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| discrete math proof calculator: 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 calculator: 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 calculator: 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 calculator: 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 calculator: 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 calculator: 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 calculator: 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 calculator: 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 calculator: 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 calculator: Discrete Mathematics Richard Johnsonbaugh, 2009 For a one- or two-term introductory course in discrete mathematics. Focused on helping students understand and construct proofs and expanding their mathematical maturity, this best-selling text is an accessible introduction to discrete mathematics. Johnsonbaugh's algorithmic approach emphasizes problem-solving techniques. The Seventh Edition reflects user and reviewer feedback on both content and organization. |
| discrete math proof calculator: 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 calculator: 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 calculator: Java Programming Ralph Bravaco, Shai Simonson, 2009-02-01 Java Programming, From The Ground Up, with its flexible organization, teaches Java in a way that is refreshing, fun, interesting and still has all the appropriate programming pieces for students to learn. The motivation behind this writing is to bring a logical, readable, entertaining approach to keep your students involved. Each chapter has a Bigger Picture section at the end of the chapter to provide a variety of interesting related topics in computer science. The writing style is conversational and not overly technical so it addresses programming concepts appropriately. Because of the flexibile organization of the text, it can be used for a one or two semester introductory Java programming class, as well as using Java as a second language. The text contains a large variety of carefully designed exercises that are more effective than the competition. |
| discrete math proof calculator: Discrete Mathematics L. Lovász, J. Pelikán, K. Vesztergombi, 2003-01-27 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 calculator: Mathematics and Computation Avi Wigderson, 2019-10-29 From the winner of the Turing Award and the Abel Prize, an introduction to computational complexity theory, its connections and interactions with mathematics, and its central role in the natural and social sciences, technology, and philosophy Mathematics and Computation provides a broad, conceptual overview of computational complexity theory—the mathematical study of efficient computation. With important practical applications to computer science and industry, computational complexity theory has evolved into a highly interdisciplinary field, with strong links to most mathematical areas and to a growing number of scientific endeavors. Avi Wigderson takes a sweeping survey of complexity theory, emphasizing the field’s insights and challenges. He explains the ideas and motivations leading to key models, notions, and results. In particular, he looks at algorithms and complexity, computations and proofs, randomness and interaction, quantum and arithmetic computation, and cryptography and learning, all as parts of a cohesive whole with numerous cross-influences. Wigderson illustrates the immense breadth of the field, its beauty and richness, and its diverse and growing interactions with other areas of mathematics. He ends with a comprehensive look at the theory of computation, its methodology and aspirations, and the unique and fundamental ways in which it has shaped and will further shape science, technology, and society. For further reading, an extensive bibliography is provided for all topics covered. Mathematics and Computation is useful for undergraduate and graduate students in mathematics, computer science, and related fields, as well as researchers and teachers in these fields. Many parts require little background, and serve as an invitation to newcomers seeking an introduction to the theory of computation. Comprehensive coverage of computational complexity theory, and beyond High-level, intuitive exposition, which brings conceptual clarity to this central and dynamic scientific discipline Historical accounts of the evolution and motivations of central concepts and models A broad view of the theory of computation's influence on science, technology, and society Extensive bibliography |
