HOWARD ANTON BERNARD KOLMAN Drexel University Applied -nite at eat-os ACADEMIC PRESS New York San Francisco London A Subsidiary of Harcourt Brace Jovanovich, Publishers COPYRIGHT © 1974, BY ACADEMIC PRESS, I N C 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 A C A D E M I C PRESS, I N C Ill Fifth Avenue, New York, New York 10003 United Kingdom Edition published by A C A D E M I C PRESS, I N C ( L O N D O N ) L T D 24/28 Oval Road, London NW1 Library of Congress Cataloging in Publication Data Anton, Howard Applied finite mathematics Includes bibliographical references Mathematics-1961I Kolman, Bernard, Date joint author II Title QA39.2.A56 510 73-18972 ISBN - - 5 - 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 ËÊÉiéàỴêKÈ Prerequisites 6.1 6.6 -6.5 • I • I· ! • I· I • I· I· I· I Topic to be covered 6.1-6.5 6.6 8.1 8.2 8.3 8.4-8.5 8.6 •I •I I I I I Γ I I I I I I·I • I· I • • • • I I I I I · [ I· I • I· I I· I · I · I·I I· I I·I I·I I· I 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 discusses the elementary set theory needed in later chapters Chapter 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 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 Since Chapter is technically more difficult, some instructors may choose to limit their treatment of linear programming entirely to Chapter 3, omitting Chapter Chapter discusses basic material on matrices, the solution of linear systems, and applications Many of the ideas here are used in later sections Chapter 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 introduces probability for finite sample spaces This material builds on the set-theory foundation of Chapter 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 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 Chebyshev'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 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 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 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 or members of the set elements 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 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 460 / ANSWERS TO SELECTED EXERCISES CHAPTER Exercise Set 6.1, page 208 (a) {0,1,2,3,4,5,6,7,8,9,10} (b) {0,1,2, } (c) {x\0< (d) (e) x< 100} {x\x>0} {(Ä, h, h), (Ä, h, t), (A, t, h), (t, h, h), (Ä, i, t), (t, Ä, , («, t, Ä), (t, t91)} ; £ , {a,6},{a,c},{6,c},{a},{6},{c} (a) the same number is obtained on both tosses (b) the first number tossed is (c) a is tossed both times (a) {(m, h, d), (m,h,r), (m,h,i), (m,a,d), (m,a,r), (τη,α,ί), (m,l,r), (m,l,i), (/, M ) , ( / , Μ ) , ( Λ Μ ) , (/, M)> (/,α,0,(/,Μ),(/,^),(/,Ζ,0} (f,a,r), (τη,Ι,ά), (f,a,r), (b) {(m,Ä,r), (ro,a,r), (m,Z,r), (f,h,r), (f,l,r)} (c) (d) { ( / , M ) , ( / , * , i ) , (/,*,10} (b) {*|0 Σ■> - I * no CHAPTER / 463 11 (a) finite discrete (b) infinite discrete (c) (d) infinite discrete continuous Exercise Set 7.2, page 282 3.4 2.888 (a) 3.6 11 $4.60 - 0.2 (b) 13 (a) (b) Exercise Set 7.3, page 293 σχ2 ~ 0.067, σχ~· 0.26 0.81 f y Exercise Set 7.4, page 301 (a) (b) (a) 16 (b) -0.5 (c) (a) at least (b) at least at least 0.91 (c) 1.5 (d) 2.75 24.1 (d) (c) at least 15/16 9.1 at least - ^ 11 0.64 Exercise Set 7.5, page 311 (a) 256/625 (e) 1/625 (b) 256/625 (c) 96/625 (f) (g) 113/625 624/625 (a) 0.042 (b) 0.017 (c) 0.000 (d) 0.083 (e) 0.157 (f) 0.002 0.1- ( ( 2> _L 3 = 0, ci = f, q2 = ì , q* = 0, E = J ψ 11 Pl = 0, p2 = +, P3 = $, = 0, g2 = TfV,ffs= H» ^ = f 13 Columbus should keep going with probability 627 15 ^ male, § female 6 / ANSWERS TO SELECTED EXERCISES Exercise Set 8.6, page 402 Next state 1 Present |~5 si state Lf fJ (a) ς Γ29 If the system is in state 1, the probability that at the next