| counterexample definition in math: Counterexamples in Analysis Bernard R. Gelbaum, John M. H. Olmsted, 2012-07-12 These counterexamples deal mostly with the part of analysis known as real variables. Covers the real number system, functions and limits, differentiation, Riemann integration, sequences, infinite series, functions of 2 variables, plane sets, more. 1962 edition. |
| counterexample definition in math: Counterexamples in Topology Lynn Arthur Steen, J. Arthur Seebach, 2013-04-22 Over 140 examples, preceded by a succinct exposition of general topology and basic terminology. Each example treated as a whole. Numerous problems and exercises correlated with examples. 1978 edition. Bibliography. |
| counterexample definition in math: Counterexamples in Probability Jordan M. Stoyanov, 2014-01-15 While most mathematical examples illustrate the truth of a statement, counterexamples demonstrate a statement's falsity. Enjoyable topics of study, counterexamples are valuable tools for teaching and learning. The definitive book on the subject in regards to probability, this third edition features the author's revisions and corrections plus a substantial new appendix. 2013 edition-- |
| counterexample definition in math: Proofs and Refutations Imre Lakatos, 1976 Proofs and Refutations is for those interested in the methodology, philosophy and history of mathematics. |
| counterexample definition in math: 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. |
| counterexample definition in math: Handbook of Process Algebra J.A. Bergstra, A. Ponse, S.A. Smolka, 2001-03-16 Process Algebra is a formal description technique for complex computer systems, especially those involving communicating, concurrently executing components. It is a subject that concurrently touches many topic areas of computer science and discrete math, including system design notations, logic, concurrency theory, specification and verification, operational semantics, algorithms, complexity theory, and, of course, algebra.This Handbook documents the fate of process algebra since its inception in the late 1970's to the present. It is intended to serve as a reference source for researchers, students, and system designers and engineers interested in either the theory of process algebra or in learning what process algebra brings to the table as a formal system description and verification technique. The Handbook is divided into six parts spanning a total of 19 self-contained Chapters. The organization is as follows. Part 1, consisting of four chapters, covers a broad swath of the basic theory of process algebra. Part 2 contains two chapters devoted to the sub-specialization of process algebra known as finite-state processes, while the three chapters of Part 3 look at infinite-state processes, value-passing processes and mobile processes in particular. Part 4, also three chapters in length, explores several extensions to process algebra including real-time, probability and priority. The four chapters of Part 5 examine non-interleaving process algebras, while Part 6's three chapters address process-algebra tools and applications. |
| counterexample definition in math: 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. |
| counterexample definition in math: 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. |
| counterexample definition in math: The Riemann Hypothesis Peter B. Borwein, 2008 The Riemann Hypothesis has become the Holy Grail of mathematics in the century and a half since 1859 when Bernhard Riemann, one of the extraordinary mathematical talents of the 19th century, originally posed the problem. While the problem is notoriously difficult, and complicated even to state carefully, it can be loosely formulated as the number of integers with an even number of prime factors is the same as the number of integers with an odd number of prime factors. The Hypothesis makes a very precise connection between two seemingly unrelated mathematical objects, namely prime numbers and the zeros of analytic functions. If solved, it would give us profound insight into number theory and, in particular, the nature of prime numbers. This book is an introduction to the theory surrounding the Riemann Hypothesis. Part I serves as a compendium of known results and as a primer for the material presented in the 20 original papers contained in Part II. The original papers place the material into historical context and illustrate the motivations for research on and around the Riemann Hypothesis. Several of these papers focus on computation of the zeta function, while others give proofs of the Prime Number Theorem, since the Prime Number Theorem is so closely connected to the Riemann Hypothesis. The text is suitable for a graduate course or seminar or simply as a reference for anyone interested in this extraordinary conjecture. |
| counterexample definition in math: An Introduction to Measure Theory Terence Tao, 2021-09-03 This is a graduate text introducing the fundamentals of measure theory and integration theory, which is the foundation of modern real analysis. The text focuses first on the concrete setting of Lebesgue measure and the Lebesgue integral (which in turn is motivated by the more classical concepts of Jordan measure and the Riemann integral), before moving on to abstract measure and integration theory, including the standard convergence theorems, Fubini's theorem, and the Carathéodory extension theorem. Classical differentiation theorems, such as the Lebesgue and Rademacher differentiation theorems, are also covered, as are connections with probability theory. The material is intended to cover a quarter or semester's worth of material for a first graduate course in real analysis. There is an emphasis in the text on tying together the abstract and the concrete sides of the subject, using the latter to illustrate and motivate the former. The central role of key principles (such as Littlewood's three principles) as providing guiding intuition to the subject is also emphasized. There are a large number of exercises throughout that develop key aspects of the theory, and are thus an integral component of the text. As a supplementary section, a discussion of general problem-solving strategies in analysis is also given. The last three sections discuss optional topics related to the main matter of the book. |
