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| discrete mathematics questions and answers: 2000 Solved Problems in Discrete Mathematics Seymour Lipschutz, 2012-09-17 Master discrete mathematics with Schaum's--the high-performance solved-problem guide. It will help you cut study time, hone problem-solving skills, and achieve your personal best on exams! Students love Schaum's Solved Problem Guides because they produce results. Each year, thousands of students improve their test scores and final grades with these indispensable guides. Get the edge on your classmates. Use Schaum's! If you don't have a lot of time but want to excel in class, use this book to: Brush up before tests Study quickly and more effectively Learn the best strategies for solving tough problems in step-by-step detail Review what you've learned in class by solving thousands of relevant problems that test your skill Compatible with any classroom text, Schaum's Solved Problem Guides let you practice at your own pace and remind you of all the important problem-solving techniques you need to remember--fast! And Schaum's are so complete, they're perfect for preparing for graduate or professional exams. Inside you will find: 2,000 solved problems with complete solutions--the largest selection of solved problems yet published on this subject An index to help you quickly locate the types of problems you want to solve Problems like those you'll find on your exams Techniques for choosing the correct approach to problems Guidance toward the quickest, most efficient solutions If you want top grades and thorough understanding of discrete mathematics, this powerful study tool is the best tutor you can have! |
| discrete mathematics questions and answers: 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 mathematics questions and answers: 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 mathematics questions and answers: Concrete Mathematics Ronald L. Graham, Donald E. Knuth, Oren Patashnik, 1994-02-28 This book introduces the mathematics that supports advanced computer programming and the analysis of algorithms. The primary aim of its well-known authors is to provide a solid and relevant base of mathematical skills - the skills needed to solve complex problems, to evaluate horrendous sums, and to discover subtle patterns in data. It is an indispensable text and reference not only for computer scientists - the authors themselves rely heavily on it! - but for serious users of mathematics in virtually every discipline. Concrete Mathematics is a blending of CONtinuous and disCRETE mathematics. More concretely, the authors explain, it is the controlled manipulation of mathematical formulas, using a collection of techniques for solving problems. The subject matter is primarily an expansion of the Mathematical Preliminaries section in Knuth's classic Art of Computer Programming, but the style of presentation is more leisurely, and individual topics are covered more deeply. Several new topics have been added, and the most significant ideas have been traced to their historical roots. The book includes more than 500 exercises, divided into six categories. Complete answers are provided for all exercises, except research problems, making the book particularly valuable for self-study. Major topics include: Sums Recurrences Integer functions Elementary number theory Binomial coefficients Generating functions Discrete probability Asymptotic methods This second edition includes important new material about mechanical summation. In response to the widespread use of the first edition as a reference book, the bibliography and index have also been expanded, and additional nontrivial improvements can be found on almost every page. Readers will appreciate the informal style of Concrete Mathematics. Particularly enjoyable are the marginal graffiti contributed by students who have taken courses based on this material. The authors want to convey not only the importance of the techniques presented, but some of the fun in learning and using them. |
| discrete mathematics questions and answers: Discrete and Combinatorial Mathematics: An applied Introduction ( For VTU) Grimaldi Ralph P., 2013 |
| discrete mathematics questions and answers: Set Theory and Logic Robert R. Stoll, 2012-05-23 Explores sets and relations, the natural number sequence and its generalization, extension of natural numbers to real numbers, logic, informal axiomatic mathematics, Boolean algebras, informal axiomatic set theory, several algebraic theories, and 1st-order theories. |
| discrete mathematics questions and answers: 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 mathematics questions and answers: 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 mathematics questions and answers: Discrete Mathematics and Its Applications Kenneth H. Rosen, 2018-05 A precise, relevant, comprehensive approach to mathematical concepts... |
