SUMS04 Elements of Logic via Numbers and Sets, D. L. Johnson (1998).pdf

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Springer Undergraduate Mathematics Series(SUMS)
D L. Johnson Elements of Logic via Numbers and sets Springer D, L Joh BSc MSc PhD Department of Mathematics, University of Nottingham, University Park, Nottingham NG7 2RD,UK Cover ilustration elemen 's reproduced by kind permission of Aptech Systems, Inc, Publishers of the GAUSS Mathematical and StatisticalSystem, 23E04 S.E. Kent-Kangley Road, Maple valley, WA98038, Usa.Tel:(206)432-7855Far(206)432-7832emaie:info@aptech.camUrl:www.aptechcom American Statistical Association: Chance Vol& No 1, 1995 article by KS and Kw Heiner Tree Rings of the Northern Shawangunks page 32 fg 2 Springer-Verlag: Mathematica in Education and Research Vcl4 Tasue 3 1995 articte by Roman E Maeder, Beatrice Amrhein and oliver Gloor llustrated Mathematics: Visualization of Mathematical Objects ' page 9 fig 11, originally published as a CD ROM Strated Mathematics'by TEL0 S ISBN0387-142243 ton by birkh BN3753-5100 Mathematica in EducaTon and Research vol 4 Issue 3 1995 article by Richard Gaylord and Kazcme Nishidate Traffic Engineering with Cellular Y Automata page 35 fg 2. Mathematica in Education and Research Vol 5 Issue 2 1996 article by Michael Trott The Implicitization of a Trefoil Mathcmatica in Education and Research Val 5 Issue 2 1996 article by Lee de Cola'Coins, Trees, Ears and Bes: Simulation of the Binomial Process page 19 fg 3. Mathematica in Education and Research Vol 5 Issue 2 1996 article by Richard Gaylord end Kazume Nishidate'contagious Spreading'page 3 fg 1. Mathematica n Fd cation and Research Vols Issue 2 1996 article by jue Buhler and Stan Wagcn'Sec page50鸱g ch Library Cataloguing in Publication Da Johnson, David lawrence abers and sets. -(Springer undergraduate mathematics series) 1. Number thed ITitle ISBN978-3-540-7612 ISBN978-1-447-0603-6( eBook) DOI10.1007978-1-4471-0603-6 Library of Congress Cataloging-in-Publication Data Johnson, DL Elements of logic via numbers and sets /D L. Johnson pcm-(Springer undergraduate mathematics series) Includes bibliographical references(p. 165)and index 1. Logic, Symbolic and mathematical I. Title II Series QA9J631998 97-28662 511.3~dc21 IP Apart from any fair dealing for the purposes of research or private study, or criticism or review, as permitted under the Copyright, Designs and Patents Act 1988, this publication may only be reproduced, stored or transmitted, in any form or by any means, with the prior permission in writing of the publishers, or in the case of reprographic reproduction in accordance with the terms of licences issued by the Copyright Licensing Agency. Enquiries concerning reproduction outside those terms should be sent to the publishers Springer-Verlag Berlin Heidelberg 1998 Originally published by Springer-Verlag Herlin Heidelberg New York in 1998 2nd printing, with corrections 1998 Brd printing 2001 The use of registered names, trademarks etc in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant laws and regulations and therefore free for general use. The publisher makes no representation, express or implied, with regard to the accuracy of the information contained in this book and cannot accept any legal responsibility or liability for any errors or omissions that may be made Typeset by Focal Image, London 12/3830-5432 Printed on acid-free paper SPIN 10831990 Fo Contents Introductio umbers 1.1 Arithmetic progressions 1.2 Proof by Contradiction 1.3 Proof by Contraposition 1.4 Proof by Inducti 1.5 Inductive Definition 19 6 The Well-ordering Principle 27 logic ■·曹■ 21 Propositions.…… 2.2 Truth Tables 2.3 Syllogisms 2.4 Quantifiers 48 3. Set 53 3.1 Introduction 54 3.2 Operations 58 3.3 Laws 62 3.4 The power Set 65 4. relations 71 4.1 Equivalence Relations 4.2 Congruences 4.3 Number Systems 79 4.4 Orde VIll Contents 5. Maps 5.1 Terminology and Notation ..89 5. 2 Examples..,. 94 5.3 Injections, Surjections and Bijections 99 5.4 Peano's axioms 6. Cardinal numbers 6.1 Cardinal arithmetic 114 6.2 The Cantor-Schroeder-Bernstein theorem ,,118 6.3 Countable sets ,,,,,,,,,,.121 6.4 Uncountable sets Solutions to exercises ∴....131 Guide to the literature 163 Bibliography ..165 D ramatis personae .,,. 