Set theory Books

Book SynopsisThis book introduces readers to the art of doing mathematical proofs. Proofs are the glue that holds mathematics together. They make connections between math concepts and show why things work the way they do. This book teaches the art of proofs using familiar high-school concepts, such as numbers, polynomials, functions, and trigonometry. It retells math as a story, where the next chapter follows from the previous one.Readers will see how various mathematical concepts are tied and will see that mathematics is not a pile of formulas and facts; rather, it has an orderly and beautiful edifice.The author begins with basic rules of logic and then progresses through the topics already familiar to the students: numbers, inequalities, functions, polynomials, exponents, and trigonometric functions. There are also beautiful proofs for conic sections, sequences, and Fibonacci numbers. Each chapter has exercises for the reader.Reviewer Comments:I find the

Read more less

Book SynopsisAssuming little mathematical background, this short introduction to category theory is ideal for beginning graduate students or advanced undergraduates learning category theory for the first time. Suitable for independent study or as a course book, it gives extensive explanations of the key concepts along with hundreds of examples and exercises.Table of ContentsNote to the reader; Introduction; 1. Categories, functors and natural transformations; 2. Adjoints; 3. Interlude on sets; 4. Representables; 5. Limits; 6. Adjoints, representables and limits; Appendix: proof of the General Adjoint Functor Theorem; Glossary of notation; Further reading; Index.

Read more less

Book SynopsisSet theory is a rich and beautiful subject whose fundamental concepts permeate virtually every branch of mathematics. One could say that set theory is a unifying theory for mathematics, since nearly all mathematical concepts and results can be formalized within set theory. This textbook is meant for an upper undergraduate course in set theory. In this text, the fundamentals of abstract sets, including relations, functions, the natural numbers, order, cardinality, transfinite recursion, the axiom of choice, ordinal numbers, and cardinal numbers, are developed within the framework of axiomatic set theory. The reader will need to be comfortable reading and writing mathematical proofs. The proofs in this textbook are rigorous, clear, and complete, while remaining accessible to undergraduates who are new to upper-level mathematics. Exercises are included at the end of each section in a chapter, with useful suggestions for the more challenging exercises.Trade Review'… Cunningham neglects no opportunity to make the subject as accessible as possible. The mathematical development is rigorous, as it should be, but not excessively so. Although he starts from zero, that is not to say the book is easy, but any difficulty that arises is in the nature of the subject, and is no fault of the author's. Throughout the book, he offers many appropriate examples (or non-examples), and provides numerous and diverse exercises, which often prove results that are later used in the body of the text, drawing the reader into the subject.' Frederic Green, ACM SIGACT News'This book fulfills its stated goals: 'The textbook is suitable for a broad range of readers, from undergraduate to graduate students, who desire a better understanding of the fundamental topics in set theory that may have been, or will be, overlooked in their other mathematics courses'.' Shoshana Friedman, MathSciNetTable of Contents1. Introduction; 2. Basic set building axioms and operations; 3. Relations and functions; 4. The natural numbers; 5. On the size of sets; 6. Transfinite recursion; 7. The axiom of choice (revisited); 8. Ordinals; 9. Cardinals.

Read more less

Book SynopsisThis book is the first modern introduction to the logic of infinitary languages in forty years, and is aimed at graduate students and researchers in all areas of mathematical logic. Connections between infinitary model theory and other branches of mathematical logic, and applications to algebra and algebraic geometry are both comprehensively explored.Table of ContentsIntroduction; Part I. Classical Results in Infinitary Model Theory: 1. Infinitary languages; 2. Back and forth; 3. The space of countable models; 4. The model existence theorem; 5. Hanf numbers and indiscernibles; Part II. Building Uncountable Models: 6. Elementary chains; 7. Vaught counterexamples; 8. Quasinimal excellence; Part III. Effective Considerations: 9. Effective descriptive set theory; 10. Hyperarithmetic sets; 11. Effective aspects of Lω1,ω; 12. Spectra of Vaught counterexamples; Appendix A. N1-free abelian groups; Appendix B. Admissibility; References; Index.