| discrete mathematics questions and answers: 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 mathematics questions and answers: Discrete Mathematics Gary Chartrand, Ping Zhang, 2011-03-31 Chartrand and Zhangs Discrete Mathematics presents a clearly written, student-friendly introduction to discrete mathematics. The authors draw from their background as researchers and educators to offer lucid discussions and descriptions fundamental to the subject of discrete mathematics. Unique among discrete mathematics textbooks for its treatment of proof techniques and graph theory, topics discussed also include logic, relations and functions (especially equivalence relations and bijective functions), algorithms and analysis of algorithms, introduction to number theory, combinatorics (counting, the Pascal triangle, and the binomial theorem), discrete probability, partially ordered sets, lattices and Boolean algebras, cryptography, and finite-state machines. This highly versatile text provides mathematical background used in a wide variety of disciplines, including mathematics and mathematics education, computer science, biology, chemistry, engineering, communications, and business. Some of the major features and strengths of this textbook Numerous, carefully explained examples and applications facilitate learning. More than 1,600 exercises, ranging from elementary to challenging, are included with hints/answers to all odd-numbered exercises. Descriptions of proof techniques are accessible and lively. Students benefit from the historical discussions throughout the textbook. |
| discrete mathematics questions and answers: 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 mathematics questions and answers: Essentials of Discrete Mathematics David J. Hunter, 2015-08-21 Written for the one-term course, the Third Edition of Essentials of Discrete Mathematics is designed to serve computer science majors as well as students from a wide range of disciplines. The material is organized around five types of thinking: logical, relational, recursive, quantitative, and analytical. This presentation results in a coherent outline that steadily builds upon mathematical sophistication. Graphs are introduced early and referred to throughout the text, providing a richer context for examples and applications. tudents will encounter algorithms near the end of the text, after they have acquired the skills and experience needed to analyze them. The final chapter contains in-depth case studies from a variety of fields, including biology, sociology, linguistics, economics, and music. |
| discrete mathematics questions and answers: Pearls in Graph Theory Nora Hartsfield, Gerhard Ringel, 2013-04-15 Stimulating and accessible, this undergraduate-level text covers basic graph theory, colorings of graphs, circuits and cycles, labeling graphs, drawings of graphs, measurements of closeness to planarity, graphs on surfaces, and applications and algorithms. 1994 edition. |
| discrete mathematics questions and answers: Discrete Mathematics Quiz Book S.R. Subramanya, This is a quiz /exercise / self–assessment book. It has a vast collection of questions in Discrete Mathematics. The topical coverage includes: Logic and Proof methods, Sets, Functions, Relations, Properties of integers, Sequences, Induction and Recursion, Basic and advanced counting methods, Discrete probability, Graph theory, Modeling computation, and Boolean algebra. |
| discrete mathematics questions and answers: Invitation to Discrete Mathematics Jiří Matoušek, Jaroslav Nešetřil, 2009 A clear and self-contained introduction to discrete mathematics for undergraduates and early graduates. |
| discrete mathematics questions and answers: Discrete Structures, Logic, and Computability James L. Hein, 2001 Discrete Structure, Logic, and Computability introduces the beginning computer science student to some of the fundamental ideas and techniques used by computer scientists today, focusing on discrete structures, logic, and computability. The emphasis is on the computational aspects, so that the reader can see how the concepts are actually used. Because of logic's fundamental importance to computer science, the topic is examined extensively in three phases that cover informal logic, the technique of inductive proof; and formal logic and its applications to computer science. |
| discrete mathematics questions and answers: 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 mathematics questions and answers: 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 mathematics questions and answers: 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 mathematics questions and answers: Discrete Mathematics with Graph Theory (Classic Version) Edgar Goodaire, Michael Parmenter, 2017-03-20 This title is part of the Pearson Modern Classics series. Pearson Modern Classics are acclaimed titles at a value price. Please visit www.pearsonhighered.com/math-classics-series for a complete list of titles. Far more user friendly than the vast majority of similar books, this text is truly written with the beginning reader in mind. The pace is tight, the style is light, and the text emphasizes theorem proving throughout. The authors emphasize Active Reading, a skill vital to success in learning how to think mathematically (and write clean, error-free programs). |
| discrete mathematics questions and answers: Problems and Exercises in Discrete Mathematics G.P. Gavrilov, A.A. Sapozhenko, 2013-03-09 Many years of practical experience in teaching discrete mathematics form the basis of this text book. Part I contains problems on such topics as Boolean algebra, k-valued logics, graphs and networks, elements of coding theory, automata theory, algorithms theory, combinatorics, Boolean minimization and logical design. The exercises are preceded by ample theoretical background material. For further study the reader is referred to the extensive bibliography. Part II follows the same structure as Part I, and gives helpful hints and solutions. Audience:This book will be of great value to undergraduate students of discrete mathematics, whereas the more difficult exercises, which comprise about one-third of the material, will also appeal to postgraduates and researchers. |
