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The Nuts and Bolts of ProofsThird EditionANTONELLA CUPILLARIPenn State Erie,The Behrend CollegeELSEVIERACADEMICPRESSAmsterdam Boston Heidelberg London New York OxfordParis San Diego San Francisco Singapore Sydney Tokyo
Acquisitions Editor: Thomas SingerProject Manager: Paul GottehrerMarketing Manager: Linda Beattie and Philip PritchardCover Design: Gate BarrComposition: Cepha Imaging Pvt. Ltd.Cover Printer: Phoenix Color CorporationInterior Printer: The Maple-Vail Book Manufacturing GroupElsevier Academic Press30 Corporate Drive, Suite 400, Burlington, MA 01803, USA525 B Street, Suite 1900, San Diego, California 92101-4495, USA84 Theobald's Road, London WCIX 8RR, UKThis book is printed on acid-free paper. @Copyright © 2005, Elsevier Inc. All rights reserved.No part of this publication may be reproduced or transmitted in any form or by any means,electronic or mechanical, including photocopy, recording, or any information storage andretrieval system, without permission in writing from the publisher.Permissions may be sought directly from Elsevier's Science & Technology Rights Department inOxford, UK: phone: (+44) 1865 843830, fax: (+44) 1865 853333, e-mail: permissions@elsevier.co.uk.You may also complete your request on-line via the Elsevier homepage(http://elsevier.com), by selecting "Customer Support" and then "Obtaining Permissions."Library of Congress Cataloging-in-Publication DataBritish Library Cataloguing in Publication DataA catalogue record for this book is available from the British LibraryISBN 13: 978-0-12-088509-1ISBN 10: 0-12-088509-3For all information on all Elsevier Academic Press Publicationsvisit our Web site at www.books.elsevier.comPrinted in the United States of America05 06 07 08 09 10 9 8 7 6 5 4 3 2 1Working together to growlibraries in developing countrieswww.elsevier.com | www.bookaid.org | www.sabre.orgELSEVIER
Page 2 and 3: The Nuts and Bolts of ProofsThird E
Page 6 and 7: List of SymbolsNatural or counting
Page 8 and 9: List of Symbolsvii(relatively prime
Page 10 and 11: ContentsList of Symbols vPreface xi
Page 12 and 13: PrefaceThis book is addressed to th
Page 14 and 15: Introduction and BasicTerminologyHa
Page 16 and 17: Introduction and Basic Terminology
Page 18 and 19: General SuggestionsThe first step,
Page 22 and 23: Basic Techniques To ProveIf/Then St
Page 48 and 49: special Kinds of TheoremsThere are
Page 50 and 51: Special Kinds of Theorems 37Parti.
Page 52 and 53: Special Kinds of Theorems39Diagram
Page 54 and 55:
Special Kinds of Theorems 41anda <
Page 56 and 57:
Special Kinds of Theorems 43Suppose
Page 58 and 59:
Special Kinds of Theorems 45and^aix
Page 60 and 61:
Special Kinds of Theorems 47EXAMPLE
Page 62 and 63:
Special Kinds of Theorems 493. Usin
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Special Kinds of Theorems 51use the
Page 66 and 67:
Special Kinds of Theorems 53As both
Page 68 and 69:
Special Kinds of Theorems 55Using t
Page 70 and 71:
Special Kinds of Theorems 57If we s
Page 72 and 73:
Special Kinds of Theorems 59B. Ther
Page 74 and 75:
Special Kinds of Theorems 61EXERCIS
Page 76 and 77:
Special Kinds of Theorems 63EXAMPLE
Page 78 and 79:
Special Kinds of Theorems 65If /c >
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Special Kinds of Theorems 67for all
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Special Kinds of Theorems 69conside
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Special Kinds of Theorems 71To cons
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Special Kinds of Theorems 73We will
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Special Kinds of Theorems 75both A
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Special Kinds of Theorems 773. Prov
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Special Kinds of Theorems 79(For de
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Special Kinds of Theorems 81This im
Page 96 and 97:
Special Kinds of Theorems 83Let us
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Special Kinds of Theorems 85Indeed,
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Special Kinds of Theorems 87of the
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Special Kinds of Theorems 89where p
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Special Kinds of Theorems 91which h
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Special Kinds of Theorems 93Thus, w
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Special Kinds of Theorems 95The exi
Page 110 and 111:
Special Kinds of Theorems 97Thus,\x
Page 112 and 113:
Special Kinds of Theorems2n-lpEXAMP
Page 114 and 115:
Special Kinds of Theorems 101that l
Page 116 and 117:
Review ExercisesDiscuss the truth o
Page 118 and 119:
Review Exercises 10519. Let y4 be a
Page 120 and 121:
Exercises Without SolutionsDiscuss
Page 122 and 123:
Exercises Without Solutions 10922.
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Exercises Without Solutions 111(b)
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Exercises Without Solutions 113(Thi
Page 128 and 129:
Collection of ProofsThe following c
Page 130 and 131:
Collection of Proofs 117Alleged Pro
Page 132 and 133:
Collection of Proofs 119where k is
Page 134 and 135:
Collection of Proofs 121THEOREM 10.
Page 136 and 137:
Solutions for the Exercises atthe E
Page 138 and 139:
Solutions for the Exercises at the
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Other Books on theSubject of Proofs
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Other Books on the Subject of Proof
Page 190 and 191:
IndexAlgorithms, Division, 16Argume
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Index 179TTerm, 99Theoremsbasic hyp
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