Read more less

Book SynopsisThe BanachTarski Paradox is the most surprising result in mathematics. This new edition of a classic book offers a comprehensive, accessible presentation, with many related results, especially connections to non-Euclidean geometry, to squaring the circle, and even to some art by Escher. This material is suited to projects for undergraduates or masters students.Trade Review'The new edition of The Banach–Tarski Paradox, by Grzegorz Tomkowicz and Stan Wagon, is a welcome revisiting and extensive reworking of the first edition of the book. Whether you are new to the topic of paradoxical decompositions, or have studied the phenomenon for years, this book has a lot to offer. I recommend buying two copies of the book, one for the office and one for the home, because studying the book carefully (perhaps in a series of working seminars) will be worthwhile, and casually browsing through the book in your spare time will be simply a lot of fun.' Joseph Rosenblatt, Department Chair, Department of Mathematical Sciences, Indiana University-Purdue University Indianapolis'This is the second edition of this classic and comprehensive monograph on paradoxical decompositions. What adds to the special appeal of this topic is the diversity of methods and the connection to several fields including set theory, group theory, measure theory, geometry, algebra, and discrete mathematics. The previous edition of this book stimulated a large amount of research. The present volume also includes these developments and furthermore discusses the solutions to some of the problems that were solved in the past thirty years, including the realization of the Banach–Tarski paradox with pieces having the Baire property and Tarski's circle squaring problem.' Miklos Laczkovich, University College London'Wagon's classic book on the Banach–Tarski paradox has been updated with Tomkowicz to include major advances over the last thirty years. It remains the definitive source for both newcomers to the subject and experts who want to broaden their knowledge. The book provides a basic introduction to the field with clear exposition and important historical background. It includes complete proofs of the Banach–Tarski paradox and related results. It continues with an extensive survey of more advanced topics. This is far and away the best resource for beginners and experts on the strangest result in all of mathematics.' Matthew Foreman, University of California, Irvine'Several spectacular results have been proved since the first edition of this book … All these results and problems are presented in a penetrating and lucid way in this new edition.' Jan Mycielski, University of Colorado, Boulder, from the ForewordReview of previous edition: '… a readable and stimulating book.' Ward Henson, American Scientist'In 1985 Stan Wagon wrote The Banach-Tarski Paradox, which not only became the classic text on paradoxical mathematics, but also provided vast new areas for research. The new second edition, co-written with Grzegorz Tomkowicz, a Polish mathematician who specializes in paradoxical decompositions, exceeds any possible expectation I might have had for expanding a book I already deeply treasured. The meticulous research of the original volume is still there, but much new research has also been included … I should also mention that this book is beautifully illustrated.' John J. Watkins, MAA Reviews'For some people the book will be over by page 36, because by then one has seen full treatments of the results of Hausdorff and of Banach and Tarski. These people are short-sighted; there is much fascinating mathematics to be learned from the further developments. As the recent result of Marks and Unger shows, there is probably still much to discover. Indeed, the book contains some very interesting questions that still await solution.' Klaas Pieter Hart, Mathematical ReviewsTable of ContentsPart I. Paradoxical Decompositions, or the Nonexistence of Finitely Additive Measures: 1. Introduction; 2. The Hausdorff paradox; 3. The Banach–Tarski paradox: duplicating spheres and balls; 4. Hyperbolic paradoxes; 5. Locally commutative actions: minimizing the number of pieces in a paradoxical decomposition; 6. Higher dimensions; 7. Free groups of large rank: getting a continuum of spheres from one; 8. Paradoxes in low dimensions; 9. Squaring the circle; 10. The semigroup of equidecomposability types; Part II: Finitely Additive Measures, or the Nonexistence of Paradoxical Decompositions: 11. Transition; 12. Measures in groups; 13. Applications of amenability; 14. Growth conditions in groups and supramenability; 15. The role of the axiom of choice.

Read more less

Book SynopsisProofs play a central role in advanced mathematics and theoretical computer science, yet many students struggle the first time they take a course in which proofs play a significant role. This bestselling text''s third edition helps students transition from solving problems to proving theorems by teaching them the techniques needed to read and write proofs. Featuring over 150 new exercises and a new chapter on number theory, this new edition introduces students to the world of advanced mathematics through the mastery of 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 an analysis of techniques that can be used to build up complex proofs step by step, using detailed ''scratch work'' sections to expose the machinery of proofs about numbers, sets, relations, and functions. Assuming no background beyond standard high school mathematics, this booTrade Review'Not only does this book help students learn how to prove results, it highlights why we care so much. It starts in the introduction with some simple conjectures and gathering data, quickly disproving the first but amassing support for the second. Will that pattern persist? How can these observations lead us to a proof? The book is engagingly written, and covers - in clear and great detail - many proof techniques. There is a wealth of good exercises at various levels. I've taught problem solving before (at The Ohio State University and Williams College), and this book has been a great addition to the resources I recommend to my students.' Steven J. Miller, Williams College, Massachusetts'This book is my go-to resource for students struggling with how to write mathematical proofs. Beyond its plentiful examples, Velleman clearly lays out the techniques and principles so often glossed over in other texts.' Rafael Frongillo, University of Colorado, Boulder'I've been using this book religiously for the last eight years. It builds a strong foundation in proof writing and creates the axiomatic framework for future higher-level mathematics courses. Even when teaching more advanced courses, I recommend students to read chapter 3 (Proofs) since it is, in my opinion, the best written exposition of proof writing techniques and strategies. This third edition brings a new chapter (Number Theory), which gives the instructor a few more topics to choose from when teaching a fundamental course in mathematics. I will keep using it and recommending it to everyone, professors and students alike.' Mihai Bailesteanu, Central Connecticut State University'Professor Velleman sets himself the difficult task of bridging the gap between algorithmic and proof-based mathematics. By focusing on the basic ideas, he succeeded admirably. Many similar books are available, but none are more treasured by beginning students. In the Third Edition, the constant pursuit of excellence is further reinforced.' Taje Ramsamujh, Florida International University'Proofs are central to mathematical development. They are the tools used by mathematicians to establish and communicate their results. The developing mathematician often learns what constitutes a proof and how to present it by osmosis. How to Prove It aims at changing that. It offers a systematic introduction to the development, structuring, and presentation of logical mathematical arguments, i.e. proofs. The approach is based on the language of first-order logic and supported by proof techniques in the style of natural deduction. The art of proving is exercised with naive set theory and elementary number theory throughout the book. As such, it will prove invaluable to first-year undergraduate students in mathematics and computer science.' Marcelo Fiore, University of Cambridge'Overall, this is an engagingly-written and effective book for illuminating thinking about and building a careful foundation in proof techniques. I could see it working in an introduction to proof course or a course introducing discrete mathematics topics alongside proof techniques. As a self-study guide, I could see it working as it so well engages the reader, depending on how able they are to navigate the cultural context in some examples.' Peter Rowlett, LMS Newsletter'Altogether this is an ambitious and largely very successful introduction to the writing of good proofs, laced with many good examples and exercises, and with a pleasantly informal style to make the material attractive and less daunting than the length of the book might suggest. I particularly liked the many discussions of fallacious or incomplete proofs, and the associated challenges to readers to untangle the errors in proofs and to decide for themselves whether a result is true.' Peter Giblin, University of Liverpool, The Mathematical GazetteTable of Contents1. Sentential logic; 2. Quantificational logic; 3. Proofs; 4. Relations; 5. Functions; 6. Mathematical induction; 7. Number theory; 8. Infinite sets.