| counterexample definition in math: How to Prove It Daniel J. Velleman, 2006-01-16 Many students have trouble the first time they take a mathematics course in which proofs play a significant role. This new edition of Velleman's successful text will prepare students to make the transition from solving problems to proving theorems by teaching them the techniques needed to read and write proofs. The book begins with the basic concepts of logic and set theory, to familiarize students with the language of mathematics and how it is interpreted. These concepts are used as the basis for a step-by-step breakdown of the most important techniques used in constructing proofs. The author shows how complex proofs are built up from these smaller steps, using detailed 'scratch work' sections to expose the machinery of proofs about the natural numbers, relations, functions, and infinite sets. To give students the opportunity to construct their own proofs, this new edition contains over 200 new exercises, selected solutions, and an introduction to Proof Designer software. No background beyond standard high school mathematics is assumed. This book will be useful to anyone interested in logic and proofs: computer scientists, philosophers, linguists, and of course mathematicians. |
| counterexample definition in math: Handbook of Algebraic Topology I.M. James, 1995-07-18 Algebraic topology (also known as homotopy theory) is a flourishing branch of modern mathematics. It is very much an international subject and this is reflected in the background of the 36 leading experts who have contributed to the Handbook. Written for the reader who already has a grounding in the subject, the volume consists of 27 expository surveys covering the most active areas of research. They provide the researcher with an up-to-date overview of this exciting branch of mathematics. |
| counterexample definition in math: Counterexamples in Calculus Sergiy Klymchuk, 2010-12-31 Counterexamples in Calculus serves as a supplementary resource to enhance the learning experience in single variable calculus courses. This book features carefully constructed incorrect mathematical statements that require students to create counterexamples to disprove them. Methods of producing these incorrect statements vary. At times the converse of a well-known theorem is presented. In other instances crucial conditions are omitted or altered or incorrect definitions are employed. Incorrect statements are grouped topically with sections devoted to: Functions, Limits, Continuity, Differential Calculus and Integral Calculus. This book aims to fill a gap in the literature and provide a resource for using counterexamples as a pedagogical tool in the study of introductory calculus. |
| counterexample definition in math: Mathematics for Human Flourishing Francis Su, 2020-01-07 Winner of the Mathematics Association of America's 2021 Euler Book Prize, this is an inclusive vision of mathematics—its beauty, its humanity, and its power to build virtues that help us all flourish“This is perhaps the most important mathematics book of our time. Francis Su shows mathematics is an experience of the mind and, most important, of the heart.”—James Tanton, Global Math ProjectA good book is an entertaining read. A great book holds up a mirror that allows us to more clearly see ourselves and the world we live in. Francis Su’s Mathematics for Human Flourishing is both a good book and a great book.—MAA Reviews For mathematician Francis Su, a society without mathematical affection is like a city without concerts, parks, or museums. To miss out on mathematics is to live without experiencing some of humanity’s most beautiful ideas.In this profound book, written for a wide audience but especially for those disenchanted by their past experiences, an award‑winning mathematician and educator weaves parables, puzzles, and personal reflections to show how mathematics meets basic human desires—such as for play, beauty, freedom, justice, and love—and cultivates virtues essential for human flourishing. These desires and virtues, and the stories told here, reveal how mathematics is intimately tied to being human. Some lessons emerge from those who have struggled, including philosopher Simone Weil, whose own mathematical contributions were overshadowed by her brother’s, and Christopher Jackson, who discovered mathematics as an inmate in a federal prison. Christopher’s letters to the author appear throughout the book and show how this intellectual pursuit can—and must—be open to all. |
| counterexample definition in math: An Introduction to Mathematical Reasoning Peter J. Eccles, 2013-06-26 This book eases students into the rigors of university mathematics. The emphasis is on understanding and constructing proofs and writing clear mathematics. The author achieves this by exploring set theory, combinatorics, and number theory, topics that include many fundamental ideas and may not be a part of a young mathematician's toolkit. This material illustrates how familiar ideas can be formulated rigorously, provides examples demonstrating a wide range of basic methods of proof, and includes some of the all-time-great classic proofs. The book presents mathematics as a continually developing subject. Material meeting the needs of readers from a wide range of backgrounds is included. The over 250 problems include questions to interest and challenge the most able student but also plenty of routine exercises to help familiarize the reader with the basic ideas. |
| counterexample definition in math: 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. |
| counterexample definition in math: 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. |
| counterexample definition in math: 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. |