observation the system will be in state is f (b) [i f] D (c) [| fl 51Ί · l_-g"ö [.722 278] [.320 258 11 F"öJ 422] (a) No power of P has all positive entries (b) P2 has all positive entries 13 (a) 15 279 spaces at Kennedy 115 spaces at LaGuardia 107 spaces at Newark (b) 167 CHAPTER Exercise Set 9.2, page 413 (a) 5627 (a) 001 (e) 101 (b) 8796 (c) 3579 (b) 010 (c) 011 (f) 110 (g) 111 (d) 100 ΌΟΟΟ Exercise Set 9.3, page 423 It directs the computer to store the integer value 84 in a memory location designated by the name MAX (a) Increase the real value stored in X by and store the new result back into X (b) Replace the real value stored in X by its square (c) Replace the integer value stored in I by K minus I (a) 5.0 (b) 15.625 (a) (A + B)/C (b) (d) (C + E ) / ( D * * ) (c) 5.75 C/(D + E) (e) (d) (c) 12.5 C/(D**2.0) (A + B ) / or 0.5*(A + B) CHAPTER / 467 READ, I READ, J READ, K READ, L M = I*J*K*L PRINT, M STOP END Exercise Set 9.4, page 430 The program reads a card containing two integers and prints the larger number 1=1 SUM = READ, X SUM = SUM + X 1=1 + IF (I.LE.1000) GO TO AVE = SUM/1000 PRINT, AVE STOP END TIME = XRAY = 0.0728 * TIME + 0.534 PRINT, TIME, XRAY TIME = TIME + IF (TIME.LE.100.0) GO TO STOP END Exercise Set 9.5, page 437 The program prints all the odd integers from to 85 inclusive The program reads a card and prints a if the number on the card is odd and prints a if the number on the card is even The program prints out the tax to be paid, based on the salary read from a card The tax is computed to be 40% of the salary if the salary is greater than $30,000 and 20% of the salary if the salary is $20,000 or less If the salary is between $20,000 and $30,000 the tax is 25% of the salary / ANSWERS TO SELECTED EXERCISES f START J \READ EMPLOYEE/ CODE AND SALARY YES ·/ STOP J NO A1 = SALARY - 10,000.00 DUES = 150.00 NO A2 = S A L A R Y - 15,000.00 DUES = 150.00+ * A1 NO DUES = 150.00+ * A1 PRINT \EMPLOYEE CODE/ AND DUES CHAPTER / f START J READ CUSTOMER ACCOUNT NUMBER NUMBER OF SHARES PRICE YES Y STOP J SETK1 = K1 = K1 = K1 + y ^ v IS K1 = YES PERCENT = 3.25 '' NO y ^ v is K1 = i r NO PERCEN T ~ 2.75 YES PERCENT = 2.50 '' COMMISSION = (NUMBER OF SHARES) * (PRICE) * (PERCENT) INDEX Abscissa, 45 Addition principle, 212 Addition of matrices, 124 Addresses, 412 ALGOL, 440 Alleles, 344 Analog computer, 406 Areas under standard normal table, 441 Arithmetic average, 276 Arithmetic unit, 409 Associative law for matrix addition, 126 for matrix multiplication, 136 Augmented matrix, 107 Average, 276 Axis coordinate, 45 x, 45 y, 45 B Babbage, Charles, 406 BASIC, 440 Basic feasible solution, 165 Basic variables, 169 Bayes' formula, 259, 263 Bayes, Thomas, 259 Beginning variables, 113 curves, Bernoulli experiments, 303 trials, 303 Bernoulli, Jacob, 303 Binary number system, 410 Binomial experiment, 308 Binomial probabilities, 442, 443 table Binomial random variable, 308 Bit, 411 Borei, Emil, 359 Card reader, 408 Carrier, 352 Cartesian coordinate system, 45 Cartesian product, 25-27 number of elements in, 38, 39 Central Processing Unit, 409 Certain event, 205 Chebyshev, Pafnuti Liwowich, 297 Chebyshev's Theorem, 297 Chi-square (χ ), 330 curve, 332 test, 333 COBOL, 440 Coefficient matrix, 139 Column maximum, 363 Combination, 237 Commutative law for matrix addition, 125 Compilation, 422 Complement of a set, 20 471 472 J INDEX Computer (s) analog, 406 digital, 406 general purpose, 406 program, 408 programmer, 408 programming, 408 special purpose, 406 Conditional probability, 247, 248 Consistent, 105 Constant sum games, 361 Constraints, 89 Continuous random variable, 271 Convex set, 93 bounded, 93 corner point in, 94 unbounded, 93 Coordinate, 44 x, 45 V, 45 Coordinate axes, 