Table Of ContentTHE RANDOM HBUSEIBIRKHAUSER MATHEMAT
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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 uni-
versity 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 impor-
tant 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 in-
troduction is provided for each chapter, and for many articles within chap-
ters, 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 ma-
terial 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