Read more less

Book SynopsisModel theory begins with an audacious idea: to consider statements about mathematical structures as mathematical objects of study in their own right. While inherently important as a tool of mathematical logic, it also enjoys connections to and applications in diverse branches of mathematics, including algebra, number theory and analysis. Despite this, traditional introductions to model theory assume a graduate-level background of the reader. In this innovative textbook, Jonathan Kirby brings model theory to an undergraduate audience. The highlights of basic model theory are illustrated through examples from specific structures familiar from undergraduate mathematics, paying particular attention to definable sets throughout. With numerous exercises of varying difficulty, this is an accessible introduction to model theory and its place in mathematics.Table of ContentsPreface; Part I. Languages and Structures: 1. Structures; 2. Terms; 3. Formulas; 4. Definable sets; 5. Substructures and quantifiers; Part II. Theories and Compactness: 6. Theories and axioms; 7. The complex and real fields; 8. Compactness and new constants; 9. Axiomatisable classes; 10. Cardinality considerations; 11. Constructing models from syntax; Part III. Changing Models: 12. Elementary substructures; 13. Elementary extensions; 14. Vector spaces and categoricity; 15. Linear orders; 16. The successor structure; Part IV. Characterising Definable Sets: 17. Quantifier elimination for DLO; 18. Substructure completeness; 19. Power sets and Boolean algebras; 20. The algebras of definable sets; 21. Real vector spaces and parameters; 22. Semi-algebraic sets; Part V. Types: 23. Realising types; 24. Omitting types; 25. Countable categoricity; 26. Large and small countable models; 27. Saturated models; Part VI. Algebraically Closed Fields: 28. Fields and their extensions; 29. Algebraic closures of fields; 30. Categoricity and completeness; 31. Definable sets and varieties; 32. Hilbert's Nullstellensatz; Bibliography; Index.

Read more less

Book Synopsis

Read more less

Book SynopsisThis book is designed for readers who knowelementary mathematical logic and axiomatic settheory, and who want to learn more about set theory.The primary focus of the book is on the independenceproofs. Most famous among these is the independenceof the Continuum Hypothesis (CH); that is, there aremodels of the axioms of set theory (ZFC) in whichCH is true, and other models in which CH is false.More generally, cardinal exponentiation on the regularcardinals can consistently be anything not contradictingthe classical theorems of Cantor and König.The basic methods for the independence proofs arethe notion of constructibility, introduced by Gödel, andthe method of forcing, introduced by Cohen. This bookdescribes these methods in detail, verifi es the basicindependence results for cardinal exponentiation, andalso applies these methods to prove the independenceof various mathematical questions in measure theoryand general topology.Before the chapters on forcing, there is a fairly longchapter on "infi nitary combinatorics". This consistsof just mathematical theorems (not independenceresults), but it stresses the areas of mathematicswhere set-theoretic topics (such as cardinal arithmetic)are relevant.There is, in fact, an interplay between infi nitarycombinatorics and independence proofs. Infi nitarycombinatorics suggests many set-theoretic questionsthat turn out to be independent of ZFC, but it alsoprovides the basic tools used in forcing arguments. Inparticular, Martin''s Axiom, which is one of the topicsunder infi nitary combinatorics, introduces many of thebasic ingredients of forcing.

Read more less

Book SynopsisThis comprehensive two-volume work is devoted to the most general beginnings of mathematics. It goes back to Hausdorff’s classic Set Theory (2nd ed., 1927), where set theory and the theory of functions were expounded as the fundamental parts of mathematics in such a way that there was no need for references to other sources. Along the lines of Hausdorff’s initial work (1st ed., 1914), measure and integration theory is also included here as the third fundamental part of contemporary mathematics.The material about sets and numbers is placed in Volume 1 and the material about functions and measures is placed in Volume 2. 　 Contents Fundamentals of the theory of classes, sets, and numbers Characterization of all natural models of Neumann – Bernays – Godel and Zermelo – Fraenkel set theories Local theory of sets as a foundation for category theory and its connection with the Zermelo – Fraenkel set theory Compactness theorem for generalized second-order language

Read more less

Book SynopsisThis comprehensive two-volume work is devoted to the most general beginnings of mathematics. It goes back to Hausdorff’s classic Set Theory (2nd ed., 1927), where set theory and the theory of functions were expounded as the fundamental parts of mathematics in such a way that there was no need for references to other sources. Along the lines of Hausdorff’s initial work (1st ed., 1914), measure and integration theory is also included here as the third fundamental part of contemporary mathematics. The material about sets and numbers is placed in Volume 1 and the material about functions and measures is placed in Volume 2. Contents Historical foreword on the centenary after Felix Hausdorff’s classic Set Theory Fundamentals of the theory of functions Fundamentals of the measure theory Historical notes on the Riesz – Radon – Frechet problem of characterization of Radon integrals as linear functionals

Read more less

Book SynopsisThis book presents articles of L.V. Kantorovich on the descriptive theory of sets and function and on functional analysis in semi-ordered spaces, to demonstrate the unity of L.V. Kantorovich's creative research. It also includes two papers on the extension of Hilbert space.

