An Introduction to Mathematical Reasoning

Numbers, Sets and Functions

Author: Peter J. Eccles

Publisher: Cambridge University Press

ISBN: 9780521597180

Category: Mathematics

Page: 350

View: 8176

This book eases students into the rigors of university mathematics. The emphasis is on understanding and constructing proofs and writing clear mathematics. The author achieves this by exploring set theory, combinatorics, and number theory, topics that include many fundamental ideas and may not be a part of a young mathematician's toolkit. This material illustrates how familiar ideas can be formulated rigorously, provides examples demonstrating a wide range of basic methods of proof, and includes some of the all-time-great classic proofs. The book presents mathematics as a continually developing subject. Material meeting the needs of readers from a wide range of backgrounds is included. The over 250 problems include questions to interest and challenge the most able student but also plenty of routine exercises to help familiarize the reader with the basic ideas.
Posted in Mathematics

Principia Mathematica.

Author: Alfred North Whitehead,Bertrand Russell

Publisher: N.A


Category: Logic, Symbolic and mathematical

Page: 167

View: 7023

Posted in Logic, Symbolic and mathematical

Das BUCH der Beweise

Author: Martin Aigner,Günter M. Ziegler

Publisher: Springer-Verlag

ISBN: 3662577674

Category: Mathematics

Page: 360

View: 7103

Diese fünfte deutsche Auflage enthält ein ganz neues Kapitel über van der Waerdens Permanenten-Vermutung, sowie weitere neue, originelle und elegante Beweise in anderen Kapiteln. Aus den Rezensionen: “... es ist fast unmöglich, ein Mathematikbuch zu schreiben, das von jedermann gelesen und genossen werden kann, aber Aigner und Ziegler gelingt diese Meisterleistung in virtuosem Stil. [...] Dieses Buch erweist der Mathematik einen unschätzbaren Dienst, indem es Nicht-Mathematikern vorführt, was Mathematiker meinen, wenn sie über Schönheit sprechen.” Aus der Laudatio für den “Steele Prize for Mathematical Exposition” 2018 "Was hier vorliegt ist eine Sammlung von Beweisen, die in das von Paul Erdös immer wieder zitierte BUCH gehören, das vom lieben (?) Gott verwahrt wird und das die perfekten Beweise aller mathematischen Sätze enthält. Manchmal lässt der Herrgott auch einige von uns Sterblichen in das BUCH blicken, und die so resultierenden Geistesblitze erhellen den Mathematikeralltag mit eleganten Argumenten, überraschenden Zusammenhängen und unerwarteten Volten.", Mai 2002 "Eine einzigartige Sammlung eleganter mathematischer Beweise nach der Idee von Paul Erdös, verständlich geschrieben von exzellenten Mathematikern. Dieses Buch gibt anregende Lösungen mit Aha-Effekt, auch für Nicht-Mathematiker." "Ein prächtiges, äußerst sorgfältig und liebevoll gestaltetes Buch! Erdös hatte die Idee DES BUCHES, in dem Gott die perfekten Beweise mathematischer Sätze eingeschrieben hat. Das hier gedruckte Buch will eine "very modest approximation" an dieses BUCH sein.... Das Buch von Aigner und Ziegler ist gelungen ..." Mathematische Semesterberichte, November 1999 "Wer (wie ich) bislang vergeblich versucht hat, einen Blick ins BUCH zu werfen, wird begierig in Aigners und Zieglers BUCH der Beweise schmökern.", Mai 2002
Posted in Mathematics

The Tools of Mathematical Reasoning

Author: Tamara J. Lakins

Publisher: American Mathematical Soc.

ISBN: 1470428997

Category: General -- Instructional exposition (textbooks, tutorial papers, etc.)

Page: 217

View: 6566

This accessible textbook gives beginning undergraduate mathematics students a first exposure to introductory logic, proofs, sets, functions, number theory, relations, finite and infinite sets, and the foundations of analysis. The book provides students with a quick path to writing proofs and a practical collection of tools that they can use in later mathematics courses such as abstract algebra and analysis. The importance of the logical structure of a mathematical statement as a framework for finding a proof of that statement, and the proper use of variables, is an early and consistent theme used throughout the book.
Posted in General -- Instructional exposition (textbooks, tutorial papers, etc.)

