Discrete Mathematics

k Discrete Mathematics k k k k This page intentionally left blank k k k k Discrete Mathematics Eighth Edition Richard Johnsonbaugh DePaul University, Chicago k k 330 Hudson Street, NY, NY 10013 k k Director, Portfolio Management Deirdre Lynch Executive Editor Jeff Weidenaar Editorial Assistant Jennifer Snyder Content Producer Lauren Morse Managing Producer Scott Disanno Media Producer Nicholas Sweeney Product Marketing Manager Yvonne Vannatta Field Marketing Manager Evan St. Cyr Marketing Assistant Jennifer Myers Senior Author Support/Technology Specialist Joe Vetere Rights and Permissions Project Manager Gina Cheselka Manufacturing Buyer Carol Melville, LSC Communications Associate Director of Design Blair Brown Text Design, Production Coordination, and Composition SPi Global Cover Design Laurie Entringer Cover and Chapter Opener Helenecanada/iStock/Getty Images c 2018, 2009, 2005 by Pearson Education, Inc. All Rights Reserved. Printed in the United Copyright  States of America. This publication is protected by copyright, and permission should be obtained from the publisher prior to any prohibited reproduction, storage in a retrieval system, or transmission in any form or by any means, electronic, mechanical, photocopying, recording, or otherwise. For information regarding permissions, request forms and the appropriate contacts within the Pearson Education Global Rights & Permissions department, please visit www.pearsoned.com/permissions/. PEARSON and ALWAYS LEARNING are exclusive trademarks owned by Pearson Education, Inc. or its affiliates in the U.S. and/or other countries. Unless otherwise indicated herein, any third-party trademarks that may appear in this work are the property of their respective owners and any references to third-party trademarks, logos or other trade dress are for demonstrative or descriptive purposes only. Such references are not intended to imply any sponsorship, endorsement, authorization, or promotion of Pearson’s products by the owners of such marks, or any relationship between the owner and Pearson Education, Inc. or its affiliates, authors, licensees or distributors. k Microsoft and/or its respective suppliers make no representations about the suitability of the information contained in the documents and related graphics published as part of the services for any purpose. All such documents and related graphics are provided "as is" without warranty of any kind. Microsoft and/or its respective suppliers hereby disclaim all warranties and conditions with regard to this information, including all warranties and conditions of merchantability, whether express, implied or statutory, fitness for a particular purpose, title and non-infringement. In no event shall microsoft and/or its respective suppliers be liable for any special, indirect or consequential damages or any damages whatsoever resulting from loss of use, data or profits, whether in an action of contract, negligence or other tortious action, arising out of or in connection with the use or performance of information available from the services. The documents and related graphics contained herein could include technical inaccuracies or typographical errors. Changes are periodically added to the information herein. Microsoft and/or its respective suppliers may make improvements and/or changes in the product(s) and/or the program(s) described herein at any time. Partial screen shots may be viewed in full within the software version specified. Microsoft Windows Explorer , and Microsoft Excel® are registered trademarks of the microsoft corporation in the U.S.A. and other countries. This book is not sponsored or endorsed by or affiliated with the microsoft corporation. Johnsonbaugh