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THE RANDOM HBUSEIBIRKHAUSER MATHEMAT . -..,.. !,.- -.-,. . , . . - .- ,.- <:. . 7 . .-1 . - MATH EI-YTICS MATHEMATICAL PROOF AND STRUCTURES BRIDGE TO ABSTRACT MATHEMATICS Mathematical Proof and Structures The Random House/Birkhaoser Mathematics Series: INVITATION TO COMPLEX ANALYSIS by R. P. Boas BRIDGE TO ABSTRACT MATHEMATICS: MATHEMATICAL PROOF AND STRUCTURES by Ronald P. Morash ELEMENTARY NUMBER THEORY by Charles Vanden Eynden INTRODUCTION TO ABSTRACT ALGEBRA by Elbert Walker BRIDGE TO ABSTRACT MATHEMATICS Mathematical Proof and Structures Ronald E Morash University of Michigan, Dearborn The Random House/Birkhauser Mathematics Series & Random House, Inc. New York First Edition Copyright @ 1987 by Random House, Inc. All rights reserved under International and Pan-American Copyright Conventions. No part of this book may be reproduced in any form or by any means, electronic or mechanical, including photocopying, without permission in writing from the publisher. All inquiries should be addressed to Random House, Inc., 201 East 50th Street, New York, N.Y. 10022. Published in the United States by Random House, Inc., and simultaneously in Canada by Random House of Canada Limited, Toronto. i Library of Congress Cataloging in Publication Data Morash, Ronald P. Bridge to abstract mathematics. Includes index. 1. Logic, Symbolic and mathematical. 2. Mathematics -1961- . I. Title. QA9.M74 1987 511.3 86-21931 ISBN 0-394-35429-X Manufactured in the United States of America to my family Preface This text is directed toward the sophomore through senior levels of university mathematics, with a tilt toward the former. It presumes that the student has completed at least one semester, and preferably a full year, of calculus. The text is a product of fourteen years of experience, on the part of the author, in teaching a not-too-common course to students with a very common need. The course is taken predominantly by sophomores and juniors from various fields of concentration who expect to enroll in junior- senior mathematics courses that include significant abstract content. It endeavors to provide a pathway, or bridge, to the level of mathematical sophistication normally desired by instructors in such courses, but generally not provided by the standard freshman-sophomore program. Toward this end, the course places strong emphasis on mathematical reasoning and ex- position. Stated differently, it endeavors to serve as a significant first step toward the goal of precise thinking and effective communication of one's thoughts in the language of science. Of central importance in any overt attempt to instill "mathematical ma- turity" in students is the writing and comprehension of proofs. Surely, the requirement that students deal seriously with mathematical proofs is the single factor that most strongly differentiates upper-division courses from the calculus sequence and other freshman-sophomore classes. Accordingly, the centerpiece of this text is a substantial body of material that deals \ explicitly and systematically with mathematical proof (Article 4.1, Chapters 5 and 6). A primary feature of this material is a recognition of and reliance on the student's background in mathematics (e.g., algebra, trigonometry, calculus, set theory) for a context in which to present proof-writing tech- niques. The first three chapters of the text deal with material that is important in its own right (sets, logic), but their major role is to lay groundwork for the coverage of proofs. Likewise, the material in Chapters 7 through 10 (relations, number systems) is of independent value to any student going on in mathematics. It is not inaccurate, however, in the context of this book, to view it primarily as a vehicle by which students may develop further the incipient ability to read and write proofs. PREFACE vll IMPORTANT FEATURES Readability. The author's primary pedagogical goal in writing the text was to produce a book that students can read. Since many colleges and uni- versities in the United States do not currently have a "bridging" course in mathematics, it was a goal to make the book suitable for the individual student who might want to study it independently. Toward this end, an introduction is provided for each chapter, and for many articles within chapters, to place content in perspective and relate it to other parts of the book, providing both an overall point of view and specific suggestions for work- ing through the unit. Solved examples are distributed liberally through- out the text. Abstract definitions are amplified, whenever appropriate, by a number of concrete examples. Occasionally, the presentation of material is interrupted, so the author can "talk to" the reader and explain various mathematical "facts of life." The numerous exercises at the end of articles have been carefully selected and placed to illustrate and supplement material in the article. In addition, exercises are often used to anticipate results or concepts in the next article. Of course, most students who use the text will do so under the direction of an instructor. Both instructor and students reap the benefit of enhanced opportunity for efficient classroom coverage of material when students are able to read a text. Organization. In Chapter 1, we introduce basic terminology and notation of set theory and provide an informal study of the algebra of sets. Beyond this, we use set theory as a device to indicate to the student what serious mathematics is really about, that is, the discovery of general theorems. Such discovery devices as examples, pictures, analogies, and counterexam- ples are brought into play. Rhetorical questions are employed often in this chapter to instill in the student the habit of thinking aggressively, of looking for questions as well as answers. Also, a case is made at this stage for both the desirability of a systematic approach to manipulating statements (i.e., logic) and the necessity of abstract proof to validate our mathematical beliefs. In Chapters 2 and 3, we study logic from a concrete and common-sense point of view. Strong emphasis is placed on those logical principles that are most commonly used in everyday mathematics (i.e., tautologies of the propositional calculus and theorems of the predicate calculus). The goal of these chapters is to integrate principles of logic into the student's way of thinking so that they are applied correctly, though most often only implicitly, to the solving of mathematical problems, including the writing of proofs. In Chapter 4, we begin to some mathematics, with an emphasis on topics whose understanding is enhanced by a knowledge of elementary logic. Most important, we begin in this chapter to deal with proofs, limiting ourselves at this stage to theorems of set theory, including properties of