| counterexample definition in math: Theorems and Counterexamples in Mathematics Bernard R. Gelbaum, John M.H. Olmsted, 2012-12-06 The gratifying response to Counterexamples in analysis (CEA) was followed, when the book went out of print, by expressions of dismay from those who were unable to acquire it. The connection of the present volume with CEA is clear, although the sights here are set higher. In the quarter-century since the appearance of CEA, mathematical education has taken some large steps reflected in both the undergraduate and graduate curricula. What was once taken as very new, remote, or arcane is now a well-established part of mathematical study and discourse. Consequently the approach here is designed to match the observed progress. The contents are intended to provide graduate and ad vanced undergraduate students as well as the general mathematical public with a modern treatment of some theorems and examples that constitute a rounding out and elaboration of the standard parts of algebra, analysis, geometry, logic, probability, set theory, and topology. The items included are presented in the spirit of a conversation among mathematicians who know the language but are interested in some of the ramifications of the subjects with which they routinely deal. Although such an approach might be construed as demanding, there is an extensive GLOSSARY jlNDEX where all but the most familiar notions are clearly defined and explained. The object ofthe body of the text is more to enhance what the reader already knows than to review definitions and notations that have become part of every mathematician's working context. |
| counterexample definition in math: Experimental Mathematics in Action David Bailey, Jonathan Borwein, Neil Calkin, Russell Luke, Roland Girgensohn, Victor Moll, 2007-05-31 With the continued advance of computing power and accessibility, the view that real mathematicians don't compute no longer has any traction for a newer generation of mathematicians. The goal in this book is to present a coherent variety of accessible examples of modern mathematics where intelligent computing plays a significant role and in so doi |
| counterexample definition in math: Mathematical Writing Donald E. Knuth, Tracy Larrabee, Paul M. Roberts, 1989 This book will help those wishing to teach a course in technical writing, or who wish to write themselves. |
| counterexample definition in math: Basic Category Theory Tom Leinster, 2014-07-24 A short introduction ideal for students learning category theory for the first time. |
| counterexample definition in math: An Invitation to Abstract Mathematics Béla Bajnok, 2020-10-27 This undergraduate textbook promotes an active transition to higher mathematics. Problem solving is the heart and soul of this book: each problem is carefully chosen to demonstrate, elucidate, or extend a concept. More than 300 exercises engage the reader in extensive arguments and creative approaches, while exploring connections between fundamental mathematical topics. Divided into four parts, this book begins with a playful exploration of the building blocks of mathematics, such as definitions, axioms, and proofs. A study of the fundamental concepts of logic, sets, and functions follows, before focus turns to methods of proof. Having covered the core of a transition course, the author goes on to present a selection of advanced topics that offer opportunities for extension or further study. Throughout, appendices touch on historical perspectives, current trends, and open questions, showing mathematics as a vibrant and dynamic human enterprise. This second edition has been reorganized to better reflect the layout and curriculum of standard transition courses. It also features recent developments and improved appendices. An Invitation to Abstract Mathematics is ideal for those seeking a challenging and engaging transition to advanced mathematics, and will appeal to both undergraduates majoring in mathematics, as well as non-math majors interested in exploring higher-level concepts. From reviews of the first edition: Bajnok’s new book truly invites students to enjoy the beauty, power, and challenge of abstract mathematics. ... The book can be used as a text for traditional transition or structure courses ... but since Bajnok invites all students, not just mathematics majors, to enjoy the subject, he assumes very little background knowledge. Jill Dietz, MAA Reviews The style of writing is careful, but joyously enthusiastic.... The author’s clear attitude is that mathematics consists of problem solving, and that writing a proof falls into this category. Students of mathematics are, therefore, engaged in problem solving, and should be given problems to solve, rather than problems to imitate. The author attributes this approach to his Hungarian background ... and encourages students to embrace the challenge in the same way an athlete engages in vigorous practice. John Perry, zbMATH |
| counterexample definition in math: Category Theory in Context Emily Riehl, 2017-03-09 Introduction to concepts of category theory — categories, functors, natural transformations, the Yoneda lemma, limits and colimits, adjunctions, monads — revisits a broad range of mathematical examples from the categorical perspective. 2016 edition. |
| counterexample definition in math: Book of Proof Richard H. Hammack, 2016-01-01 This book is an introduction to the language and standard proof methods of mathematics. It is a bridge from the computational courses (such as calculus or differential equations) that students typically encounter in their first year of college to a more abstract outlook. It lays a foundation for more theoretical courses such as topology, analysis and abstract algebra. Although it may be more meaningful to the student who has had some calculus, there is really no prerequisite other than a measure of mathematical maturity. |
