Basic Set Theory

Author: Nikolai Konstantinovich Vereshchagin,Alexander Shen

Publisher: American Mathematical Soc.

ISBN: 0821827316

Category: Mathematics

Page: 116

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The main notions of set theory (cardinals, ordinals, transfinite induction) are fundamental to all mathematicians, not only to those who specialize in mathematical logic or set-theoretic topology. Basic set theory is generally given a brief overview in courses on analysis, algebra, or topology, even though it is sufficiently important, interesting, and simple to merit its own dedicated treatment. This book provides just that in the form of a leisurely exposition for a diversified audience. It is suitable for a broad range of readers, from undergraduate students to professional mathematicians who want to finally find out what transfinite induction is and why it is always replaced by Zorn's Lemma. The text introduces all main subjects of ``naive'' (nonaxiomatic) set theory: functions, cardinalities, ordered and well-ordered sets, transfinite induction and its applications, ordinals, and operations on ordinals. Included are discussions and proofs of the Cantor-Bernstein Theorem, Cantor's diagonal method, Zorn's Lemma, Zermelo's Theorem, and Hamel bases. With over 150 problems, the book is a complete and accessible introduction to the subject.

The Discrete Math Workbook

A Companion Manual for Practical Study

Author: Sergei Kurgalin,Sergei Borzunov

Publisher: Springer

ISBN: 3319926454

Category: Computers

Page: 485

View: 3846

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This practically-oriented textbook presents an accessible introduction to discrete mathematics through a substantial collection of classroom-tested exercises. Each chapter opens with concise coverage of the theory underlying the topic, reviewing the basic concepts and establishing the terminology, as well as providing the key formulae and instructions on their use. This is then followed by a detailed account of the most common problems in the area, before the reader is invited to practice solving such problems for themselves through a varied series of questions and assignments. Topics and features: provides an extensive set of exercises and examples of varying levels of complexity, suitable for both laboratory practical training and self-study; offers detailed solutions to many problems, applying commonly-used methods and computational schemes; introduces the fundamentals of mathematical logic, the theory of algorithms, Boolean algebra, graph theory, sets, relations, functions, and combinatorics; presents more advanced material on the design and analysis of algorithms, including asymptotic analysis, and parallel algorithms; includes reference lists of trigonometric and finite summation formulae in an appendix, together with basic rules for differential and integral calculus. This hands-on study guide is designed to address the core needs of undergraduate students training in computer science, informatics, and electronic engineering, emphasizing the skills required to develop and implement an algorithm in a specific programming language.

Moscow Mathematical Olympiads, 1993-1999

Author: Roman Mikhaĭlovich Fedorov,Silvio Levy

Publisher: American Mathematical Soc.

ISBN: 0821853635

Category: Mathematics

Page: 220

View: 7809

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The Moscow Mathematical Olympiad has been challenging high school students with stimulating, original problems of different degrees of difficulty for over 75 years. The problems are nonstandard; solving them takes wit, thinking outside the box, and, sometimes, hours of contemplation. Some are within the reach of most mathematically competent high school students, while others are difficult even for a mathematics professor. Many mathematically inclined students have found that tackling these problems, or even just reading their solutions, is a great way to develop mathematical insight. In 2006 the Moscow Center for Continuous Mathematical Education began publishing a collection of problems from the Moscow Mathematical Olympiads, providing for each an answer (and sometimes a hint) as well as one or more detailed solutions. This volume represents the years 1993-1999. The problems and the accompanying material are well suited for math circles. They are also appropriate for problem-solving classes and practice for regional and national mathematics competitions. In the interest of fostering a greater awareness and appreciation of mathematics and its connections to other disciplines and everyday life, MSRI and the AMS are publishing books in the Mathematical Circles Library series as a service to young people, their parents and teachers, and the mathematics profession. Titles in this series are co-published with the Mathematical Sciences Research Institute (MSRI).

Moscow Mathematical Olympiads, 2000-2005

Author: Roman Vasilʹevich Fedorov,Silvio Levy,Alexander Kovaldzhi,Ivan Yashchenko

Publisher: American Mathematical Soc.

ISBN: 082186906X

Category: Mathematics

Page: 176

View: 3408

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The Moscow Mathematical Olympiad has been challenging high school students with stimulating, original problems of different degrees of difficulty for over 75 years. The problems are nonstandard; solving them takes wit, thinking outside the box, and, sometimes, hours of contemplation. Some are within the reach of most mathematically competent high school students, while others are difficult even for a mathematics professor. Many mathematically inclined students have found that tackling these problems, or even just reading their solutions, is a great way to develop mathematical insight. In 2006 the Moscow Center for Continuous Mathematical Education began publishing a collection of problems from the Moscow Mathematical Olympiads, providing for each an answer (and sometimes a hint) as well as one or more detailed solutions. This volume represents the years 2000-2005. The problems and the accompanying material are well suited for math circles. They are also appropriate for problem-solving classes and practice for regional and national mathematics competitions. In the interest of fostering a greater awareness and appreciation of mathematics and its connections to other disciplines and everyday life, MSRI and the AMS are publishing books in the Mathematical Circles Library series as a service to young people, their parents and teachers, and the mathematics profession. Titles in this series are co-published with the Mathematical Sciences Research Institute (MSRI).

