Table Of ContentHOWARD ANTON
BERNARD KOLMAN
Drexel University
Applied
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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
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PERMISSION IN WRITING FROM THE PUBLISHER.
ACADEMIC PRESS, INC.
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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
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Prerequisites
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5 • I · I · I · I
6.1-6.5 • I I I I I I
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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