| counterexample definition in math: Geometric Measure Theory Frank Morgan, 2014-05-10 Geometric Measure Theory: A Beginner's Guide provides information pertinent to the development of geometric measure theory. This book presents a few fundamental arguments and a superficial discussion of the regularity theory. Organized into 12 chapters, this book begins with an overview of the purpose and fundamental concepts of geometric measure theory. This text then provides the measure-theoretic foundation, including the definition of Hausdorff measure and covering theory. Other chapters consider the m-dimensional surfaces of geometric measure theory called rectifiable sets and introduce the two basic tools of the regularity theory of area-minimizing surfaces. This book discusses as well the fundamental theorem of geometric measure theory, which guarantees solutions to a wide class of variational problems in general dimensions. The final chapter deals with the basic methods of geometry and analysis in a generality that embraces manifold applications. This book is a valuable resource for graduate students, mathematicians, and research workers. |
| counterexample definition in math: How to Write Mathematics Norman Earl Steenrod, 1973-12-31 This classic guide contains four essays on writing mathematical books and papers at the research level and at the level of graduate texts. The authors are all well known for their writing skills, as well as their mathematical accomplishments. The first essay, by Steenrod, discusses writing books, either monographs or textbooks. He gives both general and specific advice, getting into such details as the need for a good introduction. The longest essay is by Halmos, and contains many of the pieces of his advice that are repeated even today: In order to say something well you must have something to say; write for someone; think about the alphabet. Halmos's advice is systematic and practical. Schiffer addresses the issue by examining four types of mathematical writing: research paper, monograph, survey, and textbook, and gives advice for each form of exposition. Dieudonne's contribution is mostly a commentary on the earlier essays, with clear statements of where he disagrees with his coauthors. The advice in this small book will be useful to mathematicians at all levels. |
| counterexample definition in math: 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. |
| counterexample definition in math: 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 |
| counterexample definition in math: Introduction to Mathematical Logic Elliot Mendelsohn, 2012-12-06 This is a compact mtroduction to some of the pnncipal tOpICS of mathematical logic . In the belief that beginners should be exposed to the most natural and easiest proofs, I have used free-swinging set-theoretic methods. The significance of a demand for constructive proofs can be evaluated only after a certain amount of experience with mathematical logic has been obtained. If we are to be expelled from Cantor's paradise (as nonconstructive set theory was called by Hilbert), at least we should know what we are missing. The major changes in this new edition are the following. (1) In Chapter 5, Effective Computability, Turing-computabIlity IS now the central notion, and diagrams (flow-charts) are used to construct Turing machines. There are also treatments of Markov algorithms, Herbrand-Godel-computability, register machines, and random access machines. Recursion theory is gone into a little more deeply, including the s-m-n theorem, the recursion theorem, and Rice's Theorem. (2) The proofs of the Incompleteness Theorems are now based upon the Diagonalization Lemma. Lob's Theorem and its connection with Godel's Second Theorem are also studied. (3) In Chapter 2, Quantification Theory, Henkin's proof of the completeness theorem has been postponed until the reader has gained more experience in proof techniques. The exposition of the proof itself has been improved by breaking it down into smaller pieces and using the notion of a scapegoat theory. There is also an entirely new section on semantic trees. |
| counterexample definition in math: Applied Analysis John K. Hunter, Bruno Nachtergaele, 2001 This book provides an introduction to those parts of analysis that are most useful in applications for graduate students. The material is selected for use in applied problems, and is presented clearly and simply but without sacrificing mathematical rigor. The text is accessible to students from a wide variety of backgrounds, including undergraduate students entering applied mathematics from non-mathematical fields and graduate students in the sciences and engineering who want to learn analysis. A basic background in calculus, linear algebra and ordinary differential equations, as well as some familiarity with functions and sets, should be sufficient. |
| counterexample definition in math: Humanizing Mathematics and its Philosophy Bharath Sriraman, 2017-11-07 This Festschrift contains numerous colorful and eclectic essays from well-known mathematicians, philosophers, logicians, and linguists celebrating the 90th birthday of Reuben Hersh. The essays offer, in part, attempts to answer the following questions set forth by Reuben himself as a focus for this volume: Can practicing mathematicians, as such, contribute anything to the philosophy of math? Can or should philosophers of math, as such, say anything to practicing mathematicians? Twenty or fifty years from now, what will be similar, and what will, or could, or should be altogether different: About the philosophy of math? About math education? About math research institutions? About data processing and scientific computing? The essays also offer glimpses into Reuben’s fertile mind and his lasting influence on the mathematical community, as well as revealing the diverse roots, obstacles and philosophical dispositions that characterize the working lives of mathematicians. With contributions from a veritable “who’s who” list of 20th century luminaries from mathematics and philosophy, as well as from Reuben himself, this volume will appeal to a wide variety of readers from curious undergraduates to prominent mathematicians. |