| discrete mathematics questions and answers: A Walk Through Combinatorics Mikl¢s B¢na, 2006 This is a textbook for an introductory combinatorics course that can take up one or two semesters. An extensive list of problems, ranging from routine exercises to research questions, is included. In each section, there are also exercises that contain material not explicitly discussed in the preceding text, so as to provide instructors with extra choices if they want to shift the emphasis of their course. Just as with the first edition, the new edition walks the reader through the classic parts of combinatorial enumeration and graph theory, while also discussing some recent progress in the area: on the one hand, providing material that will help students learn the basic techniques, and on the other hand, showing that some questions at the forefront of research are comprehensible and accessible for the talented and hard-working undergraduate. The basic topics discussed are: the twelvefold way, cycles in permutations, the formula of inclusion and exclusion, the notion of graphs and trees, matchings and Eulerian and Hamiltonian cycles. The selected advanced topics are: Ramsey theory, pattern avoidance, the probabilistic method, partially ordered sets, and algorithms and complexity. As the goal of the book is to encourage students to learn more combinatorics, every effort has been made to provide them with a not only useful, but also enjoyable and engaging reading. |
| discrete mathematics questions and answers: Sets, Logic and Maths for Computing David Makinson, 2012-02-27 This easy-to-follow textbook introduces the mathematical language, knowledge and problem-solving skills that undergraduates need to study computing. The language is in part qualitative, with concepts such as set, relation, function and recursion/induction; but it is also partly quantitative, with principles of counting and finite probability. Entwined with both are the fundamental notions of logic and their use for representation and proof. Features: teaches finite math as a language for thinking, as much as knowledge and skills to be acquired; uses an intuitive approach with a focus on examples for all general concepts; brings out the interplay between the qualitative and the quantitative in all areas covered, particularly in the treatment of recursion and induction; balances carefully the abstract and concrete, principles and proofs, specific facts and general perspectives; includes highlight boxes that raise common queries and clear confusions; provides numerous exercises, with selected solutions. |
| discrete mathematics questions and answers: 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 mathematics questions and answers: Discrete Mathematics and Its Applications Kenneth Rosen, 2016-07-19 |
| discrete mathematics questions and answers: 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 mathematics questions and answers: Discrete Mathematics Kenneth A. Ross, Charles R. B. Wright, 1988 |
| discrete mathematics questions and answers: Mathematics for Computer Science Eric Lehman, F. Thomson Leighton, Albert R. Meyer, 2017-03-08 This book covers elementary discrete mathematics for computer science and engineering. It emphasizes mathematical definitions and proofs as well as applicable methods. Topics include formal logic notation, proof methods; induction, well-ordering; sets, relations; elementary graph theory; integer congruences; asymptotic notation and growth of functions; permutations and combinations, counting principles; discrete probability. Further selected topics may also be covered, such as recursive definition and structural induction; state machines and invariants; recurrences; generating functions. |
| discrete mathematics questions and answers: 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 mathematics questions and answers: Mathematics of Choice Ivan Niven, 1965 |
| discrete mathematics questions and answers: Essentials of Discrete Mathematics David Hunter, 2010-12-29 This is the ideal text for a one-term discrete mathematics course to serve computer scientists as well as other students. It introduces students to the mathematical way of thinking, and also to many important modern applications. |
| discrete mathematics questions and answers: 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 mathematics questions and answers: Mathematical Structures for Computer Science Judith L. Gersting, 2014-03-01 Judith Gerstings Mathematical Structures for Computer Science has long been acclaimed for its clear presentation of essential concepts and its exceptional range of applications relevant to computer science majors. Now with this new edition, it is the first discrete mathematics textbook revised to meet the proposed new ACM/IEEE standards for the course. |
| discrete mathematics questions and answers: Extremal Combinatorics Stasys Jukna, 2013-03-09 This is a concise, up-to-date introduction to extremal combinatorics for non-specialists. Strong emphasis is made on theorems with particularly elegant and informative proofs which may be called the gems of the theory. A wide spectrum of the most powerful combinatorial tools is presented, including methods of extremal set theory, the linear algebra method, the probabilistic method and fragments of Ramsey theory. A thorough discussion of recent applications to computer science illustrates the inherent usefulness of these methods. |