Infinite Dimensional Analysis

A Hitchhiker's Guide

Author: Charalambos D. Aliprantis,Kim Border

Publisher: Springer Science & Business Media

ISBN: 3540295879

Category: Business & Economics

Page: 704

View: 6385

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What you’ll find in this monograph is nothing less than a complete and rigorous study of modern functional analysis. It is intended for the student or researcher who could benefit from functional analytic methods, but who does not have an extensive background in the subject and does not plan to make a career as a functional analyst. It develops the topological structures in connection with a number of topic areas such as measure theory, convexity, and Banach lattices, as well as covering the analytic approach to Markov processes. Many of the results were previously available only in works scattered throughout the literature.

Naive Set Theory

Author: P. R. Halmos

Publisher: Springer Science & Business Media

ISBN: 1475716451

Category: Mathematics

Page: 104

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Every mathematician agrees that every mathematician must know some set theory; the disagreement begins in trying to decide how much is some. This book contains my answer to that question. The purpose of the book is to tell the beginning student of advanced mathematics the basic set theoretic facts of life, and to do so with the minimum of philosophical discourse and logical formalism. The point of view throughout is that of a prospective mathematician anxious to study groups, or integrals, or manifolds. From this point of view the concepts and methods of this book are merely some of the standard mathematical tools; the expert specialist will find nothing new here. Scholarly bibliographical credits and references are out of place in a purely expository book such as this one. The student who gets interested in set theory for its own sake should know, however, that there is much more to the subject than there is in this book. One of the most beautiful sources of set-theoretic wisdom is still Hausdorff's Set theory. A recent and highly readable addition to the literature, with an extensive and up-to-date bibliography, is Axiomatic set theory by Suppes.

Differentialgeometrie

Kurven - Flächen - Mannigfaltigkeiten

Author: Wolfgang Kühnel

Publisher: Springer-Verlag

ISBN: 332292808X

Category: Mathematics

Page: 256

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Eine Einführung in die Differentialgeometrie. Es wird großer Wert auf Anschaulichkeit gelegt, was auch durch zahlreiche Abbildungen unterstützt wird. Für die 2. Auflage wurden alle Kapitel gründlich überarbeitet. Hinzu kamen einige neue Übungsaufgaben und zusätzliche Abbildungen.

Elementary Set Theory, Part I

Author: K.T. Leung,Doris Lai-chue Chen

Publisher: Hong Kong University Press

ISBN: 9789622090132

Category: Mathematics

Page: 80

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This book provides students of mathematics with the minimum amount of knowledge in logic and set theory needed for a profitable continuation of their studies. There is a chapter on statement calculus, followed by eight chapters on set theory.

Notes on Set Theory

Author: Yiannis N. Moschovakis

Publisher: Springer Science & Business Media

ISBN: 9783540941804

Category: Axiomatische Mengenlehre

Page: 272

View: 3852

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The axiomatic theory of sets is a vibrant part of pure mathematics, with its own basic notions, fundamental results, and deep open problems. It is also viewed as a foundation of mathematics so that "to make a notion precise" simply means "to define it in set theory." This book gives a solid introduction to "pure set theory" through transfinite recursion and the construction of the cumulative hierarchy of sets, and also attempts to explain how mathematical objects can be faithfully modeled within the universe of sets. In this new edition the author has added solutions to the exercises, and rearranged and reworked the text to improve the presentation.

A Course on Set Theory

Author: Ernest Schimmerling

Publisher: Cambridge University Press

ISBN: 1139501488

Category: Mathematics

Page: N.A

View: 560

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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.

The Joy of Sets

Fundamentals of Contemporary Set Theory

Author: Keith Devlin

Publisher: Springer Science & Business Media

ISBN: 146120903X

Category: Mathematics

Page: 194

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This text covers the parts of contemporary set theory relevant to other areas of pure mathematics. After a review of "naïve" set theory, it develops the Zermelo-Fraenkel axioms of the theory before discussing the ordinal and cardinal numbers. It then delves into contemporary set theory, covering such topics as the Borel hierarchy and Lebesgue measure. A final chapter presents an alternative conception of set theory useful in computer science.

