathematics Jean Gallier, 2011-02-01 This books gives an introduction to discrete mathematics for beginning undergraduates. One of original features of this book is that it begins with a presentation of the rules of logic as used in mathematics. Many examples of formal and informal proofs are given. With this logical framework firmly in place, the book describes the major axioms of set theory and introduces the natural numbers. The rest of the book is more standard. It deals with functions and relations, directed and undirected graphs, and an introduction to combinatorics. There is a section on public key cryptography and RSA, with complete proofs of Fermat's little theorem and the correctness of the RSA scheme, as well as explicit algorithms to perform modular arithmetic. The last chapter provides more graph theory. Eulerian and Hamiltonian cycles are discussed. Then, we study flows and tensions and state and prove the max flow min-cut theorem. We also discuss matchings, covering, bipartite graphs. |
| discrete math proof examples: Invitation to Discrete Mathematics Jiří Matoušek, Jaroslav Nešetřil, 2009 A clear and self-contained introduction to discrete mathematics for undergraduates and early graduates. |
| discrete math proof examples: Discrete Mathematics Robert C. Penner, 2003 |
| discrete math proof examples: 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 proof examples: 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 proof examples: A Comprehensive Treatment of q-Calculus Thomas Ernst, 2012-09-08 To date, the theoretical development of q-calculus has rested on a non-uniform basis. Generally, the bulky Gasper-Rahman notation was used, but the published works on q-calculus looked different depending on where and by whom they were written. This confusion of tongues not only complicated the theoretical development but also contributed to q-calculus remaining a neglected mathematical field. This book overcomes these problems by introducing a new and interesting notation for q-calculus based on logarithms.For instance, q-hypergeometric functions are now visually clear and easy to trace back to their hypergeometric parents. With this new notation it is also easy to see the connection between q-hypergeometric functions and the q-gamma function, something that until now has been overlooked. The book covers many topics on q-calculus, including special functions, combinatorics, and q-difference equations. Apart from a thorough review of the historical development of q-calculus, this book also presents the domains of modern physics for which q-calculus is applicable, such as particle physics and supersymmetry, to name just a few. |
| discrete math proof examples: 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 proof examples: 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 proof examples: 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 proof examples: 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 proof examples: 100% Mathematical Proof Rowan Garnier, John Taylor, 1996-08 Proof has been and remains one of the concepts which characterises mathematics. Covering basic propositional and predicate logic as well as discussing axiom systems and formal proofs, the book seeks to explain what mathematicians understand by proofs and how they are communicated. The authors explore the principle techniques of direct and indirect proof including induction, existence and uniqueness proofs, proof by contradiction, constructive and non-constructive proofs, etc. Many examples from analysis and modern algebra are included. The exceptionally clear style and presentation ensures that the book will be useful and enjoyable to those studying and interested in the notion of mathematical proof. |
| discrete math proof examples: 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 proof examples: Discrete and Combinatorial Mathematics: An applied Introduction ( For VTU) Grimaldi Ralph P., 2013 |
| discrete math proof examples: 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 proof examples: Models and Computability S. Barry Cooper, John K. Truss, Association for Symbolic Logic, 1999-06-17 Second of two volumes providing a comprehensive guide to the current state of mathematical logic. |
| discrete math proof examples: Exploring Mathematics John Meier, Derek Smith, 2017-08-07 With exercises and projects, Exploring Mathematics supports an active approach to the transition to upper-level theoretical math courses. |