| discrete mathematics questions and answers: Computing the Continuous Discretely Matthias Beck, Sinai Robins, 2015-11-14 This richly illustrated textbook explores the amazing interaction between combinatorics, geometry, number theory, and analysis which arises in the interplay between polyhedra and lattices. Highly accessible to advanced undergraduates, as well as beginning graduate students, this second edition is perfect for a capstone course, and adds two new chapters, many new exercises, and updated open problems. For scientists, this text can be utilized as a self-contained tooling device. The topics include a friendly invitation to Ehrhart’s theory of counting lattice points in polytopes, finite Fourier analysis, the Frobenius coin-exchange problem, Dedekind sums, solid angles, Euler–Maclaurin summation for polytopes, computational geometry, magic squares, zonotopes, and more. With more than 300 exercises and open research problems, the reader is an active participant, carried through diverse but tightly woven mathematical fields that are inspired by an innocently elementary question: What are the relationships between the continuous volume of a polytope and its discrete volume? Reviews of the first edition: “You owe it to yourself to pick up a copy of Computing the Continuous Discretely to read about a number of interesting problems in geometry, number theory, and combinatorics.” — MAA Reviews “The book is written as an accessible and engaging textbook, with many examples, historical notes, pithy quotes, commentary integrating the mate rial, exercises, open problems and an extensive bibliography.” — Zentralblatt MATH “This beautiful book presents, at a level suitable for advanced undergraduates, a fairly complete introduction to the problem of counting lattice points inside a convex polyhedron.” — Mathematical Reviews “Many departments recognize the need for capstone courses in which graduating students can see the tools they have acquired come together in some satisfying way. Beck and Robins have written the perfect text for such a course.” — CHOICE |
| discrete mathematics questions and answers: Discrete Mathematics and Applications Kevin Ferland, 2017-09-19 Discrete Mathematics and Applications, Second Edition is intended for a one-semester course in discrete mathematics. Such a course is typically taken by mathematics, mathematics education, and computer science majors, usually in their sophomore year. Calculus is not a prerequisite to use this book. Part one focuses on how to write proofs, then moves on to topics in number theory, employing set theory in the process. Part two focuses on computations, combinatorics, graph theory, trees, and algorithms. Emphasizes proofs, which will appeal to a subset of this course market Links examples to exercise sets Offers edition that has been heavily reviewed and developed Focuses on graph theory Covers trees and algorithms |
| discrete mathematics questions and answers: Schaum's Outline of Discrete Mathematics, 3rd Ed. Seymour Lipschutz, Marc Lipson, 2007-06-01 This is a topic that becomes increasingly important every year as the digital age extends and grows more encompassing in every facet of life Discrete mathematics, the study of finite systems has become more important as the computer age has advanced, as computer arithmetic, logic, and combinatorics have become standard topics in the discipline. For mathematics majors it is one of the core required courses. This new edition will bring the outline into synch with Rosen, McGraw-Hill’s bestselling textbook in the field as well as up to speed in the current curriculum. New material will include expanded coverage of logic, the rules of inference and basic types of proofs in mathematical reasoning. This will give students a better understanding of proofs of facts about sets and functions. There will be increased emphasis on discrete probability and aspects of probability theory, and greater accessibility to counting techniques. This new edition features: Counting chapter will have new material on generalized combinations New chapter on computer arithmetic, with binary and hexagon addition and multiplication New Cryptology chapter including substitution and RSA method This outline is the perfect supplement to any course in discrete math and can also serve as a stand-alone textbook |
| discrete mathematics questions and answers: 6th Grade Math MCQs Arshad Iqbal, 2017-10-04 6th grade math multiple choice questions has 448 MCQs. Grade 6 math quiz questions and answers, MCQs on integers, rational numbers, sequence and series, factors and multiples, volume and surface area, functions, graphs, angle properties of polygons, class 6 mathematics MCQs with answers, estimation and approximation, fundamental algebra, algebraic equations and simple inequalities, arithmetical problems and percentages, ratio rate and speed, geometrical concepts and properties, perimeter and area of geometrical figures MCQs and quiz worksheets to practice exam prep tests.6th grade math multiple choice quiz questions and answers, math exam revision and study guide with practice tests for online exam prep and interviews. Math interview questions and answers to ask, to prepare and to study for jobs interviews and career MCQs with answer keys.Algebraic equations and simple inequalities quiz has 69 multiple choice questions. Angle properties of polygons quiz has 17 multiple choice questions. Arithmetical problems and percentages quiz has 48 multiple choice questions with answers. Estimation and approximation quiz has 31 multiple choice questions. Factors and multiples quiz