| discrete math proof calculator: 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 calculator: Mathematical Structures for Computer Science Judith L. Gersting, 2007 This edition offers a pedagogically rich and intuitive introduction to discrete mathematics structures. It meets the needs of computer science majors by being both comprehensive and accessible. |
| discrete math proof calculator: 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) |
| discrete math proof calculator: Discrete Mathematics Norman Biggs, 2002-12-19 Discrete mathematics is a compulsory subject for undergraduate computer scientists. This new edition includes new chapters on statements and proof, logical framework, natural numbers and the integers and updated exercises from the previous edition. |
| discrete math proof calculator: 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 proof calculator: Gamma Julian Havil, 2017-10-31 Among the myriad of constants that appear in mathematics, p, e, and i are the most familiar. Following closely behind is g, or gamma, a constant that arises in many mathematical areas yet maintains a profound sense of mystery. In a tantalizing blend of history and mathematics, Julian Havil takes the reader on a journey through logarithms and the harmonic series, the two defining elements of gamma, toward the first account of gamma's place in mathematics. Introduced by the Swiss mathematician Leonhard Euler (1707-1783), who figures prominently in this. |
| discrete math proof calculator: Introduction to Probability Joseph K. Blitzstein, Jessica Hwang, 2014-07-24 Developed from celebrated Harvard statistics lectures, Introduction to Probability provides essential language and tools for understanding statistics, randomness, and uncertainty. The book explores a wide variety of applications and examples, ranging from coincidences and paradoxes to Google PageRank and Markov chain Monte Carlo (MCMC). Additional application areas explored include genetics, medicine, computer science, and information theory. The print book version includes a code that provides free access to an eBook version. The authors present the material in an accessible style and motivate concepts using real-world examples. Throughout, they use stories to uncover connections between the fundamental distributions in statistics and conditioning to reduce complicated problems to manageable pieces. The book includes many intuitive explanations, diagrams, and practice problems. Each chapter ends with a section showing how to perform relevant simulations and calculations in R, a free statistical software environment. |
| discrete math proof calculator: Refinement Calculus Ralph-Johan Back, Joakim Wright, 2012-12-06 Much current research in computer science is concerned with two questions: is a program correct? And how can we improve a correct program preserving correctness? This latter question is known as the refinement of programs and the purpose of this book is to consider these questions in a formal setting. In fact, correctness turns out to be a special case of refinement and so the focus is on refinement. Although a reasonable background knowledge is assumed from mathematics and CS, the book is a self-contained introduction suitable for graduate students and researchers coming to this subject for the first time. There are numerous exercises provided of varying degrees of challenge. |
| discrete math proof calculator: A Friendly Introduction to Mathematical Logic Christopher C. Leary, Lars Kristiansen, 2015 At the intersection of mathematics, computer science, and philosophy, mathematical logic examines the power and limitations of formal mathematical thinking. In this expansion of Leary's user-friendly 1st edition, readers with no previous study in the field are introduced to the basics of model theory, proof theory, and computability theory. The text is designed to be used either in an upper division undergraduate classroom, or for self study. Updating the 1st Edition's treatment of languages, structures, and deductions, leading to rigorous proofs of Gödel's First and Second Incompleteness Theorems, the expanded 2nd Edition includes a new introduction to incompleteness through computability as well as solutions to selected exercises. |
| discrete math proof calculator: Algebraic Methodology and Software Technology Michael Johnson, 1997-12 This book constitutes the refereed proceedings of the 6th International Conference on Algebraic Methodology and Software Engineering, AMAST'97, held in Sydney, Australia, in December 1997. The volume presents 48 revised full papers selected from an unusually high number of submissions. One of the outstanding features of AMAST is its mix of serious mathematical development of formal methods in software engineering with practical concerns, tools, case studies, and industrial development. The volume addresses all current aspects of formal methods in software engineering and programming methodology, with a certain emphasis on algebraic and logical foundations. |