Read more less

Book SynopsisHave you ever played the addictive card game SET? Have you ever wondered about the connections between games and mathematics? If the answer to either question is yes, then The Joy of SET is the book for you! The Joy of SET takes readers on a fascinating journey into this seemingly simple card game and reveals its surprisingly deep and diverse matTrade Review"[A] model of mathematical exposition. The quality of writing is consistently high: clear but not condescending, humorous, chatty, and a genuine pleasure to read... I doubt it will be very long before I find something from [The Joy of SET] to use in one of my classes."--Mark Hunacek, MAA Reviews "[The Joy of SET] shows how budding interest in mathematics can be fostered and developed... If [middle and high school teachers] ever try to enliven their classes or just interaction with curious students, this book is one they may depend on."--Alexander Bogomolny, Cut the Knot blog "[A]mazing... What I love about The Joy of SET is that it is written in such a way that it can be read and enjoyed by both SET enthusiasts and someone that has never played SET before... Really and truly, there is enough math in this book to keep you busy for a lifetime! ... I definitely recommend this book."--Sarah Carter, Math Equals Love blog "[The Joy of SET] takes readers on a fascinating journey into this seemingly simple card game... The book is in my view just the right way to talk about math as fun, and intellectually challenging."--Robert Harington, Scholarly Kitchen "[A]s the authors convincingly demonstrate ... the mathematics behind SET actually goes very deep... [The Joy of SET] would make a fantastic resource for a middle school, high school, or undergraduate math club."--Brent Yorgey, Math Less Traveled blog "This book, written by a mathematically inclined family, is the first and only work to explore the connection between the game and mathematics... [The Joy of SET] will attract those who play SET and those who want to explore mathematically related subjects."--ChoiceTable of ContentsPreface vii 1 SET and You 1 2 Counting Fun! 27 3 Probability! 52 4 SET and Modular Arithmetic 72 5 SET and Geometry 98 Interlude: How to Improve at SET 136 6 More Combinatorics 149 7 Probability and Statistics 171 8 Vectors and Linear Algebra 197 9 Affine Geometry Plus 229 10 Computing and Simulations 256 Conclusion 288 Solutions to Exercises 289 Bibliography 303 Index 307

Read more less

Book SynopsisProvides an introduction to algebraic cycles on complex algebraic varieties, to the major conjectures relating them to cohomology, and even more precisely to Hodge structures on cohomology. This title delves into arguments originating in Nori's work that have been further developed by others.Trade Review"This dense, fascinating book by Voisin is a report of some of the exciting discoveries she has made in the quest of the secrets of algebraic cycles."--Alberto Collino, Zentralblatt MATH "[An advanced] reader will find a rich collection of ideas as well as detailed machinery with which to attack difficult problems in the field. Any complex geometer interested in the interplay between algebraic cycles, Hodge theory and algebraic topology should have this book on his or her shelf."--C. A. M. Peters, Mathematical Reviews ClippingsTable of ContentsPreface vii 1Introduction 1 1.1 Decomposition of the diagonal and spread 3 1.2 The generalized Bloch conjecture 7 1.3 Decomposition of the small diagonal and application to the topology of families 9 1.4 Integral coefficients and birational invariants 11 1.5 Organization of the text 13 2Review of Hodge theory and algebraic cycles 15 2.1 Chow groups 15 2.2 Hodge structures 24 3Decomposition of the diagonal 36 3.1 A general principle 36 3.2 Varieties with small Chow groups 44 4Chow groups of large coniveau complete intersections 55 4.1 Hodge coniveau of complete intersections 55 4.2 Coniveau 2 complete intersections 64 4.3 Equivalence of generalized Bloch and Hodge conjectures for general complete intersections 67 4.4 Further applications to the Bloch conjecture on 0-cycles on surfaces 86 5On the Chow ring of K3 surfaces and hyper-Kahler manifolds 88 5.1 Tautological ring of a K3 surface 88 5.2 A decomposition of the small diagonal 96 5.3 Deligne's decomposition theorem for families of K3 surfaces 106 6Integral coefficients 123 6.1 Integral Hodge classes and birational invariants 123 6.2 Rationally connected varieties and the rationality problem 127 6.3 Integral decomposition of the diagonal and the structure of the Abel-Jacobi map 139 Bibliography 155 Index 163

Read more less

a huge range and FREE tracked UK delivery on ALL orders.