Learning to Reason

An Introduction to Logic, Sets, and Relations

Author: Nancy Rodgers

Publisher: John Wiley & Sons

ISBN: 1118165705

Category: Mathematics

Page: 454

View: 6488

Learn how to develop your reasoning skills and how to writewell-reasoned proofs Learning to Reason shows you how to use the basic elements ofmathematical language to develop highly sophisticated, logicalreasoning skills. You'll get clear, concise, easy-to-followinstructions on the process of writing proofs, including thenecessary reasoning techniques and syntax for constructingwell-written arguments. Through in-depth coverage of logic, sets,and relations, Learning to Reason offers a meaningful, integratedview of modern mathematics, cuts through confusing terms and ideas,and provides a much-needed bridge to advanced work in mathematicsas well as computer science. Original, inspiring, and designed formaximum comprehension, this remarkable book: * Clearly explains how to write compound sentences in equivalentforms and use them in valid arguments * Presents simple techniques on how to structure your thinking andwriting to form well-reasoned proofs * Reinforces these techniques through a survey of sets--thebuilding blocks of mathematics * Examines the fundamental types of relations, which is "where theaction is" in mathematics * Provides relevant examples and class-tested exercises designed tomaximize the learning experience * Includes a mind-building game/exercise space
Posted in Mathematics

The Nuts and Bolts of Proofs

An Introduction to Mathematical Proofs

Author: Antonella Cupillari

Publisher: Academic Press

ISBN: 0123822181

Category: Mathematics

Page: 296

View: 5561

The Nuts and Bolts of Proofs: An Introduction to Mathematical Proofs provides basic logic of mathematical proofs and shows how mathematical proofs work. It offers techniques for both reading and writing proofs. The second chapter of the book discusses the techniques in proving if/then statements by contrapositive and proofing by contradiction. It also includes the negation statement, and/or. It examines various theorems, such as the if and only-if, or equivalence theorems, the existence theorems, and the uniqueness theorems. In addition, use of counter examples, mathematical induction, composite statements including multiple hypothesis and multiple conclusions, and equality of numbers are covered in this chapter. The book also provides mathematical topics for practicing proof techniques. Included here are the Cartesian products, indexed families, functions, and relations. The last chapter of the book provides review exercises on various topics. Undergraduate students in engineering and physical science will find this book invaluable. Jumps right in with the needed vocabulary—gets students thinking like mathematicians from the beginning Offers a large variety of examples and problems with solutions for students to work through on their own Includes a collection of exercises without solutions to help instructors prepare assignments Contains an extensive list of basic mathematical definitions and concepts needed in abstract mathematics
Posted in Mathematics

How to Prove It

A Structured Approach

Author: Daniel J. Velleman

Publisher: Cambridge University Press

ISBN: 1139450972

Category: Mathematics

Page: N.A

View: 2955

Many students have trouble the first time they take a mathematics course in which proofs play a significant role. This new edition of Velleman's successful text will prepare students to make the transition from solving problems to proving theorems by teaching them the techniques needed to read and write proofs. The book begins with the basic concepts of logic and set theory, to familiarize students with the language of mathematics and how it is interpreted. These concepts are used as the basis for a step-by-step breakdown of the most important techniques used in constructing proofs. The author shows how complex proofs are built up from these smaller steps, using detailed 'scratch work' sections to expose the machinery of proofs about the natural numbers, relations, functions, and infinite sets. To give students the opportunity to construct their own proofs, this new edition contains over 200 new exercises, selected solutions, and an introduction to Proof Designer software. No background beyond standard high school mathematics is assumed. This book will be useful to anyone interested in logic and proofs: computer scientists, philosophers, linguists, and of course mathematicians.
Posted in Mathematics