45 Coordinate system Cartesian, 45 rectangular, 45 Corner point, 94 Critical level, 332 D Dantzig, George B., 175 Data, 405 Degeneracy, 189 Degrees of freedom, 333 De Morgan, Augustus, 22 De Morgan's laws, 22 Departing variable, 179 selection of, 183 Depreciation, 66 Descartes, René, 25, 43 Deviation, 287 Difference of matrices, 127 Digital computer, 406 Disjoint sets, Distributive law for matrices, 136 for sets, 13, 14 Dominant trait, 346 Dual problem, 195 Elementary events, 211 Elements of a set, E m p t y set, Entering variable, 178 selection of, 179 Entries, 122 Equal matrices, 124 Equal sets, Equation (s) linear, 59 system of, 60, 103 Event(s), 204 certain, 205 elementary, 211 impossible, 205 independent, 253, 255 probability of, 202 mutually exclusive, 207 Execution, 422 Expectation of a random variable, 281 Expected frequencies, 330 Expected value of a random variable, 281 Expected winnings, 373 Explicit variables, 169 Feasible solution, 92 Federalist papers, 265 Finite sample space, 211 Finite discrete random variable, 270 % critical level (s), 332 table of, 444 Flow chart(s), 431 F O R T R A N , 414 Frequencies, 278 expected, 330 observed, 330 G Games constant sum, 361 matrix, 361 saddle point for, 365 strictly determined, 365 two-person, 361 zero sum, 361 Gauss, Carl Friedrich, 113 Gaussian curves, 315 Gauss-Jordan elimination, 113 General purpose computer, 406 Genes, 344 Genetics, 329, 343-351, 400 Genotype, 344 Graph, 48 of a probability function, 274 INDEX / 473 H Half plane, 75 Hardy, G H., 347 Hardy-Weinberg Stability Principle, 350 Heredity, 343 Histogram, 314 I Identity matrix, 138 Implicit variables, 169 Impossible event, 205 Inconsistent, 105 Independent events, 253, 255 Infinite discrete random variable, 270 Inflection points, 317 Initial basic feasible solution, 176 Initial state, 390 Integer constant, 416 Interpreter, 409 Intersection of sets, 8, Inverse of a matrix, 143 Invertible matrix, 143 Jordan, Camille, 113 Language (s), 414 (See also Programming languages) Least squares, 70, 71 Lethal gene, 352 Life insurance, 352-359 Linear depreciation, 67 Linear equation (s), 59, 102, 103 system of, 60, 103 Linear inequalities, 75 system of, 75 Linear programming, 89 Linear programming problem (s) dual, 195 nonstandard, 193 primal, 195 standard, 156 Linear system, 103 consistent, 105 inconsistent, 105 solution of, 103 Logic unit, 409 Loop, 426 M Machine language, 414 Magnetic tape, 409 Main diagonal, 138 Markov, Andrei Andreevich, 390 Markov chains, 390 Markov processes, 390 Matrices difference of, 127 distributive laws for, 136 equal, 124 product of, 132 subtraction, 127 sum of, 124 Matrix, 107, 122 addition, 124 augmented, 107 coefficient, 139 entries of, 122 games, 361 identity, 138 inverse of, 143 invertible, 143 main diagonal of, 138 multiplication, 132 negative of, 127 nonsingular, 143 payoff, 361 reduced row echelon form, 111 row operations, 107 size of, 122 square, 122 state, 393 steady state, 397 transition, 387 transpose of, 140 zero, 126 Matrix games, 361 (See also Games) Matrix multiplication, 132 associative law for, 136 Maximality test, 177 Mean, 276 of a random variable, 281 Medical diagnosis, 338-343 Members of a set, Memory, 408 Mendel, Gregor, Johann, 329, 343 Mixed strategies, 373 Morgenstern, Oskar, 359 Mortality table(s), 352 Commissioners 1958 Standard Ordinary, 353 Mosteller, F., 265 Multiplication principle, 229, 230 Mutually exclusive events, 207 472 J INDEX N Negative direction, 44 Negative of a matrix, 127 Nonbasic variables, 169 Nonsingular matrix, 143 Normal curves, 315 standard, 318 n(S), 31 Null set, Number of elements in a set, 31, 38, 39 O Objective function, 89 Observed frequencies, 330 One-to-one