Sets and Mappings
IT, as it is often said, mathematics is the queen of science then algebra is surely the jewel in her crown. In the course of its vast development over the last half-century, algebra has emerged as the subject in which one can observe pure mathe matical r
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VOLUME ONE
Sets and Mappings
Essential Student Algebra
VOLUME ONE
Sets and Mappings T. S. BLYTH & E. F. ROBERTSON University of St Andrews
SPRINGER-SCIENCE+BUSINESS MEDIA, B.V.
© 1986 T. S. Blyth and E. F. Robertson Originally published by Chapman & Hall in 1986
ISBN 978-0-412-27880-8 ISBN 978-94-015-7713-7 (eBook) DOI 10.1007/978-94-015-7713-7 This paperback edition is sold subject to the condition that it shall not, by way of trade or otherwise, be lent, resold, hired out, or otherwise circulated without the publisher's prior consent in any form of binding or cover other than that in which it is published and without a similar condition including this condition being imposed on the subsequent purchaser. All rights reserved. No part of this book may be reprinted or reproduced, or utilized in any form or by any electronic, mechanical or other means, now known or hereafter invented, including photocopying and recording, or in any information storage and retrieval system, without permission in writing from the publisher. British Library Cataloguing in Publication Data Blyth, T. S. Essential student algebra. Vol1: Sets and mappings 1. Algebra I. Title II. Robertson, E. F. 512 QA155 ISBN 978-0-412-27880-8
Contents
Preface Chapter One : Sets
1
Chapter Two : Mappings
16
Chapter Three : Equivalence relations
40
Chapter Four : The integers
49
Chapter Five : Permutations
76
Chapter Six : Cardinals and the natural numbers
98
Index
119
Preface
IT, as it is often said, mathematics is the queen of science then algebra is surely the jewel in her crown. In the course of its vast development over the last half-century, algebra has emerged as the subject in which one can observe pure mathematical reasoning at its best. Its elegance is matched only by the ever-increasing number of its applications to an extraordinarily wide range of topics in areas other than 'pure' mathematics. Here our objective is to present, in the form of a series of five concise volumes, the fundamentals of the subject. Broadly speaking, we have covered in all the now traditional syllabus that is found in first and second year university courses, as well as some third year material. Further study would be at the level of 'honours options'. The reasoning that lies behind this modular presentation is simple, namely to allow the student (be he a mathematician or not) to read the subject in a way that is more appropriate to the length, content, and extent, of the various courses he has to take. Although we have taken great pains to include a wide selection of illustrative examples, we have not included any exercises. For a suitable companion collection of worked examples, we would refer the reader to our series Algebra through practice (Cambridge University Press), the first five books of which are appropriate to the material covered here. T.S.B., E.F.R.
CHAPTER ONE
Sets
Our approach to the theory of sets will be naive in the sense that, rather than give a precise definition of the concept of a 'set', we shall rely heavily on intuit
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