| counterexample definition in math: Mathematical Methods in Linguistics Barbara B.H. Partee, A.G. ter Meulen, R. Wall, 1990-04-30 Elementary set theory accustoms the students to mathematical abstraction, includes the standard constructions of relations, functions, and orderings, and leads to a discussion of the various orders of infinity. The material on logic covers not only the standard statement logic and first-order predicate logic but includes an introduction to formal systems, axiomatization, and model theory. The section on algebra is presented with an emphasis on lattices as well as Boolean and Heyting algebras. Background for recent research in natural language semantics includes sections on lambda-abstraction and generalized quantifiers. Chapters on automata theory and formal languages contain a discussion of languages between context-free and context-sensitive and form the background for much current work in syntactic theory and computational linguistics. The many exercises not only reinforce basic skills but offer an entry to linguistic applications of mathematical concepts. For upper-level undergraduate students and graduate students in theoretical linguistics, computer-science students with interests in computational linguistics, logic programming and artificial intelligence, mathematicians and logicians with interests in linguistics and the semantics of natural language. |
| counterexample definition in math: High-Dimensional Probability Roman Vershynin, 2018-09-27 An integrated package of powerful probabilistic tools and key applications in modern mathematical data science. |
| counterexample definition in math: A Concise Course in Algebraic Topology J. P. May, 1999-09 Algebraic topology is a basic part of modern mathematics, and some knowledge of this area is indispensable for any advanced work relating to geometry, including topology itself, differential geometry, algebraic geometry, and Lie groups. This book provides a detailed treatment of algebraic topology both for teachers of the subject and for advanced graduate students in mathematics either specializing in this area or continuing on to other fields. J. Peter May's approach reflects the enormous internal developments within algebraic topology over the past several decades, most of which are largely unknown to mathematicians in other fields. But he also retains the classical presentations of various topics where appropriate. Most chapters end with problems that further explore and refine the concepts presented. The final four chapters provide sketches of substantial areas of algebraic topology that are normally omitted from introductory texts, and the book concludes with a list of suggested readings for those interested in delving further into the field. |
| counterexample definition in math: Theory and Computation of Tensors Yimin Wei, Weiyang Ding, 2016-08-28 Theory and Computation of Tensors: Multi-Dimensional Arrays investigates theories and computations of tensors to broaden perspectives on matrices. Data in the Big Data Era is not only growing larger but also becoming much more complicated. Tensors (multi-dimensional arrays) arise naturally from many engineering or scientific disciplines because they can represent multi-relational data or nonlinear relationships. - Provides an introduction of recent results about tensors - Investigates theories and computations of tensors to broaden perspectives on matrices - Discusses how to extend numerical linear algebra to numerical multi-linear algebra - Offers examples of how researchers and students can engage in research and the applications of tensors and multi-dimensional arrays |
| counterexample definition in math: Lectures on Elliptic and Parabolic Equations in Holder Spaces Nikolaĭ Vladimirovich Krylov, 1996 These lectures concentrate on fundamentals of the modern theory of linear elliptic and parabolic equations in H older spaces. Krylov shows that this theory - including some issues of the theory of nonlinear equations - is based on some general and extremely powerful ideas and some simple computations. The main object of study is the first boundary-value problems for elliptic and parabolic equations, with some guidelines concerning other boundary-value problems such as the Neumann or oblique derivative problems or problems involving higher-order elliptic operators acting on the boundary. Numerical approximations are also discussed. This book, containing 200 exercises, aims to provide a good understanding of what kind of results are available and what kinds of techniques are used to obtain them. |
| counterexample definition in math: Berkeley Problems in Mathematics Paulo Ney de Souza, Jorge-Nuno Silva, 2004-01-08 This book collects approximately nine hundred problems that have appeared on the preliminary exams in Berkeley over the last twenty years. It is an invaluable source of problems and solutions. Readers who work through this book will develop problem solving skills in such areas as real analysis, multivariable calculus, differential equations, metric spaces, complex analysis, algebra, and linear algebra. |
| counterexample definition in math: Lectures on Symplectic Geometry Ana Cannas da Silva, 2004-10-27 The goal of these notes is to provide a fast introduction to symplectic geometry for graduate students with some knowledge of differential geometry, de Rham theory and classical Lie groups. This text addresses symplectomorphisms, local forms, contact manifolds, compatible almost complex structures, Kaehler manifolds, hamiltonian mechanics, moment maps, symplectic reduction and symplectic toric manifolds. It contains guided problems, called homework, designed to complement the exposition or extend the reader's understanding. There are by now excellent references on symplectic geometry, a subset of which is in the bibliography of this book. However, the most efficient introduction to a subject is often a short elementary treatment, and these notes attempt to serve that purpose. This text provides a taste of areas of current research and will prepare the reader to explore recent papers and extensive books on symplectic geometry where the pace is much faster. For this reprint numerous corrections and clarifications have been made, and the layout has been improved. |
| counterexample definition in math: 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. |
What is an example of a proof by minimal counterexample?
