Classic Set Theory

For Guided Independent Study

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Author: D.C. Goldrei

Publisher: CRC Press

ISBN: 9780412606106

Category: Mathematics

Page: 296

View: 5278

Designed for undergraduate students of set theory, Classic Set Theory presents a modern perspective of the classic work of Georg Cantor and Richard Dedekin and their immediate successors. This includes: The definition of the real numbers in terms of rational numbers and ultimately in terms of natural numbers Defining natural numbers in terms of sets The potential paradoxes in set theory The Zermelo-Fraenkel axioms for set theory The axiom of choice The arithmetic of ordered sets Cantor's two sorts of transfinite number - cardinals and ordinals - and the arithmetic of these. The book is designed for students studying on their own, without access to lecturers and other reading, along the lines of the internationally renowned courses produced by the Open University. There are thus a large number of exercises within the main body of the text designed to help students engage with the subject, many of which have full teaching solutions. In addition, there are a number of exercises without answers so students studying under the guidance of a tutor may be assessed. Classic Set Theory gives students sufficient grounding in a rigorous approach to the revolutionary results of set theory as well as pleasure in being able to tackle significant problems that arise from the theory.

Introduction to the Theory of Sets

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Author: Joseph Breuer

Publisher: Courier Corporation

ISBN: 0486154874

Category: Mathematics

Page: 128

View: 9507

This undergraduate text develops its subject through observations of the physical world, covering finite sets, cardinal numbers, infinite cardinals, and ordinals. Includes exercises with answers. 1958 edition.

Propositional and Predicate Calculus: A Model of Argument

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Author: Derek Goldrei

Publisher: Springer Science & Business Media

ISBN: 9781846282294

Category: Mathematics

Page: 315

View: 7246

Designed specifically for guided independent study. Features a wealth of worked examples and exercises, many with full teaching solutions, that encourage active participation in the development of the material. It focuses on core material and provides a solid foundation for further study.

Discovering Modern Set Theory: The basics

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Author: Winfried Just,Martin Weese

Publisher: American Mathematical Soc.

ISBN: 0821802666

Category: Mathematics

Page: 210

View: 9119

This book is an introduction to set theory for beginning graduate students who want to get a sound grounding in those aspects of set theory used extensively throughout other areas of mathematics. Topics covered include formal languages and models, the power and limitation of the Axiomatic Method, the Axiom of Choice, including the fascinating Banach-Tarski Paradox, applications of Zorn's Lemma, ordinal arithmetic, including transfinite induction, and cardinal arithmetic. The style of writing, more a dialogue with the reader than that of the Master indoctrinating the pupil, makes this also very suitable for self-study.

Set Theory: An Introduction

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Author: Robert L. Vaught

Publisher: Springer Science & Business Media

ISBN: 9780817642563

Category: Mathematics

Page: 167

View: 6841

An excellent undergraduate text on set theory that could be used in courses taught in mathematics and philosophy departments. The intuitive development in the first chapters also makes the book suitable for self study.

Set Theory and Metric Spaces

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Author: Irving Kaplansky

Publisher: American Mathematical Soc.

ISBN: 0821826948

Category: Mathematics

Page: 140

View: 4299

This is a book that could profitably be read by many graduate students or by seniors in strong major programs ... has a number of good features. There are many informal comments scattered between the formal development of theorems and these are done in a light and pleasant style. ... There is a complete proof of the equivalence of the axiom of choice, Zorn's Lemma, and well-ordering, as well as a discussion of the use of these concepts. There is also an interesting discussion of the continuum problem ... The presentation of metric spaces before topological spaces ... should be welcomed by most students, since metric spaces are much closer to the ideas of Euclidean spaces with which they are already familiar. --Canadian Mathematical Bulletin Kaplansky has a well-deserved reputation for his expository talents. The selection of topics is excellent. -- Lance Small, UC San Diego This book is based on notes from a course on set theory and metric spaces taught by Edwin Spanier, and also incorporates with his permission numerous exercises from those notes. The volume includes an Appendix that helps bridge the gap between metric and topological spaces, a Selected Bibliography, and an Index.