Naive Mengenlehre

Author: Paul R. Halmos

Publisher: Vandenhoeck & Ruprecht

ISBN: 9783525405277

Category: Arithmetic

Page: 132

View: 7390

Posted in Arithmetic

Doing Mathematics

An Introduction to Proofs and Problem Solving

Author: Steven Galovich

Publisher: Brooks/Cole Publishing Company


Category: Mathematics

Page: 307

View: 7132

This book introduces students to the process of doing mathematics and prepares them to succeed in higher-level mathematics courses. By discussing proof techniques, problem solving methods, and the understanding of mathematical ideas, the book provides a solid foundation for students majoring in mathematics, science, and engineering. Students will learn to grasp the underlying concepts of a subject and how to apply these concepts to solving problems. While being able to understand and reproduce proofs of theorems, they will also gain the ability to comprehend the connections among the important concepts and techniques of each subject. This book is intended for a shorter course on proofs and mathematical reasoning, and could also be used as a supplemental text in courses such as algebra, analysis, and linear algebra.
Posted in Mathematics

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: 527

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.
Posted in Mathematics

Discrete Mathematics: Introduction to Mathematical Reasoning

Author: Susanna S. Epp

Publisher: Cengage Learning

ISBN: 1133417078

Category: Mathematics

Page: 648

View: 5978

Susanna Epp's DISCRETE MATHEMATICS: AN INTRODUCTION TO MATHEMATICAL REASONING, provides the same clear introduction to discrete mathematics and mathematical reasoning as her highly acclaimed DISCRETE MATHEMATICS WITH APPLICATIONS, but in a compact form that focuses on core topics and omits certain applications usually taught in other courses. The book is appropriate for use in a discrete mathematics course that emphasizes essential topics or in a mathematics major or minor course that serves as a transition to abstract mathematical thinking. The ideas of discrete mathematics underlie and are essential to the science and technology of the computer age. This book offers a synergistic union of the major themes of discrete mathematics together with the reasoning that underlies mathematical thought. Renowned for her lucid, accessible prose, Epp explains complex, abstract concepts with clarity and precision, helping students develop the ability to think abstractly as they study each topic. In doing so, the book provides students with a strong foundation both for computer science and for other upper-level mathematics courses. Important Notice: Media content referenced within the product description or the product text may not be available in the ebook version.
Posted in Mathematics

Introduction to Abstract Mathematics

Author: John F. Lucas

Publisher: Rowman & Littlefield

ISBN: 9780912675732

Category: Mathematics

Page: 382

View: 7673

This is a book about mathematics and mathematical thinking. It is intended for the serious learner who is interested in studying some deductive strategies in the context of a variety of elementary mathematical situations. No background beyond single-variable calculus is presumed.
Posted in Mathematics

An Introduction to Fuzzy Logic and Fuzzy Sets

Author: James J. Buckley,Esfandiar Eslami

Publisher: Springer Science & Business Media

ISBN: 3790817996

Category: Computers

Page: 285

View: 1530

This book is an excellent starting point for any curriculum in fuzzy systems fields such as computer science, mathematics, business/economics and engineering. It covers the basics leading to: fuzzy clustering, fuzzy pattern recognition, fuzzy database, fuzzy image processing, soft computing, fuzzy applications in operations research, fuzzy decision making, fuzzy rule based systems, fuzzy systems modeling, fuzzy mathematics. It is not a book designed for researchers - it is where you really learn the "basics" needed for any of the above-mentioned applications. It includes many figures and problem sets at the end of sections.
Posted in Computers

Logic, Induction and Sets

Author: Thomas Forster

Publisher: Cambridge University Press

ISBN: 9780521533614

Category: Mathematics

Page: 234

View: 7978

This is an introduction to logic and the axiomatization of set theory from a unique standpoint. Philosophical considerations, which are often ignored or treated casually, are here given careful consideration, and furthermore the author places the notion of inductively defined sets (recursive datatypes) at the center of his exposition resulting in a treatment of well established topics that is fresh and insightful. The presentation is engaging, but always great care is taken to illustrate difficult points. Understanding is also aided by the inclusion of many exercises. Little previous knowledge of logic is required of the reader, and only a background of standard undergraduate mathematics is assumed.
Posted in Mathematics