correspondence, 44 Optimal solution, 92 Optimal strategy, 375 Ordered pair, 25 Ordinate, 45 Origin, 44, 45 Output, 409 P Paper tape, 409 Payoff matrix, 361 Permutation, 230 r at a time, 234 Pivot column, 181 entry, 180 row, 181 Pivotal elimination, 181 P L / I , 440 Point-slope form, 57 Positive direction, 44 Premiums, 352 Primal problem, 195 Principal, 66 Probability, 202 conditional, 247, 248 Probability function, 273 graph of, 274 Probability model, 211 uniform, 214 Product of matrices, 132 Programming languages, 414 ALGOL, 440 BASIC, 440 COBOL, 440 F O R T R A N , 414 P L / I , 440 Punch card(s), 408 Punched paper tape, 409 Pure strategies, 373 Q Quadrant, 46 Quotient for a row, 183 R Random variable, 269 binomial, 308 continuous, 271 expectation of, 281 expected value of, 281 finite discrete, 270 infinite discrete, 270 mean of, 281 standard deviation of, 290 variance of, 290 Real constant, 416 Real number line, 44 Recessive trait, 347 Rectangular coordinate system, 45 Reduced row echelon form, 111 Regular transition matrix, 397 Relative frequency, 202 Remote terminal, 409 Row minimum, 363 Row operations, 107 S Saddle point, 365 Sample points, 203 Sample space, 203 finite, 211 Scalar (s), 122 Sentinel, 429 Set-builder notation, Set(s), Cartesian product of, 25-27 complement of, 20 convex, 93 disjoint, distributive laws, 13, 14 elements of, empty, equal, intersection of, 8, members of, null, number of elements in, 31, 38, 39 subset of, union of, 10, 11 universal, 19 Venn diagrams for, 14 INDEX J 475 o-units, 296 Simple interest, 65 Simplex method, 175 steps in, 176-183 Size of a matrix, 122 Slack variables, 158 Slope, 52 Slope-intercept form, 56 Solution basic feasible, 165 of an equation, 47 feasible, 92 initial basic feasible, 176 of systems of linear inequalities, 79 optimal, 92 set, 48 Solution set, 48 Source deck, 421 Source program, 421 Special purpose computer, 406 Square matrix, 122 Square roots, table, 444 Standard deviation, 290 Standard linear programming problem, 156 Standard normal curve, 318 State matrix, 393 State vector, 393 States, 386 Steady state matrix, 397 Stochastic process, 390 Store (instruction), 414 Strategies mixed, 373 optimal, 375 pure, 373 Strictly determined matrix games, 365 Subset, Subtraction of matrices, 127 Sum of matrices, 124 System(s), 386 of inequalities, 79 of linear equations, 60, 103 solutions of, 60, 103 Trait dominant, 346 recessive, 347 Transition matrix, 387 regular, 397 Transpose of a matrix, 140 Tree diagram, 26 Two-person games, 361 U Uniform probability model, 214 Union of sets, 10, 11 Universal set, 19 V Variable@) basic, 169 departing, 179 entering, 178 explicit, 169 in FORTRAN, 415 implicit, 169 nonbasic, 169 random, 269 Variance, 290 Venn, John, 14 Venn diagrams, 14 Von Neumann, John, 359 W Wallace, D L., 265 Weinberg, Wilhelm, 347 X z axis, 45 z coordinate, 45 Y T D F G H J *9 O Tableau, 169 initial, 176 Tables areas under standard normal curves, 441 binomial probabilities, 442-443 57, critical levels for x2 curves, 444 square roots, 444 y axis, 45 y coordinate, 45 Z Zero matrix, 126 Zero sum games, 361 ... (m, A, d) (m, A, r) (m, A, i) ( /, A, d) ( /, A, r) ( /, A, i) (m, a, d) (m, a, r) (m, a, i) ( /, a, d) ( /, a, r) ( /, a, i) (m, Z, d) (m, Z, r) (m, I, i) ( /, Z, d) Figure 1.22 ( /, Z, r) ( /,... values of x and y for which the given ordered pairs of integers are equal (a) (*, 7) = (3, 7) (b) (2z, 3) = (6,* /) (c) (4, y + 7) = (2s + 2, 1 4) (d) (x2, 9) = (16, ) Let A = {a, b, c, d) and B = {0,... following sets: (a) CU (SOT) (b) CO (SUT) (c) (COS) UT (d) (CUS)O(COT) In each part determine if the given sets are disjoint (a) {a,ò,d}, {e,f,g (b) {1,2,3}, {3,7,9} (c) , {1,2} (d) {book, candle, bell},