| discrete math proof examples: Introduction to Mathematical Proofs Charles Roberts, 2024-10-14 This book is designed to prepare students for higher mathematics by focusing on the development of theorems and proofs. Beginning with logic, the text discusses deductive mathematical systems and the systems of natural numbers, integers, rational numbers, and real numbers. It covers elementary topics in set theory, explores various properties of |
| discrete math proof examples: Discrete Mathematics Martin Aigner, 2023-01-24 The advent of fast computers and the search for efficient algorithms revolutionized combinatorics and brought about the field of discrete mathematics. This book is an introduction to the main ideas and results of discrete mathematics, and with its emphasis on algorithms it should be interesting to mathematicians and computer scientists alike. The book is organized into three parts: enumeration, graphs and algorithms, and algebraic systems. There are 600 exercises with hints and solutions to about half of them. The only prerequisites for understanding everything in the book are linear algebra and calculus at the undergraduate level. Praise for the German edition… This book is a well-written introduction to discrete mathematics and is highly recommended to every student of mathematics and computer science as well as to teachers of these topics. —Konrad Engel for MathSciNet Martin Aigner is a professor of mathematics at the Free University of Berlin. He received his PhD at the University of Vienna and has held a number of positions in the USA and Germany before moving to Berlin. He is the author of several books on discrete mathematics, graph theory, and the theory of search. The Monthly article Turan's graph theorem earned him a 1995 Lester R. Ford Prize of the MAA for expository writing, and his book Proofs from the BOOK with Günter M. Ziegler has been an international success with translations into 12 languages. |
| discrete math proof examples: Further Pure Mathematics Linda Bostock, Suzanne Chandler, C. Rourke, 1982 This volume continues the work covered in Core Maths or Mathematics - The Core Course for Advanced Level to provide a full two-year course in Pure Mathematics for A-Level. |
| discrete math proof examples: 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 proof examples: 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 proof examples: Mathematics for Machine Learning Marc Peter Deisenroth, A. Aldo Faisal, Cheng Soon Ong, 2020-04-23 Distills key concepts from linear algebra, geometry, matrices, calculus, optimization, probability and statistics that are used in machine learning. |
| discrete math proof examples: 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. |
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 NVIDIA …
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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 Changes. …
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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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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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 …
Introduction to Proofs - Kent
Proving Conditional Statements: p → q Direct Proof: Assume that p is true.Bring axioms, definitions, rules of inference, and logical equivalences to show that q must also be true. …
Direct Proofs - Stanford University
Our First Direct Proof Theorem: If n is an even integer, then n2 is even. Proof:Let n be an even integer. Since n is even, there is some integer k such that n = 2k. This means that n2 = (2k)2 = …
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 …
Lecture 12: Introduction to Graphs and Trees - Northeastern …
If a counterexample is hard to find, a proof might be easier CS 5002: Discrete Math ©Northeastern University Fall 2018 5. Proof by Induction ... Lots of examples CS 5002: …
Discrete Mathematics - (Functions) - Stony Brook University
Proof Directproof. Supposex 1 andx 2 areelementsofXsuchthatf(x 1) = f(x 2). Showthatx 1 = x 2. Problem Provethatafunctionfisnotone-to-one. Proof Counterexample. Findelements x 1 and x 2 …
Proof By Contraposition - University of Toronto Department …
Proof By Contraposition by L. Shorser The contrapositive of the statement \A → B" (i.e., \A implies B.") is the statement \∼ B →∼ A" (i.e., \B is not true implies that A is not true."). These two …