Die Grundlagen der Mathematik

Author: David Hilbert

Publisher: Springer-Verlag

ISBN: 3663161021

Category: Mathematics

Page: 29

View: 4115

Dieser Buchtitel ist Teil des Digitalisierungsprojekts Springer Book Archives mit Publikationen, die seit den Anfängen des Verlags von 1842 erschienen sind. Der Verlag stellt mit diesem Archiv Quellen für die historische wie auch die disziplingeschichtliche Forschung zur Verfügung, die jeweils im historischen Kontext betrachtet werden müssen. Dieser Titel erschien in der Zeit vor 1945 und wird daher in seiner zeittypischen politisch-ideologischen Ausrichtung vom Verlag nicht beworben.
Posted in Mathematics

A Bridge to Advanced Mathematics

Author: Dennis Sentilles

Publisher: Courier Corporation

ISBN: 0486277585

Category: Mathematics

Page: 416

View: 2063

This helpful "bridge" book offers students the foundations they need to understand advanced mathematics. The two-part treatment provides basic tools and covers sets, relations, functions, mathematical proofs and reasoning, more. 1975 edition.
Posted in Mathematics

Numbers, Groups and Codes

Author: J. F. Humphreys,M. Y. Prest

Publisher: Cambridge University Press

ISBN: 9781139451161

Category: Mathematics

Page: 338

View: 8163

This textbook is an introduction to algebra via examples. The book moves from properties of integers, through other examples, to the beginnings of group theory. Applications to public key codes and to error correcting codes are emphasised. These applications, together with sections on logic and finite state machines, make the text suitable for students of computer science as well as mathematics students. Attention is paid to historical development of the mathematical ideas. This second edition contains new material on mathematical reasoning skills and a new chapter on polynomials has been added. The book was developed from first-level courses taught in the UK and USA. These courses proved successful in developing not only a theoretical understanding but also algorithmic skills. This book can be used at a wide range of levels: it is suitable for first- or second-level university students, and could be used as enrichment material for upper-level school students.
Posted in Mathematics

Quaestiones Mathematicae

Author: N.A

Publisher: N.A


Category: Mathematics

Page: N.A

View: 4795

Posted in Mathematics

Notes on Logic and Set Theory

Author: P. T. Johnstone

Publisher: Cambridge University Press

ISBN: 9780521336925

Category: Mathematics

Page: 110

View: 5242

A succinct introduction to mathematical logic and set theory, which together form the foundations for the rigorous development of mathematics. Suitable for all introductory mathematics undergraduates, Notes on Logic and Set Theory covers the basic concepts of logic: first-order logic, consistency, and the completeness theorem, before introducing the reader to the fundamentals of axiomatic set theory. Successive chapters examine the recursive functions, the axiom of choice, ordinal and cardinal arithmetic, and the incompleteness theorems. Dr. Johnstone has included numerous exercises designed to illustrate the key elements of the theory and to provide applications of basic logical concepts to other areas of mathematics.
Posted in Mathematics

Introduction to Advanced Mathematics: A Guide to Understanding Proofs

Author: Connie M. Campbell

Publisher: Cengage Learning

ISBN: 1133168787

Category: Mathematics

Page: 144

View: 697

This text offers a crucial primer on proofs and the language of mathematics. Brief and to the point, it lays out the fundamental ideas of abstract mathematics and proof techniques that students will need to master for other math courses. Campbell presents these concepts in plain English, with a focus on basic terminology and a conversational tone that draws natural parallels between the language of mathematics and the language students communicate in every day. The discussion highlights how symbols and expressions are the building blocks of statements and arguments, the meanings they convey, and why they are meaningful to mathematicians. In-class activities provide opportunities to practice mathematical reasoning in a live setting, and an ample number of homework exercises are included for self-study. This text is appropriate for a course in Foundations of Advanced Mathematics taken by students who've had a semester of calculus, and is designed to be accessible to students with a wide range of mathematical proficiency. It can also be used as a self-study reference, or as a supplement in other math courses where additional proofs practice is needed. Important Notice: Media content referenced within the product description or the product text may not be available in the ebook version.
Posted in Mathematics