Applied Finite Mathematics PDF
Preview Applied Finite Mathematics
HOWARD ANTON BERNARD KOLMAN Drexel University Applied -nite at e a t - o s ACADEMIC PRESS New York San Francisco London A Subsidiaryof Harcourt BraceJovanovich,Publishers COPYRIGHT © 1974, BY ACADEMIC PRESS, 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 AND RETRIEVAL SYSTEM, WITHOUT PERMISSION IN WRITING FROM THE PUBLISHER. ACADEMIC PRESS, INC. Ill Fifth Avenue, New York, New York 10003 United Kingdom Edition published by ACADEMIC PRESS, INC. (LONDON) LTD. 24/28 Oval Road, London NW1 Library of Congress Cataloging in Publication Data Anton, Howard. Applied finite mathematics. Includes bibliographical references. 1. Mathematics-1961- I. Kolman, Bernard, Date joint author. II. Title. QA39.2.A56 510 73-18972 ISBN 0-12-059550-8 PRINTED IN THE UNITED STATES OF AMERICA To our mothers PREFACE This book presents the fundamentals of finite mathematics in a style tailored for beginners, but at the same time covers the subject matter in sufficient depth so that the student can see a rich variety of realistic and relevant applications. Since many students in this course have a minimal mathematics background, we have devoted considerable effort to the pedagogical aspects of this book—examples and illustrations abound. We have avoided complicated mathematical notation and have painstakingly worked to keep technical difficulties from hiding otherwise simple ideas. Where appropriate, each exercise set begins with basic computational "drill" problems and then progresses to problems with more substance. The writing style, illustrative examples, exercises, and applications have been designed with one goal in mind : To produce a textbook that the student will find readable and valuable. Since there is much more finite mathematics material available than can be included in a single reasonably sized text, it was necessary for us to be selective in the choice of material. We have tried to select those topics that we believe are most likely to prove useful to the majority of readers. Guided by this principle, we chose to omit the traditional symbolic logic material in favor of a chapter on computers and computer program ming. The computer chapter is optional and does not require access to any computer facilities. Computer programming requires the same kind of logical precision as symbolic logic, but is more likely to prove useful to most students, since computers affect our lives on a daily basis. In keeping with the title, Applied Finite Mathematics, we have included a host of applications. They range from artificial "applications" which are designed to point out situations in which the material might be used, all the way to bona fide relevant applications based on "live" data and xi xii / PREFACE mmWmìm ËÊUâÉiéàÎaÊêKÈ Prerequisites 2 3 4 5 6.1 6.6 -6.5 1 2 • I I I I I I 3 • I · ! 4 • I · I Γ 5 • I · I · I · I 6.1-6.5 • I I I I I I Topic to be covered 6.6 • I I · I 7 • I · I · I · 8.1 • I I · I 8.2 • I I · I 8.3 • I I · I 8.4-8.5 • I · [ I · I I · I 8.6 • I · I I · I I · I 9 PREFACE / xiii actual research papers. We have tried to include a balanced sampling from business, biology, behavioral sciences, and social sciences. There is enough material in this book so that each instructor can select the topics to fit his needs. To help in this selection, we have included a discussion of the structure of the book and a flow chart suggesting possible organizations of the material. The prerequisites each for topic are shown in the table below the flow chart. Chapter 1 discusses the elementary set theory needed in later chapters. Chapter 2 gives an introduction to cartesian coordinate systems and graphs. Equations of straight lines are discussed and applications are given to problems in simple interest, linear depreciation, and prediction. We also consider the least squares method for fitting a straight line to empirical data, and we discuss material on linear inequalities that will be needed for linear programming. Portions of this chapter may be familiar to some students, in which case the instructor can review this material quickly. Chapter 3 is devoted to an elementary introduction to linear program ming from a geometric point of view. A more extensive discussion of linear programming, including the simplex method, appears in Chapter 5. Since Chapter 5 is technically more difficult, some instructors may choose to limit their treatment of linear programming entirely to Chapter 3, omitting Chapter 5. Chapter 4 discusses basic material on matrices, the solution of linear systems, and