| discrete math proof calculator: Approximately Calculus Shahriar Shahriari, 2006 Is there always a prime number between $n$ and $2n$? Where, approximately, is the millionth prime? And just what does calculus have to do with answering either of these questions? It turns out that calculus has a lot to do with both questions, as this book can show you. The theme of the book is approximations. Calculus is a powerful tool because it allows us to approximate complicated functions with simpler ones. Indeed, replacing a function locally with a linear--or higher order--approximation is at the heart of calculus. The real star of the book, though, is the task of approximating the number of primes up to a number $x$. This leads to the famous Prime Number Theorem--and to the answers to the two questions about primes. While emphasizing the role of approximations in calculus, most major topics are addressed, such as derivatives, integrals, the Fundamental Theorem of Calculus, sequences, series, and so on. However, our particular point of view also leads us to many unusual topics: curvature, Pade approximations, public key cryptography, and an analysis of the logistic equation, to name a few. The reader takes an active role in developing the material by solving problems. Most topics are broken down into a series of manageable problems, which guide you to an understanding of the important ideas. There is also ample exposition to fill in background material and to get you thinking appropriately about the concepts. Approximately Calculus is intended for the reader who has already had an introduction to calculus, but wants to engage the concepts and ideas at a deeper level. It is suitable as a text for an honors or alternative second semester calculus course. |
| discrete math proof calculator: Unifying Theories of Programming Steve Dunne, 2006-06-27 This book constitutes the thoroughly refereed post-proceedings of the First International Symposium on Unifying Theories of Programming, UTP 2006, held at Walworth Castle, County Durham, UK, in February 2006. The book presents 14 revised full papers. Based on the pioneering work on unifying theories of programming by Tony Haare and Jifeng He, UTP 2006 focused on the most significant results and raised awareness of the benefits of unifying theoretical frameworks. |
| discrete math proof calculator: 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 calculator: 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 proof calculator: Street-Fighting Mathematics Sanjoy Mahajan, 2010-03-05 An antidote to mathematical rigor mortis, teaching how to guess answers without needing a proof or an exact calculation. In problem solving, as in street fighting, rules are for fools: do whatever works—don't just stand there! Yet we often fear an unjustified leap even though it may land us on a correct result. Traditional mathematics teaching is largely about solving exactly stated problems exactly, yet life often hands us partly defined problems needing only moderately accurate solutions. This engaging book is an antidote to the rigor mortis brought on by too much mathematical rigor, teaching us how to guess answers without needing a proof or an exact calculation. In Street-Fighting Mathematics, Sanjoy Mahajan builds, sharpens, and demonstrates tools for educated guessing and down-and-dirty, opportunistic problem solving across diverse fields of knowledge—from mathematics to management. Mahajan describes six tools: dimensional analysis, easy cases, lumping, picture proofs, successive approximation, and reasoning by analogy. Illustrating each tool with numerous examples, he carefully separates the tool—the general principle—from the particular application so that the reader can most easily grasp the tool itself to use on problems of particular interest. Street-Fighting Mathematics grew out of a short course taught by the author at MIT for students ranging from first-year undergraduates to graduate students ready for careers in physics, mathematics, management, electrical engineering, computer science, and biology. They benefited from an approach that avoided rigor and taught them how to use mathematics to solve real problems. Street-Fighting Mathematics will appear in print and online under a Creative Commons Noncommercial Share Alike license. |
| discrete math proof calculator: Probability Rick Durrett, 2010-08-30 This classic introduction to probability theory for beginning graduate students covers laws of large numbers, central limit theorems, random walks, martingales, Markov chains, ergodic theorems, and Brownian motion. It is a comprehensive treatment concentrating on the results that are the most useful for applications. Its philosophy is that the best way to learn probability is to see it in action, so there are 200 examples and 450 problems. The fourth edition begins with a short chapter on measure theory to orient readers new to the subject. |