has 41 multiple choice questions. Functions and graphs quiz has 17 multiple choice questions.Fundamental algebra quiz has 70 multiple choice questions. Geometrical concepts and properties quiz has 24 multiple choice questions. Integer's quiz has 42 multiple choice questions. Number sequences quiz has 12 multiple choice questions. Perimeter and area of geometrical figures quiz has 20 multiple choice questions. Ratio rate and speed quiz has 46 multiple choice questions. Rational numbers quiz has 32 multiple choice questions. Volume and surface area quiz has 19 multiple choice questions and answers.Math interview questions and answers, MCQs on tax calculations, polygons, time calculation, least common multiple, rational numbers, cylinders, complementary angles, prime factorization, significant figures, supplementary angles, math formulas, number line, adjacent angles, algebraic expressions, ratio calculations, discount calculations, types of triangles, Cartesian plane, rounding numbers, average speed, highest common factor, how to do percentages, prime and composite numbers, types of angles, convex polygons, number sequences, addition and subtraction, finding coordinates, algebra rules, factors and multiples, rounding off numbers, commission calculations, index notation, ratio examples, addition of integers, equations and inequalities, percentage of number, rules of integers, subtraction of integers, units of area, algebraic notation, examples of equations, writing algebraic expressions, average rate, geometric concepts, multiplication of integers, squares and square roots, division of integers, solving simple equations, cubes and cube roots, volume of fluids, making formula, rate calculations, absolute value of integer, evaluation of algebraic expressions, factorization by grouping, percentage comparison, distributive law of multiplication, estimation and rounding, multiplication and division of rational numbers, line rays and segments, terminating and recurring decimals, percentage fractions and decimals, ordering of rational numbers, problem solving with algebra, arithmetical operations on rational numbers, brackets in simplification, class 6 factorization, expressing quantities and percentage, idea of functions, increasing decreasing quantities, inequalities learning, linear algebraic expressions and fractional coefficients, ratio increase and decrease, real numbers calculations, round off values, simple equations solutions, grade 6 math worksheets for competitive exams preparation. |
| discrete mathematics questions and answers: Foundations of Discrete Mathematics K. D. Joshi, 1989 This Book Is Meant To Be More Than Just A Text In Discrete Mathematics. It Is A Forerunner Of Another Book Applied Discrete Structures By The Same Author. The Ultimate Goal Of The Two Books Are To Make A Strong Case For The Inclusion Of Discrete Mathematics In The Undergraduate Curricula Of Mathematics By Creating A Sequence Of Courses In Discrete Mathematics Parallel To The Traditional Sequence Of Calculus-Based Courses.The Present Book Covers The Foundations Of Discrete Mathematics In Seven Chapters. It Lays A Heavy Emphasis On Motivation And Attempts Clarity Without Sacrificing Rigour. A List Of Typical Problems Is Given In The First Chapter. These Problems Are Used Throughout The Book To Motivate Various Concepts. A Review Of Logic Is Included To Gear The Reader Into A Proper Frame Of Mind. The Basic Counting Techniques Are Covered In Chapters 2 And 7. Those In Chapter 2 Are Elementary. But They Are Intentionally Covered In A Formal Manner So As To Acquaint The Reader With The Traditional Definition-Theorem-Proof Pattern Of Mathematics. Chapters 3 Introduces Abstraction And Shows How The Focal Point Of Todays Mathematics Is Not Numbers But Sets Carrying Suitable Structures. Chapter 4 Deals With Boolean Algebras And Their Applications. Chapters 5 And 6 Deal With More Traditional Topics In Algebra, Viz., Groups, Rings, Fields, Vector Spaces And Matrices.The Presentation Is Elementary And Presupposes No Mathematical Maturity On The Part Of The Reader. Instead, Comments Are Inserted Liberally To Increase His Maturity. Each Chapter Has Four Sections. Each Section Is Followed By Exercises (Of Various Degrees Of Difficulty) And By Notes And Guide To Literature. Answers To The Exercises Are Provided At The End Of The Book. |
Practice Questions with Solutions - University of Victoria
Questions: 1.Consider the sets Aand Bwhere: A= fa2Zja= 2k, for some integer kg, B= fb2Zjb= 2j 2, for some integer jg. Does A= B? If yes, prove it. If no, explain why not. 2.Consider the sets A= f1;2;3g, B= fx;yg, and C= fu;vg. Let P(A) denote the powerset of A. Find each of the following: (a) P(A[B) (b) P(B C) (c) P(P(C)) (d) A (B\C) (e)(A B) C
Discrete Maths: Exercises and Solutions
Discrete Maths: Exercises and Solutions Basic Structures: Sets, Functions, Sequences, Sums and Matrices
Discrete Mathematics Final Exam Question Bank
Discrete Mathematics Final Exam Question Bank Note: This Question Bank may not include some topics. Final Exam Topics: 1. Set Theory (cardinality, relationships, operations, identities) 2. Cartesian Products 3. Propositional Logic & Logic Circuits 4. Truth Tables 5. Logic Laws (proofs) 6.
Discrete Maths: Exercises & Solutions
EXAMPLE 1 : We can construct examples of tautologies and contradictions using just one propositional variable. Consider the truth tables of p ∨¬p and p ∧¬p, shown in Table 1. Because p ∨¬p is always true, it is a tautology. Because p ∧¬p is always false, it is a contradiction.