A Tour Through Mathematical Logic

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Author: Robert S. Wolf

Publisher: MAA

ISBN: 9780883850367

Category: Mathematics

Page: 397

View: 1314

A guide that any interested reader with some post-calculus experience in mathematics can read, enjoy, and learn from.

A Course on Set Theory

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Author: Ernest Schimmerling

Publisher: Cambridge University Press

ISBN: 1139501488

Category: Mathematics

Page: N.A

View: 8052

Set theory is the mathematics of infinity and part of the core curriculum for mathematics majors. This book blends theory and connections with other parts of mathematics so that readers can understand the place of set theory within the wider context. Beginning with the theoretical fundamentals, the author proceeds to illustrate applications to topology, analysis and combinatorics, as well as to pure set theory. Concepts such as Boolean algebras, trees, games, dense linear orderings, ideals, filters and club and stationary sets are also developed. Pitched specifically at undergraduate students, the approach is neither esoteric nor encyclopedic. The author, an experienced instructor, includes motivating examples and over 100 exercises designed for homework assignments, reviews and exams. It is appropriate for undergraduates as a course textbook or for self-study. Graduate students and researchers will also find it useful as a refresher or to solidify their understanding of basic set theory.

Introduction to Lattices and Order

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Author: B. A. Davey,H. A. Priestley

Publisher: Cambridge University Press

ISBN: 1107717523

Category: Mathematics

Page: 309

View: 4798

This new edition of Introduction to Lattices and Order presents a radical reorganization and updating, though its primary aim is unchanged. The explosive development of theoretical computer science in recent years has, in particular, influenced the book's evolution: a fresh treatment of fixpoints testifies to this and Galois connections now feature prominently. An early presentation of concept analysis gives both a concrete foundation for the subsequent theory of complete lattices and a glimpse of a methodology for data analysis that is of commercial value in social science. Classroom experience has led to numerous pedagogical improvements and many new exercises have been added. As before, exposure to elementary abstract algebra and the notation of set theory are the only prerequisites, making the book suitable for advanced undergraduates and beginning graduate students. It will also be a valuable resource for anyone who meets ordered structures.

Set Theory

A First Course

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Author: Daniel W. Cunningham

Publisher: Cambridge University Press

ISBN: 1107120322

Category: Mathematics

Page: 262

View: 4788

Set theory can be considered a unifying theory for mathematics. This book covers the fundamentals of the subject.

A Mathematical Introduction to Logic

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Author: Herbert Enderton,Herbert B. Enderton

Publisher: Elsevier

ISBN: 0080496466

Category: Mathematics

Page: 317

View: 2303

A Mathematical Introduction to Logic, Second Edition, offers increased flexibility with topic coverage, allowing for choice in how to utilize the textbook in a course. The author has made this edition more accessible to better meet the needs of today's undergraduate mathematics and philosophy students. It is intended for the reader who has not studied logic previously, but who has some experience in mathematical reasoning. Material is presented on computer science issues such as computational complexity and database queries, with additional coverage of introductory material such as sets. * Increased flexibility of the text, allowing instructors more choice in how they use the textbook in courses. * Reduced mathematical rigour to fit the needs of undergraduate students

Abstract Algebra: An Introduction

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Author: Thomas Hungerford

Publisher: Cengage Learning

ISBN: 1111569622

Category: Mathematics

Page: 616

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Abstract Algebra: An Introduction is set apart by its thematic development and organization. The chapters are organized around two themes: arithmetic and congruence. Each theme is developed first for the integers, then for polynomials, and finally for rings and groups. This enables students to see where many abstract concepts come from, why they are important, and how they relate to one another. New to this edition is a groups first option that enables those who prefer to cover groups before rings to do so easily. Important Notice: Media content referenced within the product description or the product text may not be available in the ebook version.