| discrete math proof calculator: The American Mathematical Monthly , 1991 |
| discrete math proof calculator: Resources in Education , 1995-07 |
| discrete math proof calculator: 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 calculator: Computational Complexity Sanjeev Arora, Boaz Barak, 2009-04-20 New and classical results in computational complexity, including interactive proofs, PCP, derandomization, and quantum computation. Ideal for graduate students. |
| discrete math proof calculator: Undergraduate Catalog University of Michigan--Dearborn, 2006 |
| discrete math proof calculator: Essential Discrete Mathematics for Computer Science Harry Lewis, Rachel Zax, 2019-03-19 Discrete mathematics is the basis of much of computer science, from algorithms and automata theory to combinatorics and graph theory. Essential Discrete Mathematics for Computer Science aims to teach mathematical reasoning as well as concepts and skills by stressing the art of proof. It is fully illustrated in color, and each chapter includes a concise summary as well as a set of exercises. |
| discrete math proof calculator: Discrete Optimization E. Boros, P.L. Hammer, 2003-03-19 One of the most frequently occurring types of optimization problems involves decision variables which have to take integer values. From a practical point of view, such problems occur in countless areas of management, engineering, administration, etc., and include such problems as location of plants or warehouses, scheduling of aircraft, cutting raw materials to prescribed dimensions, design of computer chips, increasing reliability or capacity of networks, etc. This is the class of problems known in the professional literature as discrete optimization problems. While these problems are of enormous applicability, they present many challenges from a computational point of view. This volume is an update on the impressive progress achieved by mathematicians, operations researchers, and computer scientists in solving discrete optimization problems of very large sizes. The surveys in this volume present a comprehensive overview of the state of the art in discrete optimization and are written by the most prominent researchers from all over the world.This volume describes the tremendous progress in discrete optimization achieved in the last 20 years since the publication of Discrete Optimization '77, Annals of Discrete Mathematics, volumes 4 and 5, 1979 (Elsevier). It contains surveys of the state of the art written by the most prominent researchers in the field from all over the world, and covers topics like neighborhood search techniques, lift and project for mixed 0-1 programming, pseudo-Boolean optimization, scheduling and assignment problems, production planning, location, bin packing, cutting planes, vehicle routing, and applications to graph theory, mechanics, chip design, etc.Key features:• state of the art surveys• comprehensiveness• prominent authors• theoretical, computational and applied aspects.This book is a reprint of Discrete Applied Mathematics Volume 23, Numbers 1-3 |
| discrete math proof calculator: 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 proof calculator: MAA Notes , 1983 |
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 …
Math 03.150 - Discrete Mathematics - Rowan University
Math 03.150 - Discrete Mathematics . CATALOG DESCRIPTION: Math 03.150 Discrete Mathematics, 3s.h. ... The use of graphing calculator is required. OBJECTIVES: In terms of …
Discrete Mathematics - Rules of Inference and Mathematical Proofs
The proof can have various forms, e.g.: direct proof (using P to directly show C) indirect proof proof by contraposition (proving contrapostion :C ):P proof by contradiction (reductio ad …
Discrete Mathematics, Chapter 5: Induction and Recursion
Proof of the Fundamental Theorem of Arithmetic, using Strong Induction Show that if n is an integer 2, then n can be written as the product of primes. Solution: Let P(n) be the proposition …
Proof of the Master Method - Department of Computer Science
Proof of the Master Method Theorem (Master Method) Consider the recurrence T(n) = aT(n=b) + f(n); (1) where a;b are constants. Then (A)If f(n) = O(nlog b a ") for some constant " > 0, then …
CS 2336 Discrete Mathematics - National Tsing Hua University
Discrete Mathematics Lecture 4 Proofs: Methods and Strategies 1 . Outline •What is a Proof ? •Methods of Proving •Common Mistakes in Proofs •Strategies : How to Find a Proof ? 2 . What …
Lecture 35: Intro to Posets - MIT Mathematics