Read more less

Book SynopsisWilliam Watkin is Professor of Contemporary Literature and Philosophy at Brunel University, UK.Trade ReviewA remarkable achievement and a distinctive contribution to our knowledge of Badiou’s thought. For all the skill, and sometimes brilliance, of previous explications, no-one had quite captured the tenor, the character or (a heretical word, perhaps) the feel of Badiou’s ontology; not, at least, in language. * Journal of Badiou Studies *Watkin’s is a very important book. Much of the commentary on Badiou has tended to move his politics to the centre of his thought. With meticulousness, clarity and rigour, Watkin works rather through Badiou’s philosophization of mathematics, liberating the philosopher qua philosopher, the thinker whose awesome achievement has been to recast and transform our understanding of a major set of traditional philosophical terms. This book returns us to what is most gripping about Badiou, his stark, courageous and deeply uncontemporary asceticism. -- Andrew Gibson, former Research Professor of Modern Literature and Theory, Royal Holloway, University of London, UKTable of Contentspreface acknowledgements Introduction: Subtractive Being; Nonrelationality; Indifference; Set Theory; Retroactive Axiomatic Reasoning; Transmissibility, Intelligibility and Communicability; Theory of the Subject Chapter One: Being: The One and the Multiple How to Prove that the One Is-Not (Meditation One); The One as Operational Counts-as-One; The Ancient Problem of Classes; Situations and Structures; The Multiple; Presentation of Presentation; Reasoning on Being by Means of Axioms; How to Accept that Being Is-Not Chapter Two: Being: Separation, Void, Mark Meditation Two; Set Theory and Aggregation as Collection (Meditation Three); Axiom of Separation; Notation and Self-Predication; The Pure Multiple is Real; The Void: Proper Name of Being (Meditation Four); The Void and Nothing; Void as Nomination; ZF+C: The Nine Axioms of Contemporary Set Theory (Meditation 5); Axiom of Extensionality; Axiom of Replacement or Substitution; The Void Set and In-Difference; Meditation Six: Aristotle; Conclusion: Pure Multiple and the Void Chapter Three: Being and Excess Powerset Axiom (Meditation Seven); Point of Excess; Void as Name; Four Kinds of One-ness: One, Count-as-one, Unicity, forming-into-one; The State (Meditation Eight); Threat of the Void; Belonging, Inclusion and Parts; Typologies of Being; States and Indifference (Meditation Nine) Chapter Four: Nature and Infinity Nature is Normal (Meditation Eleven); Transitive Sets: Cardinal and Ordinal (Meditation Twelve); Nature and Minimality; Nature and Intrication; The Inexistence of Nature; Potential and Actual Infinity; Proving the Actual Infinite; Doubling and Dedekind Infinites; Frege and Equinumerosity Chapter Five: Infinity, Limit and Succession The Limit; Succession and Limit; The Upper or Maximal Limit; Succession; Infinite Thought: Problems of Procedure (Meditation Thirteen); In-Different Other: The Second Existential Seal; There is some infinity in natural multiples (Meditation Fourteen); Conclusion on Being Chapter Six: The Event: History and Ultra-One Historical Singularities (Meditation Sixteen); Historical Singularities, and Evental Sites: Examples; Primal Ones and the Edge of the Void; Singularity vs. Normality; Self-Predication: The Matheme of the Event (Meditation Seventeen); The Problem of Naming; Axiom of Foundation (Meditation Eighteen); Implications of Foundation; Coda: Un-Relation Chapter Seven: The Event, Intervention and Fidelity The Wager: yes or no (Meditation Twenty); Intervention; Seven Consequences of the Event; Axiom of Choice (Meditation Twenty Two); Choice is Indifferent; Due to Choice, Singularities Exist and they are Indifferent; Fidelity, Connection (Meditation Twenty Three) Chapter Eight: The Generic Continuum Hypothesis: (Meditation Twenty-Seven); The Thought of the Generic (Meditation Thirty-One); Truth and Knowledge: The Indifference of Avoidance; Generic Procedure; The Matheme of the Indiscernible (Meditation Thirty-Three); Easton’s Theorem (Meditation Twenty-Six); Conditioning the Indiscernible; Indiscernible or Generic Subsets; The Existence of the Indiscernible (Meditation Thirty-Four); Extension; Is there a name for the discernible such that it can be said to exist? Chapter Nine: Forcing: Truth and Subject Theory of the Subject (Meditation Thirty-Five); Subjectivization; Chance; Faith; Names; Forcing (Mediation Thirty-Six); Leibniz’s Identity of Indiscernibles (Meditation Thirty); The Proof of Forcing (Meditation Thirty-Six); From the Indiscernible to the Undecidable; Conclusion (Meditation Thirty-Seven); bibliography index

Read more less

Book SynopsisInformation granules, as encountered in natural language, are implicit in nature. To make them fully operational so they can be effectively used to analyze and design intelligent systems, information granules need to be made explicit. An emerging discipline, granular computing focuses on formalizing information granules and unifying them to create a coherent methodological and developmental environment for intelligent system design and analysis. Granular Computing: Analysis and Design of Intelligent Systems presents the unified principles of granular computing along with its comprehensive algorithmic framework and design practices. Introduces the concepts of information granules, information granularity, and granular computing Presents the key formalisms of information granules Builds on the concepts of information granules with discussion of higher-order and higher-type information granules Discusses the operational concept of inTrade Review"Dr. Pedrycz is an internationally acclaimed authority in the granular computing area. ... I particularly appreciate his elegant writing style. This book is the first comprehensive treatise of the granular computing techniques and their application to the design of intelligent systems. ... As an application-oriented practitioner in computational intelligence systems, I think that this book will be a welcome and strongly needed addition to this field. I cannot think of any other expert worldwide more qualified than Prof. Pedycz to write such a book."—Emil M. Petriu, University of Ottawa, Canada "This volume covers most of the interesting and important topics in granular computing. The contents may be well understood by senior or master course students in the field of computer science ... also a good textbook for engineers who are involved in developing so-called intelligent systems."—Kaoru Hirota, Tokyo Institute of Technology, Japan "Dr. Pedrycz’s latest magnum opus ... breaks new ground in many directions. [It] takes an important step toward achievement of human-level machine intelligence—a principal goal of artificial intelligence (AI) since its inception. ... [This is] a remarkably well put together and reader-friendly collection of concepts and techniques, which constitute granular computing. ... [The book] combines extraordinary breadth with extraordinary depth. It contains a wealth of new ideas, and unfolds a vast panorama of concepts, methods, and applications. ... Dr. Pedrycz’s development and description of these concepts, techniques, and their applications is a truly remarkable achievement. ... must reading for all who are concerned with the design and application of intelligent systems."—From the Foreword by Lotfi A. Zadeh, University of California, Berkeley, USA Table of ContentsInformation Granularity, Information Granules, and Granular Computing. Key Formalisms for Representation of Information Granules and Processing Mechanisms. Information Granules of Higher Type and Higher Order, and Hybrid Information Granules. Representation of Information Granules. The Design of Information Granules. Optimal Allocation of Information Granularity: Building Granular Mappings. Granular Description of Data and Pattern Classification. Granular Models: Architectures and Development. Granular Time Series. From Models to Granular Models. Collaborative and Linguistic Models of Decision Making. Index.