Nov 18, 2018 · $\begingroup$ (If I understand correctly) "proof by minimal counterexample" proves that a minimal counterexample does not exist , and the 2 comments above are not …
logic - Constructing a counterexample - Mathematics Stack Exchange
Jan 24, 2017 · Stack Exchange Network. Stack Exchange network consists of 183 Q&A communities including Stack Overflow, the largest, most trusted online community for …
logic - Proof by contradiction using counterexample - Mathematics …
Apr 6, 2016 · Why can't we use one counterexample as the contradiction to the contradicting statement? Example: Let a statement be A where a-->b. We can prove A is not true by finding …
Advice on finding counterexamples - Mathematics Stack Exchange
Apr 1, 2015 · If you don’t find a counterexample using the approaches already suggested, try to prove the theorem, and see where you get stuck: this may well give you a better handle on the …
big list - Conjectures that have been disproved with extremely large ...
Jul 23, 2010 · The first counterexample found has 96 vertices. The "least" counterexample known has 54 vertices. Heidetniemi conjecture; Chromatic number of tensor product of finite …
Ideas of finding counterexamples? - Mathematics Stack Exchange
Sometimes blindly constructing a counterexample works. In certain situations, you can look at the list of properties a counterexample needs to satisfy and just write down an example which …
logic - Using truth tables to evaluate an argument and provide ...
Dec 18, 2016 · These 3 statements aren't even related and it does not matter for a good counterexample. All that matters is that the atomic statements you choose for J, K and L are …
How are proofs formatted when the answer is a counterexample?
Prove or find a counterexample: the sum of two integers is odd. The fact that 1 + 1 = 2 is a counterexample that disproves that statement. What is the proper format in which to write this? …
Counterexample to Kunneth Formula - Mathematics Stack Exchange
Oct 7, 2021 · I'm reading Example 9.14 of Bott&Tu's Differential Forms in Algebraic Topology, which gave a counter example to the Kunneth formula when the assumption is not …
A counterexample in topology - Mathematics Stack Exchange
EDIT Also, I am looking for a counterexample which is a locally path connected space (a previous answer showed a counterexample without this property.. but @answerer it was interesting …
Everything You Need to Know About Modular Arithmetic
Ex. 2 The standard representatives for all possible numbers modulo 10 are given by 0,1,2,3,4,5,6,7,8,9 although, for example, 3 ≡ 13 ≡ 23(mod 10), we would take the smallest …
Small sets without unique products in torsion-free groups
the group Pwas used by Gardam in [8] to give a counterexample to the unit conjecture over the fieldF 2. Since any potential counterexample to Kaplansky’s zero-divisor conjecture involves a …
A Primer on Mathematical Proof - University of Michigan
Math 150s Proof and Mathematical Reasoning Jenny Wilson Common Mistake # 7. ... Example of a False Statement and Counterexample To prove that a statement is true, you must show that …
Minimal Counterexample: A Different look at Induction1
Aug 28, 2021 · picks the minimal counterexample to the statement at hand, and then tries to argue a contradiction. To make things concrete, let is give a “different” proof of something we …
1-1 Patterns and Inductive Reasoning - portal.mywccc.org
finding one counterexample.A to a conjecture is an example for which the conjecture is incorrect. Finding a Counterexample Find a counterexample for each conjecture. a. The square of any …
Disproof - Gordon College
In other words, we need to nd a counterexample. If we are able to do this, we will have disproved the universal statement. MAT231 (Transition to Higher Math) Disproof Fall 2014 10 / 16. …
The Four Color Theorem - math.uchicago.edu
Definition 1.9.A graph is connected if there exists a sequence of edges connect-ing any two vertices in the graph. A graph that is not connected is disconnected. Definition 1.10.Two or …
