Applied Finite MathematicsTomastik/EpsteinApplied Finite Mathematics, Second EditionEdmond C. Tomastik, University of ConnecticutJanice L. Epstein, Texas A&M UniversityEditor: Carolyn CrockettEditorial Assistant: Rebecca DashiellMarketing Manager: Myriah FitzgibbonTechnical Editor: Mary KanableIllustrator: Jennifer TribblePhotographs: Janice Epstein©1994, 2008 Brooks/Cole, Cengage LearningALL RIGHTS RESERVED. No part of this work covered by the copyrightherein may be reproduced, transmitted, stored, or used in any form or by anymeans graphic, electronic, or mechanical, including but not limited to photo-copying, recording, scanning, digitizing, taping, Web distribution, informationnetworks, or information storage and retrieval systems, except as permittedunder Section 107 or 108 of the 1976 United States Copyright Act, withoutthe prior written permission of the publisher.For product information and technology assistance, contact us atCengage Learning Customer & Sales Support, 1-800-354-9706.For permission to use material from this text or product, submit allrequests online at cengage.com/permissions.Further permissions questions can be e-mailed [email protected] of Congress Control Number: 2008927157ISBN-13: 978-0-495-55533-9ISBN-10: 0-495-55533-9Brooks/Cole10 Davis DriveBelmont, CA 94002-3098USACengage Learning is a leading provider of customized learning solutions withoffice locations around the globe, including Singapore, the United Kingdom,Australia, Mexico, Brazil, and Japan. Locate your local office at: interna-tional.cengage.com/region.Cengage Learning products are represented in Canada byNelson Education, Ltd.For your course and learning solutions, visit academic.cengage.com.Purchase any of our products at your local college bookstore or at ourpreferred online store www.ichapters.com.Printed in the United States of America12345671211100908LLLLLLLLLLLLLLLLLLLLLLLLLContentsL Logic 2L.1 Introduction to Logic . . . . . .................. 3Statements ............................ 3Connectives . . ......................... 3Exercises ............................ 7L.2 Truth Tables ............................ 9Introduction to Truth Tables ................... 9Exclusive Disjunction . . .................... 10Tautology and Contradiction .................. 10Exercises ............................ 11L.3 Implication and Equivalence ................... 13Conditional Statements . .................... 13Biconditional .......................... 14Logical Equivalence . . . .................... 15Exercises ............................ 17L.4 Laws of Logic ........................... 19Logical Equivalence . . . .................... 19Implication ........................... 21Exercises ............................ 23L.5 Arguments ............................. 25Valid Arguments . . . . .................... 25Using Implications and Equivalences in Arguments ...... 27Indirect Proof .......................... 29Exercises ............................ 31L.6 Switching Networks ........................ 33Exercises ............................ 37Review .............................. 39Summary Outline ........................ 39Review Exercises . . . . .................... 40Answers to Selected Exercises . .................... 42Index 49CHAPTER LLogicCONNECTIONCircuit BoardsHow should the circuits on this board be laidout so that the video card works? Logic isused in the design of circuit boards.L.1 Introduction to Logic 3L.1 Introduction to Logic✧ StatementsLogic is the science of correct reasoning and of making valid conclusions. Inlogic conclusions must be inescapable. Every concept must be clearly defined.Thus, dictionary definitions are often not sufficient since there can be no ambi-guities or vagueness.We restrict our study to declarative sentences that are unambiguous and thatcan be classified as true or false but not both. Such declarative sentences arecalled statements and form the basis of logic.StatementsA statement is a declarative sentence that is either true or false but notboth.Thus, commands, questions, exclamations, and ambiguous sentences cannotbe statements.HISTORICAL NOTEA Brief History of LogicThe Greek philosopher Aristotle(384–322 B.C.) is generally giventhe credit for the first systemicstudy of logic. His work, however,used ordinary language. Thesecond great period of logic camewith Gottfried Leibnitz(1646–1716), who initiated the useof symbols to simplify complicatedlogical arguments. This treatmentis referred to as symbolic logic ormathematical logic. In symboliclogic, symbols and prescribed rulesare used very much as in ordinaryalgebra. This frees the subjectfrom the ambiguities of ordinarylanguage and permits the subjectto proceed and develop in amethodical way. It was, however,Augustus De Morgan (1806–1871)and George Boole (1815–1864)who systemically developedsymbolic logic. The “algebra” oflogic that they developed removedlogic from philosophy and attachedit to mathematics.EXAMPLE 1 Determining if Sentences Are Statements Decide which ofthe following sentences are statements and which are not.a. Look at me.b. Do you enjoy music?c. What a beautiful sunset!d. Two plus two equals four.e. Two plus two equals five.f. The author got out of bed after 6:00 A.M. today.g. That was a great game.h. x+ 2 = 5.Solution The first three sentences are not statements since the first is a com-mand, the second is a question, and the third is an exclamation. Sentences d ande are statements; d is a true statement while e is a false statement. Sentence f is astatement, but you do not know if it is true or not. Sentence g is not a statementsince we are not told what “great” means. With a definition of great, such as “Ourteam won,” then it would be a statement. The last sentence h. is not a statementsince it cannot be classified as true or false. For example, if x = 3 it is true. Butif x = 2 it is false. ✦✧ ConnectivesA statement such as “I have money in my pocket” is called a simple statementsince it expresses a single thought. But we need to also deal with compoundstatements such as “I have money in my pocket and my gas tank is full.” Wewill let letters such as p, q, and r denote simple statements. To write compoundstatements, we need to introduce symbols for the connectives.4 Chapter L LogicConnectivesA connective is a word or words, such as “and” or “if and only if,” thatis used to combine two or more simple statements into a
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