Read more less

Book SynopsisSet Theoretical Aspects of Real Analysis is built around a number of questions in real analysis and classical measure theory, which are of a set theoretic flavor. Accessible to graduate students, and researchers the beginning of the book presents introductory topics on real analysis and Lebesgue measure theory. These topics highlight the boundary between fundamental concepts of measurability and nonmeasurability for point sets and functions. The remainder of the book deals with more specialized material on set theoretical real analysis. The book focuses on certain logical and set theoretical aspects of real analysis. It is expected that the first eleven chapters can be used in a course on Lebesque measure theory that highlights the fundamental concepts of measurability and non-measurability for point sets and functions. Provided in the book are problems of varying difficulty that range from simple observations to advanced results. Relatively difficult Table of ContentsZF theory and some point sets on the real line. Countable versions of AC and real analysis. Uncountable versions of AC and Lebesgue nonmeasurable sets. The Continuum Hypothesis and Lebesgue nonmeasurable sets. Measurability properties of sets and functions. Radon measures and nonmeasurable sets. Real-valued step functions with strange measurability properties. Relationships between certain classical constructions of Lebesgue nonmeasurable sets. Measurability properties of Vitali sets. A relationship between the measurability and continuity of real-valued functions. A relationship between absolutely nonmeasurable functions and Sierpinski-Zygmund functions. Sums of absolutely nonmeasurable injective functions. A large group of absolutely nonmeasurable additive functions. Additive properties of certain classes of pathological functions. Absolutely nonmeasurable homomorphisms of commutative groups. Measurable and nonmeasurable sets with homogeneous sections. A combinatorial problem on translation invariant extensions of the Lebesgue measure. Countable almost invariant partitions of G-spaces. Nonmeasurable unions of measure zero sections of plane sets. Measurability properties of well-orderings. Appendices. Bibliography. Subject Index.

Read more less

Book SynopsisA Bridge to Higher Mathematics is more than simply another book to aid the transition to advanced mathematics. The authors intend to assist students in developing a deeper understanding of mathematics and mathematical thought. The only way to understand mathematics is by doing mathematics. The reader will learn the language of axioms and theorems and will write convincing and cogent proofs using quantifiers. Students will solve many puzzles and encounter some mysteries and challenging problems. The emphasis is on proof. To progress towards mathematical maturity, it is necessary to be trained in two aspects: the ability to read and understand a proof and the ability to write a proof. The journey begins with elements of logic and techniques of proof, then with elementary set theory, relations and functions. Peano axioms for positive integers and for natural numbers follow, in particular mathematical and other forms of induction. Next Trade ReviewThis is one of the shorter books for a course that introduces students to the concept of mathematical proofs. The brevity is due to the "bare-bones" nature of the treatment. The number of topics covered, the number of examples, and the number of exercises are not smaller than what appears in competing textbooks; what is shorter is the text that one finds between theorems, lemmas, examples, and exercises. Besides the topics found in similar textbooks (i.e., proof techniques, logic, set theory, relations, and functions), there are chapters on (very) elementary number theory, combinatorial counting techniques, and Peano axioms on the set of positive integers. Several chapters are devoted to the construction of various kinds of numbers, such as integers, rationals, real numbers, and complex numbers. Answers to around half the exercises are included at the end of the book, and a few have complete solutions. This reviewer finds the book more enjoyable than the average competing textbook. --M. Bona, University of FloridaTable of ContentsElements of logicTrue and false statementsLogical connectives and truth tablesLogical equivalenceQuantifiersProofs: Structures and strategiesAxioms, theorems and proofsDirect proofContrapositive proofProof by equivalent statementsProof by casesExistence proofsProof by counterexampleProof by mathematical inductionElementary Theory of Sets. FunctionsAxioms for set theoryInclusion of setsUnion and intersection of setsComplement, difference and symmetric difference of setsOrdered pairs and the Cartersian productFunctionsDefinition and examples of functionsDirect image, inverse imageRestriction and extension of a functionOne-to-one and onto functionsComposition and inverse functions*Family of sets and the axiom of choiceRelationsGeneral relations and operations with relationsEquivalence relations and equivalence classesOrder relations*More on ordered sets and Zorn's lemmaAxiomatic theory of positive integersPeano axioms and additionThe natural order relation and subtractionMultiplication and divisibilityNatural numbersOther forms of inductionElementary number theoryAboslute value and divisibility of integersGreatest common divisor and least common multipleIntegers in base 10 and divisibility testsCardinality. Finite sets, infinite setsEquipotent setsFinite and infinite setsCountable and uncountable setsCounting techniques and combinatoricsCounting principlesPigeonhole principle and parityPermutations and combinationsRecursive sequences and recurrence relationsThe construction of integers and rationals Definition of integers and operationsOrder relation on integersDefinition of rationals, operations and orderDecimal representation of rational numbersThe construction of real and complex numbersThe Dedekind cuts approachThe Cauchy sequences approachDecimal representation of real numbersAlgebraic and transcendental numbersComples numbersThe trigonometric form of a complex number

Read more less

Book SynopsisThis book serves as a textbook in real analysis. It focuses on the fundamentals of the structural properties of metric spaces and analytical properties of functions defined between such spaces. Topics include sets, functions and cardinality, real numbers, analysis on R, topology of the real line, metric spaces, continuity and differentiability, sequences and series, Lebesgue integration, and Fourier series. It is primarily focused on the applications of analytical methods to solving partial differential equations rooted in many important problems in mathematics, physics, engineering, and related fields. Both the presentation and treatment of topics are fashioned to meet the expectations of interested readers working in any branch of science and technology. Senior undergraduates in mathematics and engineering are the targeted student readership, and the topical focus with applications to real-world examples will promote higher-level mathematical understanding for undergraduates in sciences and engineering.Table of Contents1. Sets, Functions and Cardinality.- 2. The Real Numbers.- 3. Sequence and Series of Numbers.- 4. Analysis on R.- 5. Topology of the Real Line.- 6. Metric Spaces.- 7. Continuity and Differentiability.- 8. Sequences and Series of Functions.- 9. Lebesgue Integration.- 10. Fourier Series.