MATH 304 Linear Algebra - Texas A&M University
MATH 304 Linear Algebra Lecture 8: Vector spaces. Subspaces. Linear operations on vectors ... Counterexample: dumb scaling Consider the set V = Rn with the standard addition and a …
2-2 Biconditionals and Definitions - portal.mywccc.org
a counterexample. Language Arts Is the given statement a good definition? Explain. a. An airplane is a vehicle that flies. The statement is not a good definition because it is not …
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Definition Of Counterexample In Math (PDF)
Definition Of Counterexample In Math and Bestseller Lists 5. Accessing Definition Of Counterexample In Math Free and Paid eBooks Definition Of Counterexample In Math Public …
1.Proving Injectivity and Surjectivity - Stanford University
The definition of right-cancellative is another implication, so again, we should assume the antecedent and show the consequent. Since we want to show a universally quantified …
Geometry Notes – Chapter 2: Reasoning and Proof - Dan …
Counterexample: 3 + 4 = 7 (odd) 2.2 – Conditional Statements . A conditional statement is a logical statement that has two parts, a hypothesis and a conclusion. This type of statement can …
Math 120A — Introduction to Group Theory - University of …
We’ll see a formal definition shortly, at which point we’ll be able to verify that(Z,+) really is a group. The simplicity of the group structure means that it is often used as a building block for more …
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Quarter 2 Module 14: Writing Proofs (Direct and Indirect)
2. ∠ = ∠ and ∠ = ∠ Definition of Congruent Angles 3. ∠ = ∠ (15) 4. ∠ ≅ ∠ Definition of Congruent Angles A. Addition Property of Equality B. Distributive Property of Equality C. Subtraction …
MATH 350: Homework #8 Due: Thursday, November 8, 2018
MATH 350: Homework #8 Due: Thursday, November 8, 2018 Solve the below problems concerning determinants and eigenvalues. A (possi- ... Prove or provide a counterexample. …
Introduction - University of California, Berkeley
Definition 3.1). This question of Slaman and Steel has been completely unstudied since the 1988 paper where it was posed, and it remains open. However, we show that it is deeply ...
RamseyRL: A Framework for Intelligent Ramsey Number …
of a graph being a counterexample. The paper also proposes algorithmic optimizations to confine a polynomial search run-time. This paper does not aim to present new counterexam-ples but …
Solutions to linear algebra, homework 1 - Stanford University
Prove or give a counterexample: If U 1;U 2;W subspaces of V such that U 1 W = U 2 W= V then U 1 = U 2. Remark. As commented in class, Axler’s use of corresponded to what I called \internal …
Name: Lesson 8.2 Conjecture and Counter-Examples - Weebly
Math 11 Foundations: Unit 8 – Logic & Geometry Sardis Secondary Foundationsmath11.weebly.com Mr. Sutcliffe Example 1: Make a conjecture about intersecting …
CS 214 Introduction to Discrete Structures Chapter 2 Proofs, …
Inductive reasoning in math and CS: not acceptable • Deductive reasoning State P →Q as conjecture Prove using logic and domain knowledge; P →Q becomes theorem Disprove by …
ELEMENTARY NUMBER THEORY AND METHODS OF PROOF
Disproving Universal Statements by Counterexample To disprove a statement means to show that it is false. ! To disprove a statement of the form ... It follows from the definition that since m and …
A COUNTEREXAMPLE TO BORSUK'S CONJECTURE
A COUNTEREXAMPLE TO BORSUK'S CONJECTURE JEFF KAHN AND GIL KALAI Abstract. Let f(d) be the smallest number so that every set in Rd of diam- ... that definition is the result of …
Introduction to Relations - Florida State University
Definition 1.4.1. Let Rbe a relation from Ato B. Then R 1 = f(b;a)j(a;b) 2 Rgis a relation from Bto A. R 1 is called the inverse of the relation R. Discussion The inverse of a relation Ris simply the …
…Definition… …Characteristics… …Definition…
exterior. angle. exterior angle. exterior. angle. exterior. angle …Definition… …Characteristics… …Example… …Counterexample… …Definition ...