Solved exercises in Discrete mathematics Sample problems - vsb.cz
Most of the problems were prepared by Michael Kubesa, Tereza Kovarova, and Petr Kovar. The English version was prepared by Tereza Kovarova and Petr Kovar. 1.1. Determine the set A B, if A = bn : n 2 N; 5 n. (6; ) and (e; 2) elements of A B? 11g and B = f ; eg. Are the ordered pairs. e); (3; ); (3; e); (4; ); (4; e); (5; ); (5; e)g.
Discrete Mathematics - University of Cambridge
The main aim is to practice the analysis and understanding of mathematical statements (e.g. by isolating the diferent components of composite statements) and exercise the art of presenting a logical argument in the form of a clear proof (e.g. by following proof strategies and patterns).
Practice Final Exam - discrete math
Let B be the set of strings of four bits, b1 b2 b3 b4 where each bit is either 0 or 1. Define relation w on B by. x w y if x has the same number of 1’s as y (for example 011 w 101, but 100 is not related to 101 by 2). (a) Explain why w is an equivalence relation. (b) Into how many equivalence classes is B divided by w?
Math 220, Discrete Mathematics, Spring 2017 Midterm 1 Practice …
Math 220, Discrete Mathematics, Spring 2017 Midterm 1 Practice Test Solutions Instructions: Please read each question carefully. No calculators, notes, books, or outside help of any kind are allowed to be used on this exam. Please turn cell phones o ! Show all of your work and explain your answers clearly. In order to receive
Sample Problems in Discrete Mathematics - Rensselaer …
This handout lists some sample problems that you should be able to solve as a pre-requisite to Design and Analysis of Algorithms. Try to solve all of them. You should also read Chapters 2 and 3 of the textbook, and look at the Exercises at the end of these chapters.
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2 long answers, 5 points each: 1. In a round-robin tournament, every player plays every other player. There is a "Hamiltonian path" through the tournament in the case that we can trace a path from player x to player y to player z where x --beats--> y --beats--> z , and so on throughout the whole tournament.
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Answering these problems as you might handle homework problems won’t necessarily give you much of a clue on what you need to work on. Instructions: Provide all steps necessary to solve the problem. Unless otherwise stated, your answer must be exact and reasonably simplified.
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Department of Mathematics MATH 222 - Discrete and Combinatorial Mathematics Answers and Hints to Practice Questions Contributors: Elise Marchessault Ashna Wright This work is licensed under the Creative Commons Attribution 4.0 International License. To …
Discrete Mathematics Midterm Exam Question Bank
Discrete Mathematics Midterm Exam Question Bank 1. Let A = {a, b, c}, B = {x, z}, and C = {0, -1}. Find a) B × A × C. b) C × B × A. c) C × A × B. d) B × B × B 2. Find the truth set of each of these predicates where the domain is the set of integers. a) P (x): 4 2-125 < 3 b) Q(x): x 2 > x
151 Math Exercises (3,1) Methods of Proof
Math 151 Discrete Mathematics [Methods of Proof] By: Malek Zein AL-Abidin Exercises 1. Use a proof by contraposition to show that if x + y ≥ 2, where x and y are real numbers, then x ≥ 1 or y ≥ 1. Solution:
Discrete Maths: Exercises & Solutions
EXAMPLE 7 Let p be the statement “Maria learns discrete mathematics” and q the statement “Maria will find a good job.” Express the statement p → q as a statement in English.
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Exam in Discrete Mathematics First Year at The TEK-NAT Faculty June 11th, 2014, 9.00–13.00 ANSWERS Part I (”regular exercises”) Exercise 1 (6%). Find the expansion of (2x y)4 using The Binomial Theorem. Answer: 16x4 232x3y+24x y2 8xy3 +y4 Exercise 2 (8%). Find witnesses proving that f(x) = 2x3 + x2 +5 is O(x3). Exercise 3 (12%).
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Basic building block for types of objects in discrete mathematics. Set operations in programming languages: Issues about data structures used to represent sets and the computational cost of
I. Practice in 1st-order predicate logic – with answers. - UMass
I. Practice in 1st-order predicate logic – with answers. 1. Mary loves everyone. [assuming D contains only humans] ∀ x love (Mary, x) Note: No further parentheses are needed here, and according to the syntax on the handout, no further parentheses are possible.