Proof. If Ais an antichain of P of maximum size and mis the size of any minimum-size chain cover of P, then the fact that every chain of P intersects Ain at most one element ensures that |A|≤m. …
MATH 113: DISCRETE STRUCTURES SUNZI’S THEOREM - Reed …
MATH 113: DISCRETE STRUCTURES SUNZI’S THEOREM The Chinese mathematician Sunzi Suanjing considered the following problem in the 3-rd cen-tury C.E. A general arrays his …
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 …
Discrete MathematicsDiscrete Mathematics CS 2610
Discrete MathematicsDiscrete Mathematics CS 2610 February 26, 2009 -- part 1. Big-O Notation Big-O notation is used to express the time complexity of an algorithm ... Proof: first note that …
Proof by Contradiction - Gordon College
A contrapositive proof seems more reasonable: assume n is odd and show that n3 +5 is even. The second approach works well for this problem. However, today we want try another …
Discrete Maths: Exercises & Solutions
Page 8 of 14 The statement “x is greater than 3” has two parts.The first part, the variable x, is the subject of the statement. The second part—the predicate, “is greater than 3”—refers to a …
Predicate Logic and Quantifiers - Computer Science and Engineering
Let P( x) be the predicate “ must take a discrete mathematics course” and let Q(x) be the predicate “x is a computer science student”. The universe of discourse for both P(x) and Q(x) …
Informal proofs. Types of proofs. - University of Pittsburgh
• Proof by contradiction – Show that (p ∧¬ q) contradicts the assumptions • Proof by cases • Proofs of equivalence –p ↔q is replaced with (p →q) ∧(q →p) Sometimes one method of proof …
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 …
Mustafa Jarrar: Lecture Notes in Discrete Mathema6cs Birzeit …
Mustafa Jarrar: Lecture Notes in Discrete Mathema6cs. Birzeit University, Palestine, 2015 mjarrar©2015 Set Theory 6.1. Basics of Set Theory 6.2 Properties of Sets and Element …
Discrete Mathematics Introduction to Proofs Definition: A …
Discrete Mathematics Introduction to Proofs Definition: A theorem is a statement that can be shown to be true. We demonstrate that a theorem is true with a proof (valid argument) using: …
TOPICS IN DISCRETE MATHEMATICS - Harvey Mudd College
example, in the discrete calculus, (presented in Chapter 3 of this book), despite many e orts, the notation is not quite so natural and suggestive. It may also just be the case that human …
Discrete Mathematics (Math 151)
Discrete Mathematics (Math 151) Department of Mathematics College of Sciences King Saud University ... results, and rules of inference to complete the proof. (King Saud University) …
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 • …
The Foundations: Logic and Proof, Sets, and Functions
Proof, Sets, and Functions The Foundations: Logic and Proof, Sets, and Functions Spring 2005 CS 233601 Discrete Math by Shun-Ren Yang, CS, NTHU 2 Outline ꅅLogic ꅅPropositional …
Proof Calculator Discrete Math (PDF) - ftp.marmaranyc.com
Proof Calculator Discrete Math Discrete Mathematics Oscar Levin,2016-08-16 This gentle introduction to discrete mathematics is written for first and second year math majors especially …
Introduction to Number Theory CS311H: Discrete Mathematics …
In nitely Many Primes I Theorem:There are in nitely many prime numbers. I Proof:(by contradiction) Suppose there are nitely many primes: p1;p2;:::;pn I Now consider the number …
Announcements CS243: Discrete Structures - University of Texas …
Combining Proofs, cont. I Now, employ proof by contradiction to show pr 2 is irrational. I Suppose pr 2 was rational. I Then, for some integers p;q: pr 2 = p q I This can be rewritten as p 2 = rq p …
Module 6.3: Contrapositives, Converses, and Counter-Examples
The primary reason is that an entire method of proof, called proof by contraposition, is based upon exactly this. Moreover, I have found that these logical exercises make me a much more e …
Discrete Maths: Exercises & Solutions
“Maria will find a good job when she learns discrete mathematics.” “For Maria to get a good job, it is sufficient for her to learn discrete mathematics.” and “Maria will find a good job unless she …
1 Quantifiers - Khoury College of Computer Sciences
Discrete Math in CS Quantifiers CS 280 Fall 2005 (Kleinberg) 1 Quantifiers To formulate more complex mathematical statements, we use the quantifiers there exists, written ∃, and for all, …
Notes on Discrete Mathematics - Yale University
8 Jun 2022 · Contents Tableofcontentsii Listoffiguresxvii Listoftablesxix Listofalgorithmsxx Prefacexxi Resourcesxxii 1 Introduction1 1.1 ...
CALCCHECK: A Proof Checker for Teaching - Springer
the “Logical Approach to Discrete Math” Wolfram Kahl(B) McMaster University, Hamilton, ON, Canada kahl@cas.mcmaster.ca Abstract. For calculational proofs as they are propagated by …
Assessing Proofs With Rubrics: the RVF Method - MAA