Read more less

a huge range and FREE tracked UK delivery on ALL orders.

Read more less

a huge range and FREE tracked UK delivery on ALL orders.

Read more less

a huge range and FREE tracked UK delivery on ALL orders.

Read more less

a huge range and FREE tracked UK delivery on ALL orders.

Read more less

a huge range and FREE tracked UK delivery on ALL orders.

Read more less

a huge range and FREE tracked UK delivery on ALL orders.

Read more less

Book SynopsisIrrationality and Transcendence in Number Theory tells the story of irrational numbers from their discovery in the days of Pythagoras to the ideas behind the work of Baker and Mahler on transcendence in the 20th century. It focuses on themes of irrationality, algebraic and transcendental numbers, continued fractions, approximation of real numbers by rationals, and relations between automata and transcendence. This book serves as a guide and introduction to number theory for advanced undergraduates and early postgraduates. Readers are led through the developments in number theory from ancient to modern times. The book includes a wide range of exercises, from routine problems to surprising and thought-provoking extension material.Features Uses techniques from widely diverse areas of mathematics, including number theory, calculus, set theory, complex analysis, linear algebra, and the theory of computation Suitable as a primary textbook forTrade Review"Exceptionally informative, impressively organized and presented, Irrationality and Transcendence in Number Theory is an ideal selection as a curriculum textbook."- Midwest Books Review"This excellent book not only helps fill a substantial gap in the undergraduate mathematics literature, but it does so in a way that most students will, I think, find interesting, inviting and accessible. [. . .] This material is, of course, very nontrivial, but Angell goes to great lengths to make it accessible. He writes slowly and clearly and spends a lot of time motivating results. As previously noted, he also includes background Appendices in each chapter.There are other useful pedagogical features. Each chapter ends with an extensive collection of exercises, most of them non-routine; a 20-page section at the end of the book offers hints to these. The book also contains a five-page bibliography (one that, surprisingly, omits the Burger/Tubbs book mentioned earlier) that directs a reader to useful sources. The subject matter of this book is interesting and beautiful and deserves to be made accessible to well-prepared senior undergraduates. Angell has done an excellent job in helping to do so."- MAA Reviews Table of Contents1. Introduction. 1.1. Irrational Surds. 1.2. Irrational Decimals. 1.3. Irrationality of the Exponential Constant. 1.4. Other Results, and Some Open Questions. Exercises. Appendix: Some Elementary Number Theory. 2. Hermite’s Method. 2.1. Irrationality of er. 2.2. Irrationality of π. 2.3. Irrational values of trigonometric functions. Exercises. Appendix: Some Results of Elementary Calculus. 3. Algebraic & Transcendental Numbers. 3.1. Definitions and Basic Properties. 3.2. Existence of Transcendental Numbers. 3.3. Approximation of Real Numbers by Rationals. 3.4. Irrationality of (3) : a sketch. Exercises. Appendix 1: Countable and Uncountable Sets. Appendix 2: The Mean Value Theorem. Appendix 3: The Prime Number Theorem. 4. Continued Fractions. Definition and Basic Properties. 4.2. Continued Fractions of Irrational Numbers. 4.3. Approximation Properties of Convergents. 4.4. Two important Approximation Problems. 4.5. A "Computational" Test for Rationality. 4.6. Further Approximation Properties of Convergents. 4.7. Computing the Continued Fraction of an Algebraic Irrational. 4.8. The Continued Fraction of e. Exercises. Appendix 1: A Property of Positive Fractions. Appendix 2: Simultaneous Equations with Integral Coefficients. Appendix 3: Cardinality of Sets of Sequences. Appendix 4: Basic Musical Terminology. 5. Hermite’s Method for Transcendence. 5.1. Transcendence of e. 5.2. Transcendence of π. 5.3. Some more Irrationality Proofs. 5.4. Transcendence of ea .5.5. Other Results. Exercises. Appendix 1: Roots and Coefficients of Polynomials. Appendix 2: Some Real and Complex Analysis. Appendix 3: Ordering Complex Numbers. 6. Automata and Transcendence. 6.1. Deterministic Finite Automata. 6.2 Mahler’s Transcendence Proof. 6.3 A More General Transcendence Result. 6.4. A Transcendence Proof for the Thue Sequence. 6.5. Automata and Functional Equations. 6.6. Conclusion. Exercises. Appendix 1: Alphabets, Languages and DFAs. Appendix 2: Some Results of Complex Analysis. Appendix 3: A Result on Linear Equations. 7. Lambert’s Irrationality Proofs. 7.1. Generalised Continued Fractions. 7.2. Further Continued Fractions. Exercises. Appendix: Some Results from Elementary Algebra and Calculus. Hints for Exercises. Bibliography. Index.

Read more less

a huge range and FREE tracked UK delivery on ALL orders.