A Guided Tour of Mathematical Methods for the Physical Sciences

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Author: Roel Snieder,Kasper van Wijk,Matthew M. Haney

Publisher: Cambridge University Press

ISBN: 1107084962

Category: Mathematics

Page: 584

View: 6819

This completely revised edition provides a tour of the mathematical knowledge and techniques needed by students across the physical sciences. There are new chapters on probability and statistics and on inverse problems. It serves as a stand-alone text or as a source of exercises and examples to complement other textbooks.

Philosophical Introduction to Set Theory

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Author: Stephen Pollard

Publisher: Courier Dover Publications

ISBN: 0486797147

Category: Mathematics

Page: 192

View: 8046

This unique approach maintains that set theory is the primary mechanism for ideological and theoretical unification in modern mathematics, and its technically informed discussion covers a variety of philosophical issues. 1990 edition.

Security and Privacy Assurance in Advancing Technologies: New Developments

New Developments

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Author: Nemati, Hamid

Publisher: IGI Global

ISBN: 1609602021

Category: Computers

Page: 494

View: 1682

"This book provides a comprehensive collection of knowledge from experts within the field of information security and privacy and explores the changing roles of information technology and how this change will impact information security and privacy"--Provided by publisher.

Abstract Algebra

An Inquiry Based Approach

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Author: Jonathan K. Hodge,Steven Schlicker,Ted Sundstrom

Publisher: CRC Press

ISBN: 1466567082

Category: Mathematics

Page: 595

View: 8312

To learn and understand mathematics, students must engage in the process of doing mathematics. Emphasizing active learning, Abstract Algebra: An Inquiry-Based Approach not only teaches abstract algebra but also provides a deeper understanding of what mathematics is, how it is done, and how mathematicians think. The book can be used in both rings-first and groups-first abstract algebra courses. Numerous activities, examples, and exercises illustrate the definitions, theorems, and concepts. Through this engaging learning process, students discover new ideas and develop the necessary communication skills and rigor to understand and apply concepts from abstract algebra. In addition to the activities and exercises, each chapter includes a short discussion of the connections among topics in ring theory and group theory. These discussions help students see the relationships between the two main types of algebraic objects studied throughout the text. Encouraging students to do mathematics and be more than passive learners, this text shows students that the way mathematics is developed is often different than how it is presented; that definitions, theorems, and proofs do not simply appear fully formed in the minds of mathematicians; that mathematical ideas are highly interconnected; and that even in a field like abstract algebra, there is a considerable amount of intuition to be found.

Forcing for Mathematicians

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Author: Nik Weaver

Publisher: World Scientific

ISBN: 9814566020

Category: Mathematics

Page: 152

View: 3124

Ever since Paul Cohen's spectacular use of the forcing concept to prove the independence of the continuum hypothesis from the standard axioms of set theory, forcing has been seen by the general mathematical community as a subject of great intrinsic interest but one that is technically so forbidding that it is only accessible to specialists. In the past decade, a series of remarkable solutions to long-standing problems in C*-algebra using set-theoretic methods, many achieved by the author and his collaborators, have generated new interest in this subject. This is the first book aimed at explaining forcing to general mathematicians. It simultaneously makes the subject broadly accessible by explaining it in a clear, simple manner, and surveys advanced applications of set theory to mainstream topics. Contents:Peano ArithmeticZermelo–Fraenkel Set TheoryWell-Ordered SetsOrdinalsCardinalsRelativizationReflectionForcing NotionsGeneric ExtensionsForcing EqualityThe Fundamental TheoremForcing CHForcing ¬ CHFamilies of Entire Functions*Self-Homeomorphisms of βℕ \ ℕ, I*Pure States on B(H)*The Diamond PrincipleSuslin's Problem, I*Naimark's problem*A Stronger DiamondWhitehead's Problem, I*Iterated ForcingMartin's AxiomSuslin's Problem, II*Whitehead's Problem, II*The Open Coloring AxiomSelf-Homeomorphisms of βℕ \ ℕ, II*Automorphisms of the Calkin Algebra, I*Automorphisms of the Calkin Algebra, II*The Multiverse Interpretation Readership: Graduates and researchers in logic and set theory, general mathematical audience. Keywords:Forcing;Set Theory;Consistency;Independence;C*-AlgebraKey Features:A number of features combine to make this thorough and rigorous treatment of forcing surprisingly easy to follow. First, it goes straight into the core material on forcing, avoiding Godel constructibility altogether; second, key definitions are simplified, allowing for a less technical development; and third, further care is given to the treatment of metatheoretic issuesEach chapter is limited to four pages, making the presentation very readableA unique feature of the book is its emphasis on applications to problems outside of set theory. Much of this material is currently only available in the primary literatureThe author is a pioneer in the application of set-theoretic methods to C*-algebra, having solved (together with various co-authors) Dixmier's “prime versus primitive” problem, Naimark's problem, Anderson's conjecture about pure states on B(H), and the Calkin algebra outer automorphism problemReviews: “The author presents the basics of the theory of forcing in a clear and stringent way by emphasizing important technical details and simplifying some definitions and arguments. Moreover, he presents the content in a way that should help beginners to understand the central concepts and avoid common mistakes.” Zentralblatt MATH