csci54 – discrete math & functional programming proofs: …
discrete math so far sets introductions to propositional and predicate logic reflections on what it means to prove something this week: proof techniques group meeting Thursday/Friday …
Math 3355 Some Review Problems Solutions - d.umn.edu
Math 3355 Fall 2018 Some Review Problems for Exam 1: Solutions Here is my quick review of proof techniques. I will focus exclusively on propositions of the form p !q; or more properly, …
5 Mathematical Induction - Carleton
Apr 5, 2021 · proof of P (0) is called the base case , and the proof that P (n 1) ) P (n ) is called the inductive case. In the same way that a recursive solution to a problem relies o n solu-tions to a …
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 …
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 …
2 The Well Ordering Principle - MIT OpenCourseWare
easy proof that every integer greater than one can be expressed as some product of primes. Theorem 2.3.1. Every positive integer greater than one can be factored as a prod-uct of …
151 Math Exercises (3,1) Methods of Proof
Math 151 Discrete Mathematics [Methods of Proof] By: Malek Zein AL-Abidin Proofs by Contradiction Suppose we want to prove that a statement p is true.Furthermore, suppose that …
Handout 01: Discrete Math Review Sheet - Department of …
Handout 01: Discrete Math Review Sheet CS Theory Fall 2022 Cyrus Illick and Walt McKelvie September 19, 2022 Intro This document is a review of Discrete Math. The following list of …
CSL105: Discrete Mathematical Structures - IIT Delhi
Vacuous proof: When proving p !q, a proof showing p to be false is called a vacuous proof. Example: Show that the proposition P(0) is true, where P(n) is \if n > 1, then n2 > n" and the …
Discrete Mathematic Chapter 1: Logic and Proof 1.5 Rules of …
is used in proof Equivalence (p ↔ q) Meaning: pis equal to q p ↔ q mean q ↔ p Only equivalence rules can be used p ↔ qcan be proved by showing p q and q p is used in proof …
Question 1. Prove using mathematical induction that for all …
Induction Examples Question 6. Let p0 = 1, p1 = cos (for some xed constant) and pn+1 = 2p1pn pn 1 for n 1.Use an extended Principle of Mathematical Induction to prove that pn = cos(n ) for …
Sets and set operations - University of Pittsburgh
2 CS 441 Discrete mathematics for CS M. Hauskrecht Set • Definition: A set is a (unordered) collection of objects. These objects are sometimes called elements or members of the set. …
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 …
Chapter 4.3 Principles of Inclusion and Exclusion - University …
6 Principle of Inclusion and Exclusion (cont’d) The principle of inclusion and exclusion calculates for given the sets of events A 1, …, A n, the total number of events “A 1 OR…,OR A n”. Note: …
CMSC 250: Weak and Strong Mathematical Induction - UMD
2.3 Examples Example 2.1. Let’s prove that 2n n2 0 for all n 4. Here P(n) is the statement 2n n2 0 and the starting value is n 0 = 4. (a)The Base Case: ... This is not a formal proof by strong …
Announcements CS243: Discrete Structures - University of …
Is l Dillig, CS243: Discrete Structures Mathematical Proof Techniques 9/38 More Direct Proof Examples I An integer a is called aperfect squareif there exists an integer b such that a = b2. I …
Informal proofs - University of Pittsburgh
CS 441 Discrete mathematics for CS M. Hauskrecht Methods of proving theorems Basic methods to prove the theorems: • Direct proof –p q is proved by showing that if p is true then q follows • …
A good source of exam problems. Provide a proof or a …
You may not use a math symbol for not (e.g., ˘). In the open sentence, you may use English words. 9x 2Z>0 x2 + x+ 41 is not prime A3. Is Conjecture A true or false? (circle one) true …
Discrete Mathematics - (Sets) - Stony Brook University
Venndiagrams Definition Relationshipbetweenasmallnumberofsetscanberepresented bypicturescalledVenndiagrams Problems Write a Venn diagram representing sets of numbers:
Proof By Counterexample - Department of Mathematics
Proof. To prove that this is true, we need to nd a pair of linear functions in one variable that are not perpendicular. We suggest the following counterexample: f 1(x) = 3x+4 f 2(x) = 2x−1 In …
Discrete Mathematics: An Open Introduction