Read more less

Book SynopsisDesigned for undergraduate students of set theory, this book presents a modern perspective of the classic work of Georg Cantor and Richard Dedekin and their immediate successors. It aims to give students a grounding to the results of set theory as well as to tackle significant problems that arise from the theory.Table of ContentsINTRODUCTIONOutline of the bookAssumed knowledgeTHE REAL NUMBERSIntroductionDedekind's constructionAlternative constructionsThe rational numbersTHE NATURAL NUMBERSIntroductionThe construction of the natural numbersArithmeticFinite setsTHE ZERMELO-FRAENKEL AXIOMSIntroductionA formal languageAxioms 1 to 3Axioms 4 to 6Axioms 7 to 9CARDINAL (Without the Axiom of Choice)IntroductionComparing SizesBasic properties of ˜ and =Infinite sets without AC-countable setsUncountable sets and cardinal arithmetic without ACORDERED SETSIntroductionLinearly ordered setsOrder arithmeticWell-ordered setsORDINAL NUMBERSIntroductionOrdinal numbersBeginning ordinal arithmeticOrdinal arithmeticThe ÀsSET THEORY WITH THE AXIOM OF CHOICEIntroductionThe well-ordering principleCardinal arithmetic and the axiom of choiceThe continuum hypothesisBIBLIOGRAPHYINDEX

Read more less

Book SynopsisThoroughly revised, updated, expanded, and reorganized to serve as a primary text for mathematics courses, Introduction to Set Theory, Third Edition covers the basics: relations, functions, orderings, finite, countable, and uncountable sets, and cardinal and ordinal numbers. It also provides five additional self-contained chapters, consolidates the material on real numbers into a single updated chapter affording flexibility in course design, supplies end-of-section problems, with hints, of varying degrees of difficulty, includes new material on normal forms and Goodstein sequences, and adds important recent ideas including filters, ultrafilters, closed unbounded and stationary sets, and partitions.Trade Review"The book remains an excellent text for a senior undergraduate or first-year graduate level course. There is sufficient material for instructors of widely differing views to assemble one-semester courses. . ..the chapter on the axiom of choice is particularly strong. "---Mathematical Reviews ". . .a fine text. . ..The proofs are both elegant and readable. "---American Mathematical Monthly ". . .offers many benefits including. . .interesting applications of abstract set theory to real analysis. . .enriching standard classroom material. "---L'Enseignement mathematique ". . .an excellent and much needed book. . .Especially valuable are a number of remarks sprinkled throughout the text which afford a glimpse of further developments. "---The Mathematical Intelligencer "The authors show that set theory is powerful enough to serve as an underlying framework for mathematics by using it to develop the beginnings of the theory of natural, rational, and real numbers. "---Quarterly Review of Applied Mathematics ". . .In the third edition, Chapter 11 has been expanded, and four new chapters have been added. "---Mathematical ReviewsTable of ContentsSets; relations, functions and orderings; natural numbers; finite, countable and uncountable sets; cardinal numbers; ordinal numbers; alephs; the axiom of choice; arithmetic of cardinal numbers; sets of real numbers; filters and ultrafilters; combinatorial set theory; large cardinals; the axiom of foundation; the axiomatic set theory.

Read more less

Book SynopsisThe Language of Symmetry is a re-assessment of the structure and reach of symmetry, by an interdisciplinary group of specialists from the arts, humanities, and sciences at Oxford University.It explores, amongst other topics: order and chaos in the formation of planetary systems entropy and symmetry in physics group theory, fractals, and self-similarity symmetrical structures in western classical music how biological systems harness disorder to create order This book aims to open up the scope of interdisciplinary work in the study of symmetry and is intended for scholars of any background - whether it be science, arts, or philosophy.Table of Contents1. Planetary Systems: From Symmetry to Chaos. 2. Entropy and Symmetry in the Universe. 3. Darkness, Light, and how Symmetry might relate Them. Self-Similar 4. Self-Similarity. 5 The Language of Symmetry in Music. 6. The Interdependence of Order and Disorder: How Complexity arises in the Living and the Inanimate Universe. 7. A Philosophers Perspective on the Harnessing of Stochasticity. 8. Postscript: A Dialogue between Denis Noble and Benedict Rattigan. 9. Appendix: A Response to Professor Nobles Paper: Ordered disorder to drive Physiology.

Read more less

Book SynopsisA Beginnerâs Guide to Mathematical Proof prepares mathematics majors for the transition to abstract mathematics, as well as introducing a wider readership of quantitative science students, such as engineers, to the mathematical structures underlying more applied topics.The text is designed to be easily utilized by both instructor and student, with an accessible, step-by-step approach requiring minimal mathematical prerequisites. The book builds towards more complex ideas as it progresses, but never makes assumptions of the reader beyond the material already covered.Features No mathematical prerequisites beyond high school mathematics Suitable for an Introduction to Proofs course for mathematics majors and other students of quantitative sciences, such as engineering Replete with exercises and examples.

Read more less

Book SynopsisContemporary students of mathematics differ considerably from those of half a century ago. In spite of this, many textbooks written and now considered to be âœclassicsâ decades ago are still prescribed for students today. These texts are not suitable for todayâs students. This text is meant for and written to todayâs mathematics students. Set theory is a pure mathematics endeavour in the sense that it seems to have no immediate applications; yet the knowledge and skills developed in such a course can easily branch out to various fields of both pure mathematics and applied mathematics.Rather than transforming the reader into a practicing mathematician this book is more designed to initiate the reader to what may be called âœmathematical thinkingâ while developing knowledge about foundations of modern mathematics. Without this insight, becoming a practicing mathematician is much more daunting.The main objective is twofold. The students will develop some fun

Read more less

Book SynopsisDiscrete Mathematics: An Open Introduction, Fourth Edition aims to provide an introduction to select topics in discrete mathematics at a level appropriate for first or second year undergraduate math and computer science majors, especially those who intend to teach middle and high school mathematics. The book began as a set of notes for the Discrete Mathematics course at the University of Northern Colorado. This course serves both as a survey of the topics in discrete math and as the âœbridgeâ course for math majors. Features Uses problem-oriented and inquiry-based methods to teach the concepts. Suitable for undergraduates in mathematics and computer science. New to the 4th edition Large scale restructuring. Contains more than 750 exercises and examples. New sections on probability, relations, and discrete structures and their proofs.

Read more less