Set Theory for the Working Mathematician

Author: Krzysztof Ciesielski

Publisher: Cambridge University Press

ISBN: 9780521594653

Category: Mathematics

Page: 236

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Presents those methods of modern set theory most applicable to other areas of pure mathematics.

Basic Discrete Mathematics

Logic, Set Theory, and Probability

Author: Richard Kohar

Publisher: World Scientific Publishing Company

ISBN: 9814730416

Category: Mathematics

Page: 732

View: 701

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This lively introductory text exposes the student in the humanities to the world of discrete mathematics. A problem-solving based approach grounded in the ideas of George Pólya are at the heart of this book. Students learn to handle and solve new problems on their own. A straightforward, clear writing style and well-crafted examples with diagrams invite the students to develop into precise and critical thinkers. Particular attention has been given to the material that some students find challenging, such as proofs. This book illustrates how to spot invalid arguments, to enumerate possibilities, and to construct probabilities. It also presents case studies to students about the possible detrimental effects of ignoring these basic principles. The book is invaluable for a discrete and finite mathematics course at the freshman undergraduate level or for self-study since there are full solutions to the exercises in an appendix. "Written with clarity, humor and relevant real-world examples, Basic Discrete Mathematics is a wonderful introduction to discrete mathematical reasoning."- Arthur Benjamin, Professor of Mathematics at Harvey Mudd College, and author of The Magic of Math

Axiomatic Set Theory

Author: Patrick Suppes

Publisher: Courier Corporation

ISBN: 0486136876

Category: Mathematics

Page: 265

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Geared toward upper-level undergraduates and graduate students, this treatment examines the basic paradoxes and history of set theory and advanced topics such as relations and functions, equipollence, more. 1960 edition.

A First Course in Mathematical Logic and Set Theory

Author: Michael L. O'Leary

Publisher: John Wiley & Sons

ISBN: 0470905883

Category: Mathematics

Page: 464

View: 1042

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Rather than teach mathematics and the structure of proofssimultaneously, this book first introduces logic as the foundationof proofs and then demonstrates how logic applies to mathematicaltopics. This method ensures that readers gain a firmunderstanding of how logic interacts with mathematics and empowersthem to solve more complex problems. The study of logic andapplications is used throughout to prepare readers for further workin proof writing. Readers are first introduced tomathematical proof-writing, and then the book provides anoverview of symbolic logic that includes two-column logicproofs. Readers are then transitioned to set theory andinduction, and applications of number theory, relations, functions,groups, and topology are provided to further aid incomprehension. Topical coverage includes propositional logic,predicate logic, set theory, mathematical induction, number theory,relations, functions, group theory, and topology.

Basic Set Theory

Author: Azriel Levy

Publisher: Courier Corporation

ISBN: 0486150739

Category: Mathematics

Page: 416

View: 5414

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The first part of this advanced-level text covers pure set theory, and the second deals with applications and advanced topics (point set topology, real spaces, Boolean algebras, infinite combinatorics and large cardinals). 1979 edition.

The Real Numbers

An Introduction to Set Theory and Analysis

Author: John Stillwell

Publisher: Springer Science & Business Media

ISBN: 331901577X

Category: Mathematics

Page: 244

View: 2586

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While most texts on real analysis are content to assume the real numbers, or to treat them only briefly, this text makes a serious study of the real number system and the issues it brings to light. Analysis needs the real numbers to model the line, and to support the concepts of continuity and measure. But these seemingly simple requirements lead to deep issues of set theory—uncountability, the axiom of choice, and large cardinals. In fact, virtually all the concepts of infinite set theory are needed for a proper understanding of the real numbers, and hence of analysis itself. By focusing on the set-theoretic aspects of analysis, this text makes the best of two worlds: it combines a down-to-earth introduction to set theory with an exposition of the essence of analysis—the study of infinite processes on the real numbers. It is intended for senior undergraduates, but it will also be attractive to graduate students and professional mathematicians who, until now, have been content to "assume" the real numbers. Its prerequisites are calculus and basic mathematics. Mathematical history is woven into the text, explaining how the concepts of real number and infinity developed to meet the needs of analysis from ancient times to the late twentieth century. This rich presentation of history, along with a background of proofs, examples, exercises, and explanatory remarks, will help motivate the reader. The material covered includes classic topics from both set theory and real analysis courses, such as countable and uncountable sets, countable ordinals, the continuum problem, the Cantor–Schröder–Bernstein theorem, continuous functions, uniform convergence, Zorn's lemma, Borel sets, Baire functions, Lebesgue measure, and Riemann integrable functions.

Logic, Sets, and Recursion

Author: Robert L. Causey

Publisher: Jones & Bartlett Learning

ISBN: 9780763737849

Category: Computers

Page: 512

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The new Second Edition incorporates a wealth of exercise sets, allowing students to test themselves and review important topics discussed throughout the text."--Jacket.