Lecture 12: Introduction to Graphs and Trees - Northeastern …
Proof by Counterexample Searching for counterexamples is the best way to disprove the correctness of some things. Identify a case for which something is NOT true If the proof seems …
Definition Of Counterexample In Math Full PDF
Definition Of Counterexample In Math Robert J. Bond,William J. Keane. Content Counterexamples in Calculus Sergiy Klymchuk,2010-12-31 Counterexamples in Calculus …
1.1 Solving Problems by Inductive Reasoning - Crossroads …
4 CHAPTER 1 The Art of Problem Solving EXAMPLE 1 Identify each premise and the conclusion in each of the fol- lowing arguments. Then tell whether each argument is an example of …
Geometry: Conditionals, Converses, and Biconditionals …
Which statement provides a counterexample to the following faulty definition? A square is a figure with four congruent sides. a. A six-sided figure can have four sides congruent. b. Some …
Section 1.3: Valid and Invalid Arguments - University of …
1. The Definition of a Valid and Invalid Argument We start with the definition of an argument. Definition 1.1. Anargument(form)isasequence ofstatements (forms). All statements (forms) in …
www.math.ucsd.edu
92.Hewitt’sCondensedCorkscrew Definition: • LetT = S ∪{a+}∪{a−} denotetheTychonoffCorkscrew,and if[0,Ω) isthesetofcountableordinals,weletA = T ×[0,Ω ...
BERNSTEIN’S THEOREM AND OTHER BASICS OF MINIMAL …
Definition 2.13.Let Σ ⊂R3 be a two-dimensional differentiable manifold. Then, the continuos map N: Σ →S2 giving the unit normal vector to a point on Σ, is called the Gauss map. Another …
Chapter 2: Reasoning and Proof - portal.mywccc.org
Jan 16, 2003 · Find a Counterexample FINANCE Find a counterexample for the following statement based on the graph. The rates for CDs are at least 1.5% less than the rates a year …
Direct Proof – Divisibility
Direct Proof – Divisibility Lecture 15 Section 4.3 Robb T. Koether Hampden-Sydney College Fri, Feb 7, 2014 Robb T. Koether (Hampden-Sydney College) Direct Proof – Divisibility Fri, Feb 7, …
Section 3.2: Direct Proof and Counterexample 2
1. The Definition of a Rational Number We start with the formal definition of a rational number. Definition 1.1. A real number r is rational if and only if it can be expressed as a quotient of two …
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Proof Techniques - Computer Science
To disprove a universal statement, it su ces to nd one counterexample. (For \existence" statements, this is reversed. For example, if your statement is \there exists at least one odd …
Introduction #P THE BUNKBED CON - math.ucla.edu
Abstract. We give an explicit counterexample to the bunkbed conjecture introduced by Kaste-leyn in 1985. The counterexample is given by a planar graph on 7222 vertices, and is built on the …
Notes for Math 122 Logic and Foundations - University of …
Chapter 1 Propositional Logic 1.1 Statements A statement or proposition is an assertion which is either true or false, though you may not know which.
1 Homeomorphisms of metric spaces - dept.math.lsa.umich.edu
10 October 2024 Math 490: Worksheet #8 Jenny Wilson 1 Homeomorphisms of metric spaces Definition 1.1. (Homeomorphism). Let X and Y be metric spaces. Then a map f : X →Y is a …
Definition Of Counterexample In Geometry
Definition Of Counterexample In Geometry The Math Forum Counterexamples in Topology Lynn Arthur Steen,J. Arthur Seebach,2013-04-22 Over 140 examples, preceded by a succinct …
Definition Of Counterexample In Geometry [PDF]
Definition Of Counterexample In Geometry: Counterexamples in Topology Lynn Arthur Steen,J. Arthur Seebach,2013-04-22 Over 140 examples preceded by a ... math problems and the math …
Lecture Notes: Discrete Mathematics for Computer Science
is called a counterexample. For instance, to argue that the assertion 8x(x2 x) (1) is false, we can use the value x = 1 2 as a counterexample. The assertion that formula (1) is false can be …
Definition Of Counterexample In Geometry (Download Only)
Definition Of Counterexample In Geometry: Counterexamples in Topology Lynn Arthur Steen,J. Arthur Seebach,2013-04-22 Over 140 examples preceded by a ... discrete math course for …
COUNTEREXAMPLES IN MEASURE AND INTEGRATION
Cambridge University Press 978-1-316-51913-4 — Counterexamples in Measure and Integration René L. Schilling , Franziska Kühn Frontmatter More Information
Definition Of Counterexample In Geometry [PDF]
Definition Of Counterexample In Geometry: ... future math teachers It is open source with low cost print editions and free electronic editions E-math Iii' 2007 Ed.(geometry) , Lectures on …
Definition Of Counterexample In Geometry (2024)
Definition Of Counterexample In Geometry: ... future math teachers It is open source with low cost print editions and free electronic editions E-math Iii' 2007 Ed.(geometry) , Lectures on …
