Reading_Writing_and_Proving_Mathematics-Daepp-Gorkin.pdf

Undergraduate Texts in Mathematics

Editors

S. Axler

F.W. Gehring

K.A. Ribet

Springer

New York

Berlin

Heidelberg

Hong Kong

London

Milan

Paris

Tokyo

Ulrich Daepp Pamela Gorkin

Reading, Writing,

and Proving

A Closer Look at Mathematics

Ulrich Daepp Pamela Gorkin

Department of Mathematics Department of Mathematics

Bucknell University

Lewisburg, PA 17837

USA

udaepp@bucknell.edu

pgorkin@bucknell.edu

Editorial Board:

S. Axler

F.W. Gehring

K.A. Ribet

Mathematics Department Mathematics Department

Mathematics Department

San Francisco State

East Hall

University of California,

University

University of Michigan

Berkeley

San Francisco, CA 94132

Ann Arbor, MI 48109

Berkeley, CA 94720-3840

axler@sfsu.edu

fgehring@math.lsa.umich.edu ribet@math.berkeley.edu

Mathematics Subject Classiﬁcation (2000): 00-01, 03-01

Library of Congress Cataloging-in-Publication Data

Daepp, Ulrich.

Reading, writing, and proving : a closer look at mathematics / Ulrich Daepp, Pamela

Gorkin.

p. cm.– (Undergraduate texts in mathematics)

Includes bibliographical references and index.

ISBN 0-387-00834-9 (alk. paper)

1. Mathematics–Study and teaching (Higher)–United States. 2. Technical

writing–Study and teaching (Higher)–United States. I. Gorkin, Pamela. II. Title.

III. Series.

QA113.D34 2003

510-dc21

2003045420

ISBN 0-387-00834-9

Printed on acid-free paper.

the written permission of the publisher (Springer-Verlag New York, Inc., 175 Fifth Avenue,

New York, NY 10010, USA), except for brief excerpts in connection with reviews or scholarly

analysis. Use in connection with any form of information storage and retrieval, electronic

adaptation, computer software, or by similar or dissimilar methodology now known or

hereafter developed is forbidden.

The use in this publication of trade names, trademarks, service marks, and similar terms,

even if they are not identiﬁed as such, is not to be taken as an expression of opinion as to

whether or not they are subject to proprietary rights.

Printed in the United States of America.

987654321

SPIN 10920411

www.springer-ny.com

Springer-Verlag New York Berlin Heidelberg

A member of BertelsmannSpringer Science + Business Media GmbH

For Hannes and Madeleine

Preface

You are probably about to teach or take a “ﬁrst course in proof

techniques,” or maybe you just want to learn more about mathemat-

ics. No matter what the reason, a student who wishes to learn the

material in this book likes mathematics, and we hope to keep it that

way. At this point, students have an intuitive sense of why things

are true, but not the exposure to the detailed and critical thinking

necessary to survive in the mathematical world. We have written

this book to bridge this gap.

In our experience, students beginning this course have little

training in rigorous mathematical reasoning; they need guidance.

At the end, they are where they should be; on their own. Our aim

is to teach the students to read, write, and do mathematics inde-

pendently, and to do it with clarity, precision, and care. If we can

maintain the enthusiasm they have for the subject, or even create

some along the way, our book has done what it was intended to do.

Reading . This book was written for a course we teach to ﬁrst and

second year college students. The style is informal. A few problems

require calculus, but these are identiﬁed as such. Students will also

need to participate while reading proofs, prodded by questions (such

as, “Why?”). Many detailed examples are provided in each chapter.

vii

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