Biophysics

Searching for Principles

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Author: William Bialek

Publisher: Princeton University Press

ISBN: 1400845572

Category: Science

Page: 640

View: 3026

Interactions between the fields of physics and biology reach back over a century, and some of the most significant developments in biology--from the discovery of DNA's structure to imaging of the human brain--have involved collaboration across this disciplinary boundary. For a new generation of physicists, the phenomena of life pose exciting challenges to physics itself, and biophysics has emerged as an important subfield of this discipline. Here, William Bialek provides the first graduate-level introduction to biophysics aimed at physics students. Bialek begins by exploring how photon counting in vision offers important lessons about the opportunities for quantitative, physics-style experiments on diverse biological phenomena. He draws from these lessons three general physical principles--the importance of noise, the need to understand the extraordinary performance of living systems without appealing to finely tuned parameters, and the critical role of the representation and flow of information in the business of life. Bialek then applies these principles to a broad range of phenomena, including the control of gene expression, perception and memory, protein folding, the mechanics of the inner ear, the dynamics of biochemical reactions, and pattern formation in developing embryos. Featuring numerous problems and exercises throughout, Biophysics emphasizes the unifying power of abstract physical principles to motivate new and novel experiments on biological systems. Covers a range of biological phenomena from the physicist's perspective Features 200 problems Draws on statistical mechanics, quantum mechanics, and related mathematical concepts Includes an annotated bibliography and detailed appendixes Instructor's manual (available only to teachers)

Geometry: Euclid and Beyond

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Author: Robin Hartshorne

Publisher: Springer Science & Business Media

ISBN: 0387226761

Category: Mathematics

Page: 528

View: 4129

This book offers a unique opportunity to understand the essence of one of the great thinkers of western civilization. A guided reading of Euclid's Elements leads to a critical discussion and rigorous modern treatment of Euclid's geometry and its more recent descendants, with complete proofs. Topics include the introduction of coordinates, the theory of area, history of the parallel postulate, the various non-Euclidean geometries, and the regular and semi-regular polyhedra.

Logic: The Basics

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Author: Jc Beall,Shay Allen Logan

Publisher: Taylor & Francis

ISBN: 1317528611

Category: Philosophy

Page: 288

View: 4018

Logic: The Basics is an accessible introduction to several core areas of logic. The first part of the book features a self-contained introduction to the standard topics in classical logic, such as: · mathematical preliminaries · propositional logic · quantified logic (first monadic, then polyadic) · English and standard ‘symbolic translations’ · tableau procedures. Alongside comprehensive coverage of the standard topics, this thoroughly revised second edition also introduces several philosophically important nonclassical logics, free logics, and modal logics, and gives the reader an idea of how they can take their knowledge further. With its wealth of exercises (solutions available in the encyclopedic online supplement), Logic: The Basics is a useful textbook for courses ranging from the introductory level to the early graduate level, and also as a reference for students and researchers in philosophical logic.