the terms deduction, inference, evidence, and precedent with their notion of proof in X, where Xis one of the contexts listed. The results of these discussions has been observed and recorded …
Conditional Statement/Implication CSCI 1900 Discrete Structures
CSCI 1900 – Discrete Structures Conditional Statements – Page 9 Equivalence Truth table • The only time that the expression can evaluate as true is if both statements, p and q, are true or …
DISCRETE MATH: LECTURE 15 - Saint Louis University
DISCRETE MATH: LECTURE 15 5 1.2. Operations on Sets. Let A and B be subsets of a set U. (1) The union of A and B, denoted A [B, is the set of all elements that are in at least one of A …
Gerard O’Regan Guide to Discrete Mathematics
mon proof technique in mathematics, and there are two parts to a proof by induction (the base case and the inductive step). We discuss strong and weak induction, and we discuss how …
CHAPTER 4 DirectProof - Virginia Commonwealth University
90 DirectProof Definition4.4 Suppose aandb areintegers. Wesaythat dividesb, written aj b,if ˘ac forsome c2Z.Inthiscasewealsosaythat isa divisorof b,andthat isamultipleofa. For example, …
Schaum's Outline of Discrete Mathematics, Third Edition …
CONTENTS ix 7.8 Chebyshev’s Inequality, Law of Large Numbers 135 SolvedProblems 136 SupplementaryProblems 149 CHAPTER 8 Graph Theory 154 8.1 Introduction, Data Structures …
Proof by Contradiction - University of British Columbia
6. Proof by Contradic-tion 6.1 Proving Statements with Con-tradiction 6.2 Proving Conditional Statements by Contra-diction 6.3 Combining Techniques The square root of two is irrational. …
Discrete Mathematics for Computer Science - UH
Sets, Proof Templates, and Induction x e A x is an element ofA 1.1 x f A x is not an element ofA 1.1 Ix x E A and P(x)} Set notation 1.1 N Natural numbers 1.1.1l 2 Integers 1.1.1 Q Rationals …
Discrete Mathematics - University of Cambridge
DISCRETE MATHEMATICSEXERCISES 1 – SOLUTIONS WITH COMMENTARY This is an example of an existence proof, which tend to look a bit backwards when written formally: …
151 Math Exercises (3,1) Methods of Proof
Math 151 Discrete Mathematics [Methods of Proof] By: Malek Zein AL-Abidin Proof by Contraposition Proofs by contraposition make use of the fact that the conditional statement p …
MATH0103 Discrete Mathematics - UCL
MATH0103 Discrete Mathematics Year: 2024–2025 Code: MATH0103 Level: 5 (UG) Normal student group(s): UG Year 2 BASc and NatSci students Value: 15 credits (= 7.5 ECTS credits) …
Dijkstra’s Shortest Path Packet - Discrete Math Hub
Math-270: Discrete Mathematics November 21, 2019 Inputs: The set of vertices, V. The weights of the edges, as a two-dimensional array, where weight[i,j]isthe weight of the edge from v i to v …
Lecture 3.2: Parity, and proving existential statements
Math 4190, Discrete Mathematical Structures M. Macauley (Clemson) Lecture 3.2: Parity, & proving existential statements Discrete Mathematical Structures 1 / 7. Overview De nition ...
Syllabus - MATH 180 - Discrete Computational Structures
2. Applied Discrete Structures by Alan Doerr and Kenneth Levasseur 3. Proofs and Concepts: The Fundamentals of Mathematics by Morris & Morris Course Description, Expectations, and …
Chapter 2.2 Conditional Statements - Saint Louis University
DISCRETE MATH: LECTURE 3 3 1.4. Contrapositive, Converse, Inverse{Words that made you tremble in high school geometry. The contrapositive of a conditional statement of the form p !q …
Prime factorization - Department of Mathematics
CU, Discrete Math, March 20 & 30, 2020. Finally we’ll realize our longterm goal of proving: The Fundamental Theorem of Arithmetic ... Proof of 1. Clearly Q i2N p min( ei; f) i divides a = Q …
Permutations CS311H: Discrete Mathematics Permutations and …
Instructor: Is l Dillig, CS311H: Discrete Mathematics Permutations and Combinations 10/26 Examples I How many hands of 5 cards can be dealt from a standard deck of 52 cards? I I …
Combinatorial Proof Examples - Department of Mathematics
Combinatorial Proof Examples September 29, 2020 A combinatorial proof is a proof that shows some equation is true by ex-plaining why both sides count the same thing. Its structure should …
Discrete - openmathbooks.github.io
Acknowledgements Thisbookwouldnotexistifnotfor“DiscreteandCombinatorialMathemat-ics”byRichardGrasslandTabithaMingus. ItisthebookIlearneddiscrete
MATH1081 Discrete Mathematics INFORMATION BOOKLET
MATH1081, Discrete Mathematics, is a first year 6UOC course available in semester 1 and semester ... 1988 or “Proof in Mathematics: An Introduction”, Quakers Hill Press, 1995. …
Discrete Mathematics with Proof - GBV
3 Proof 85 3.1 Introduction to Mathematical Proof 85 3.1.1 Mathematics and Proof: The Big Picture 86 3.1.2 Mathematical Objects Related to Proofs 87 3.1.3 Exercises 91 3.2 Elementary …