applications. Many of the ideas here are used in later sections. Chapter 5 gives an elementary presentation of the simplex method for solving linear programming problems. Although our treatment is as elemen tary as possible, the material is intrinsically technical, so that some in structors may choose to omit this chapter. For this reason we have labeled this chapter with a star in the table of contents. Chapter 6 introduces probability for finite sample spaces. This material builds on the set-theory foundation of Chapter 1. We carefully explain the nature of a probability model so that the student understands the relation ship between the model and the corresponding real-world problem. Section 6.6 on Bayes' Formula is somewhat more difficult than the rest of the chapter and is starred. Instructors who omit Bayes' Formula should also omit Section 8.1 which applies the formula to problems in medical diagnosis. Chapter 7 discusses basic concepts in statistics. In Section 7.7 the student is introduced to hypothesis testing by means of the chi-square test, thereby exposing him to some realistic statistical applications. Section 7.4 on Cheby- shev's inequality is included because it helps give the student a better feel for the notions of mean and variance. We marked it as a starred section since it can be omitted from the chapter without loss of continuity. An instructor whose students will take a separate statistics course may choose to omit this chapter entirely. Chapter 8 is intended to give the student some solid, realistic applica tions of the material he has been studying. The topics in this chapter are drawn from a variety of fields so that the instructor can select those sections that best fit the needs and interests of his class. Chapter 9 introduces the student to computers and programming. There is no need to have access to any computer facilities. It is not the purpose of this chapter to make the student into a computer expert; rather we are concerned with providing him with an intelligent understanding of what a computer is and how it works. We touch on binary arithmetic and then proceed to some FORTRAN programming and flow charting. We have starred this chapter since we regard it as optional. ACKNOWLEDGMENTS We gratefully acknowledge the assistance of our reviewers, Elizabeth Berman (Rockhurst College), Daniel P. Maki (Indiana University), and J. A. Morene (San Diego City College) ; their penetrating comments greatly improved the entire manuscript. We also express our appreciation to Robert E. Beck (Villanova University), Alan I. Brooks (UNIVAC), and Leon Steinberg (Temple University) for their invaluable assistance with the computer material. We are grateful to the International Business Machines Corporation and the UNIVAC Division of the Sperry Rand Corporation for providing illustrations for the material on computers. We wish to express our thanks to our problem solvers Albert J. Herr and John Quigg. We also thank our typists Miss Susan R. Gershuni who skillfully and cheerfully typed most of the manuscript, and Mrs. Judy A. Kummerer who also helped with the typing; finally thanks are also due to the staff of Academic Press for their interest, encouragement, and cooperation. Case: Frank Stella's Hyena Stomp, 1962, courtesy of the Tate Gallery, London. A painter whose principal concern is with the formal problems generated by the canvas itself and the rigorous development of color re lationships, Stella is represented in major private and public collections. He was born in Maiden, Massachusetts in 1935. He studied at the Phillips Academy and at Princeton under Stephen Green and William Seitz. XV 1 SET THEORY A herd of buffalo, a bunch of bananas, the collection of all positive even integers, and the set of all stocks listed on the New York Stock Exchange have something in common; they are all examples of objects that have been grouped together and viewed as a single entity. This idea of grouping objects together gives rise to the mathematical notion of a set, which we shall study in this chapter. We shall use this material in later chapters to help solve a variety of important problems. 1.1 INTRODUCTION TO SETS A set is a collection of objects; the objects are called the elements or members of the set. One way of describing a set is to list the elements of the set between braces. Thus, the set of all positive integers that are less than 4 can be written {1,2,3}; the set of all positive integers can be written {1,2,3, ...}; and the set of all United States Presidents whose last names begin with the l
The list of books you might like