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DISCRETE MATHEMATICS - SAMPLE EXAM QUESTIONS 1.Using truth tables decide whether each of the following is a tautology: (a)(p →q) ↔(q →p), (b)(p →q) ↔(p∧q), (c)(p∨q) ↔(p∧q), (d)((p∨q) ∧r) ↔((p∧r) ∨(q ∧r)). 2.For each of the following sentences, negate the sentence, and say whether the sentence or its negation is true:
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VERY SHORT ANSWER QUESTIONS 1) State fundamental theorem of arithmetic hence find the prime factorization of 810. 2) Write prime numbers less than 150. 3) Write the properties of gcd. 4) State and prove Euclid’s lemma. 5) Define Fibonacci numbers recursively. 6) Explain about Mathematical Induction.
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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 “ > …
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MATH 3336 – Discrete Mathematics Recurrence Relations (8.1, 8.2) Definition: A recurrence relation ...
True-False Questions - University of Texas at Austin
CS311H: Discrete Mathematics Graph Theory III Instructor: Is l Dillig Instructor: Is l Dillig, CS311H: Discrete Mathematics Graph Theory III 1/23 Rooted Trees Subtrees I Given a rooted tree and …
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Discrete Mathematics Lecture 12 Sets, Functions, and Relations: Part IV 1 . Outline •Equivalence Relations •Partial Orderings 2 . Equivalence Relations 3 . Equivalence Relations •A relation …
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The course exercises are meant for the students of the course of Discrete Mathematics and Logic at the Free University of Bozen-Bolzano. 1 Graph Theory Exercise Set 10.2: Euler Circuits ...
Discrete Mathematics: Practice Problems - IIT Hyderabad
Discrete Mathematics: Practice Problems 1. For the two statements below, decide whether they are true or false. (i) 9n2N : 8m2N;((m
DIGITAL NOTES ON DISCRETE MATHEMATICS B.TECH II …
Discrete Mathematics Page 2 F T Logical conjunction Logical conjunction is an operation on two logical values, typically the values of two propositions, that produces a value of true if both of …
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. …
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Discrete Mathematics Questions and Answers – Counting & Pigeonhole Principle 1. A drawer contains 12 red and 12 blue socks, all unmatched. A person takes socks out at random in the …
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The life and career of Augustus De Morgan (–1871) is fascinating; his mathematics, somewhat less so. Mathematically, he worked mostly on logic and what modern mathemati …
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Algorithms (Abu Ja ’far Mohammed Ibin Musa Al-Khowarizmi, 780-850) Definition An algorithm is a finite set of precise instructions for performing a computation or for solving a problem. …
2 Marks Questions - G. S. Mandal's Marathwada Institute of …
Subject: Discrete Mathematics Subject In charge: Prof. L. B. Randive Unit -2 Fundamentals of Logic 2 Marks Questions 1. Which of these sentences are propositions? What are the truth …
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“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 …
Discrete Mathematics - Rules of Inference and Mathematical Proofs
Discrete Mathematics (c) Marcin Sydow Proofs Inference rules Proofs Set theory axioms Formal proof Let P= f1; 2;:::; m gbe a set of premises or axioms and let C be a conclusion do be …
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Venndiagrams Definition Relationshipbetweenasmallnumberofsetscanberepresented bypicturescalledVenndiagrams Problems Write a Venn diagram representing sets of numbers:
DISCRETE MATHEMATICS
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Discrete Mathematics and Its Applications - IPFS
plements the Student’s Solutions Guide for Discrete Mathematics and Its Applications, eighth edition, which contains solutions to the odd-numbered exercises. It is assumed that the user of …
Week 2 Worksheet - Equivalences, Predicates
2.Start with one statement and work towards the second. 3.Justify each step. 4.It’s ok to apply one rule multiple times in one step. Practice Prove or disprove each example below.
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Discrete Mathematics for Computer Science. Key College Publishing, Emeryville, Cali-fornia, 2006. Examinations. There will be a final exam (covering the material of the entire semester) …
Course Sta CS311H: Discrete Mathematics - University of Texas …
Class Participation I Everyone expected to attend lectures and participate I 5% of course grade for participation (attendance, asking/answering questions, being active on Ed Discussion) I …
Lecture Notes on Discrete Mathematics - IIT Kanpur
using the concept of a set to answer questions is hardly new. It has been in use since ancient times. However, the rigorous treatment of sets happened only in the 19-th century due to the …
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Mathematics is the only instructional material that can be presented in an entirely undogmatic way. The Mathematical Intelligencer, v. 5, no. 2, 1983 MAX DEHN Chapter 1 Introduction The …
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answers are provided on the final page of the chapter. It is recommended to try the practice questions soon after the relevant lectures, to help you correct any misunderstanding quickly. …
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Discrete Mathematics and its Applications, Kenneth H. Rosen, Fifth Edition.TMH. 2. Discrete Mathematical structures Theory and application-Malik & Sen, Cengage.