No. Whenever we nd an \answer" in math, we really have a (perhaps hidden) argument - given the situation we are in, we can conclude the answer is the case. Of course real mathematics is …
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 …
Discrete Structures Lecture Notes - Stanford University
Discrete Structures Lecture Notes Vladlen Koltun1 Winter 2008 1Computer Science Department, 353 Serra Mall, Gates 374, Stanford University, Stanford, CA 94305, USA; …
Functions CS311H: Discrete Mathematics Functions I
Instructor: Is l Dillig, CS311H: Discrete Mathematics Functions 26/46 Floor and Ceiling Functions I Two important functions in discrete math are oorandceiling functions, both from R to Z I The …
1. Introduction - University of California, San Diego
Mathematical Database Page 5 of 21 Theorem 3.2. (Principle of Mathematical Induction, Variation 2) Let ( )Sn denote a statement involving a variable n.Suppose (1) S(1) and S(2) are true; (2) if …
Discrete Mathematics - (Relations) - Stony Brook University
Example: Congruencemodulon Solution R isReflexive.Showthat∀a ∈Z,n |(a −a).Weknowthat a−a = 0 andn |0. Hence,n |(a−a). R is Symmetric.Show that ∀a,b ∈Z, if a ≡b (mod n), then b ≡a …
B Exercises Exercise Sheet 1: Propositional Logic
Proof Suppose the conclusion of the Theorem is false. Then x =3and y =8. But then x + y =11, which contradicts the assumption that x + y =10. Hence the conclusion must be true. 3. Give a …
MATH208: DISCRETE MATHEMATICS - University of North …
DISCRETE MATHEMATICS DEPARTMENT OF MATHEMATICS THE UNIVERSITY OF NORTH DAKOTA. ... 7 Styles of Proof 81 7.1 Direct proof 81 7.2 Indirect proof 84. 6 7.3 Proof by …
Discrete Mathematics - (Predicate Logic) - Stony Brook …
Examples ∀realx,ifx > 10,thenx2 > 100. Negation: ∃realx suchthatx > 10 andx2 ≤100. If a computer program has more than 100,000 lines, then it containsabug. Negation: There is at …
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 Proofs - Stanford University
A Simple Direct Proof Theorem: If n is an even integer, then n2 is even. Proof: Let n be an even integer. Since n is even, there is some integer k such that n = 2k. This means that n2 = (2k)2 = …
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 …
Some Sample Proofs
Proof by contraposition: We will prove the equivalent statement that if n is odd, then 5n+2 is odd. Assume n is odd. Then n = 2k + 1 for some integer k. Then we have that 5n+2 = 5(2k +1)+2 = …
Combinations with Repetition - University of Texas at Austin
Example 1 I Suppose there is a bowl containing apples, oranges, and pears I There is at least four of each type of fruit in the bowl I How many ways to select four pieces of fruit from this bowl? I …
2.1 Asymptotic Analysis - Department of Computer Science
Let’s see an example proof of the rst of these examples. Theorem 2.1.2 2n2 + 27 = O(n2) Proof: Let’s set c= 3. Now suppose that 2n2 + 27 >cn2 = 3n2. Then simple algebra implies ... You …
Chapter 14 Big-O - University of Illinois Urbana-Champaign
use expands to larger examples. For example, a small database program developed for a community college might have trouble coping if deployed to handle (say) all registration …
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, …
Predicate Logic and Quantifiers - Computer Science and …
Examples & Exercises Predicate Logic and Quantifiers Slides by Christopher M. Bourke Instructor: Berthe Y. Choueiry Fall 2007 ... Let P( x) be the predicate “ must take a discrete …
Pigeonhole Principle - Lecture Notes in Math 212 Discrete …
Proof by contrapositive (by contradiction) If each box contained k objects, there would be nk objects totally. If each box contained k objects, there would be nk objects totally. Examples …
Discrete Mathematics Predicates and Quantifiers - UH
Discrete Mathematics Predicates and Quantifiers Predicates Propositional logic is not enough to express the meaning of all statements in mathematics and natural language. Examples: Is “ > …
Quantifiers, Proofs and Sets - Department of Computer Science
Set Relations Set A is a subset of set B if and only if every element of A is also present in B (definition) – B is a superset of A Sets A and B are equal if and only if A ⊆ B and B ⊆ A …
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 …
Indirect (“Contra”) Proof Examples
CSc 144 — Discrete Structures for Computer Science I (McCann) (Last Modified: March 2023) Indirect (“Contra”) Proof Examples ... Two examples are -3, 0, 3 and 15, 18, 21. Because we …