167 Index .171 Introduction This book is based on a module given to first-year undergraduates at the uni versity of Nottingham with the aim of bridging the gap between school and university in mathematics. This is not so much a gap in the substance or na terial content of the subject but more a change in attitude and approach. In pure mathematics for example, rather than memorize a formula and be able to apply it, we want to understand that formula and be able to prove it. Physical intuition does not constitute a proof and neither does accumulated statistical evidence, what is required is a formal logical process. Since logic can sometimes appear rather a dry subject, we take as an underlying theme the concept of "number", which not only provides a rich source of illustrations but also helps to lay the foundations for many areas of more advanced mathematical study We begin in Chapter 1 with a survey of useful facts about numbers that are more or less familiar, such as the binomial theorem and Euclid s algorithm respectively, giving formal proofs of different kinds, notably proof by induction It will become increasingly evident that to gain a better understanding of what happens in the course of a proof we need a systematic language or framework within which to develop the ideas involved. Fortunately, one exists, and forms the subject of the next chapter The first half of Chapter 2 looks at propositional calculus, which treats propositions(statements that are either true of false) and the relations between them(such as implication) as mathematical objects. Ways of combining these and of operating on them, familiar from our knowledge of language, are put into formal shape governed by precise rules. The second half of Chapter 2 studies syllogisms, which are logical arguments, or processes of deduction, of the simplest kind. They thus form an ideal model or pattern for all forms of logical inference, and also serve to introduce the deceptively simple but crucially important idea, of quantification Introduction The core of the book is Chapter 3, where the development of set theory closely parallels that of logic in the previous chapter. It also lays the foundations for important results in the later chapters and introduces the terminology and notation that comprise the language of modern mathematics Two important types of relation are studied in Chapter 4: the notion of ordering forms a, familiar and fundamental element of structure in most number systems, and the seemingly artificial concept of equivalence relation. The latter finds application in many areas of advanced mathematics, especially in the construction of number systems and in abstract algebra, where the key result Theorem 4. 1 plays a decisive role The elementary concept of function is generalized to the more abstract idea. of map in Chapter 5, where many more or less familiar examples are given. A study of the basic properties of maps leads, among other things to our second key result of ubiquitous application, Theorem 5.6 The revolutionary ideas of Cantor described in Chapter 6 bring us ver. nearly into the present century. In addition to proving our third key result Theorem 6.6, we make good the claim that the concept of set is more funda- mental than that of number, and the axiomatic development of number systems reaches its crowning glory in the construction of the real numbers. The diligent reader will be rewarded with at least a glimpse of the meaning of the infinite It is a pleasure to acknowledge my gratitude to Springer-Verlag, and espe- cially to Susan Hezlet, for their courteous and efficient handling of all matters connected with the production of the book D. LJ Numbers 1+1=2. A.N. whitehead and B. Russell Principia Mathematica, Vol. Ii, p. 83 Historically, mathematics came into being to serve two purposes, counting and measuring. Both of these required the use of numbers, the positive integers N and the real numbers R, respectively. The need to solve equations such as 2x=3,m+4=0,x2+1=0 subsequently led to the appearance of more sophisticated number systems like the rational numbers @, the integers Z, and the complex numbers c. We are chiefly concerned here with properties of the positive integers and, at the same time, the means by which such properties are established This revolves around the concept of a mathematical proof, of which we give examples of four kinds finishing up with the most important for N, proof by induction 1.1 Arithmetic Progressions Definition 1.1 An arithmetic progression is a sequence of numbers C0,a1,2 whose consecutive terms differ by a constant called the common difference 01-a0=a2-a1 a,k D. L. Johnson, Elements of i ogic via Numbers and sets C Springer-Verlag Berlin Heidelberg 1998

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