10.6 Shortest-Path Problems - University of Hawaiʻi
ICS 241: Discrete Mathematics II (Spring 2015) 10.6 Shortest-Path Problems Given a graph G = (V;E), a weighting function w(e);w(e) > 0, for the edges of G, and a source vertex, v 0. We …
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SUBJECT:DISCRETE MATHEMATICS SUBCODE: MA8351 SEM/YEAR: III/II PART A UNIT I(2 Marks) LOGIC AND PROOFS 1. Using truth table show that the proposition PVד (PɅQ) is a …
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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 ... Are the answers unique? A: …
OCR A Level Further Mathematics A Y544 Discrete mathematics …
• Formulae A Level Further Mathematics A You may use: • ascientific or graphical calculator * 0 0 0 0 0 0 * INSTRUCTIONS • Use black ink. HB pencil may be used for graphs and diagrams …
CS201A: Math for CS I/Discrete Mathematics Endsem exam - IIT …
CS201A: Math for CS I/Discrete Mathematics Endsem exam Max marks:150 Time:180 mins. 23-Nov-2017 1.Answer all 8 questions. It has 4 pages + 1 page for the standard normal …
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Recurrence Relations - IIT Kharagpur
Aritra Hazra (CSE, IITKGP) CS21001 : Discrete Structures Autumn 2020 1/36. Introduction Recurrence Relations are Mathematical Equations: A recurrence relation is an equation which …
Discrete Mathematics: Algorithms - MIT OpenCourseWare
Far from being dumb, or easy, or useless, discrete mathematics asks good questions. It is important to know the fastest way across the country. It is more important to know the fastest …
Functions II - University of Pittsburgh
CS 441 Discrete mathematics for CS M. Hauskrecht CS 441 Discrete Mathematics for CS Lecture 9 Milos Hauskrecht milos@cs.pitt.edu 5329 Sennott Square Functions II M. Hauskrecht …
Discrete Mathematics R19 - SIETK
9. Given the premises “ A student of this class has not read the Discrete mathematics text book” and “Everyone in this class passed the first unit test” show that “someone who passed the first …
Question 1. Prove using mathematical induction that for all n(3n …
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 …
BCA Ist Year Examination Discrete Mathematics Paper - BCA-02
Discrete Mathematics Paper - BCA-02 Time : 3 Hours ] [ Max. Marks :- 100 Note: The question paper is divided into three sections A, B and C. Write answers as per given instructions. …
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Discrete Mathematics Comprehensive Exam Questions 1. Prove that for every integer k 1 there exists an integer Nsuch that if the subsets of f1;2;:::;Ngare colored using kcolors, then there …
Discrete Mathematics Questions And Answers (book)
Discrete Mathematics Questions and Answers: A Comprehensive Guide Description: Discrete mathematics is a branch of mathematics dealing with objects that can be counted, such as …
Discrete Mathematics & Mathematical Reasoning Chapter 7 …
Using Bayes’ Theorem Problem: There are two boxes, Box B 1 and Box B 2. Box B 1 contains 2 red balls and 8 blue balls. Box B 2 contains 7 red balls and 3 blue balls. Suppose Jane first …
AS/A Level Mathematics Discrete Random Variables - Maths Genie
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Question Paper Code : 40784 - STUCOR
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Problems on Discrete Mathematics1 - Illinois State University
Problems on Discrete Mathematics1 Chung-Chih Li2 Kishan Mehrotra3 LATEX at July 18, 2007 1No part of this book can be reproduced without permission from the authors. 2Illinois State …
Discrete Mathematics II: Set Theory for Computer Science Part …
The aim of this part of the ‘Discrete Mathematics" course is to introduce fundamental concepts and techniques in set theory in preparation for its many ... and have supervisions in. Past …
Rules of Inference - University of Texas at Austin
Encoding in Logic, cont. I "It is not raining or Kate has her umbrella." I "Kate does not have her umbrella or she does not get wet" I "It is raining or Kate does not get wet." I "Kate is grumpy …
M.Sc. (Mathematics), SEM- I Paper - V DISCRETE MATHEMATICS
1 1 DIVISIBILITY THEORY OF INTEGERS AND ARITHMETIC FUNCTIONS Unit Structure : 1.0 Objectives 1.1 Introduction 1.2 Division Algorithm 1.3 The Greatest Common divisor
