Howard anton bernard kolman applied finite mathematics academic press, geniza, elsevier inc (1978)

APPLIED FINI TE MATHEMATICS SECOND EDITION HOWARD ANTON BERNARD KOLMAN DREXEL UNIVERSITY ACADEMIC PRESS NEWYCRK SAN FRANCISCO LON[X)N A Subsidiary of Harcourt Broce Jovanovich, Publishers Cover art, granted by by Kenneth :\Toland Permission Des Moines Arts Center, Coffin Fine Arts Trust Fund , 1974 Whirl The @ 1974, 1978, COPYRIGH'r ALL NO PART OF 'rHIS BY ACADEMIC PRESS, INC RIGHTS RESERVED PUBLICATION MAY RE REPRODUCf;D OR 'fRANSMITTED I N ANY FORM OR BY ANY �EANS, ELECTRONIC OR MECHANICAL, INCLUDING PHOTOCOPY, RECORDING, OR ANY INFORMATION S'fORACE AND SYSTf:M, RETRIEVAL PERMISSION IN WRITING FROM THE WITHOUT PUBLISHER ACADEMIC PRESS, INC 111 FIFTH AVENUE, NEW YORK, NEW UNITED YORK 10003 KINGDOM EDITION PUBLISHED BY ACADEMIC PRESS, INC., 24/28 ( LONDON) LTD OVAL llOAD, LONDON NW l ISBN: 0-12-059565-6 Library of Congress Catalog Card Number: PRINTED IN 'fHE UNI1'ED STATES 77-90975 OF AMERICA To our mothers NEW FEATURES IN THE SECOND EDITION • • • • • x A new self-contained chapter on the mathematics of finance Stochastic processes are introduced Tree diagrams are used more extensively as a tool in probabilit.y problems Additional exercises BASIC replaces FORTRAN in the computer chapter 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 programming Computer programming requires the same kind of logical precision as symbolic logic, but is more likely to prove useful to most students The computer chapter is optional and does not require access to any computer facilities However, this chapter is extensive enough that the student will be able to run programs on a computer, if desired 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 actual research papers We have tried to include a balanced sampling from business, finance, biology, behavioral sciences, and social sciences xi Set Theory � Coordinate Systems and Graphs ' Linear Programming 6.1-6.5 . - (Geometric) Probability r ' r - 6.6 Matrices and Bayes' Formula and Linear Systems Stochastic Processes Statistics Linear Programming (Algebraic) Mathematics 10 Applications of Computers (See table below.) Finance Prerequisites 2 Topic to be covered 6.1-6.5 6.6 8.1 8.2 8.3 8.4-8.5 8.6 10 • • 6.1 6.6 -6.5 • r-• r• • • • • • • • • • i·I • • • • • • t • • • • • + -, - • t • • • �,I l - • - PREFACE / xiii There is enough materi:1l in this book so that each instructor can select the topics that best fit the needs of the class To help in this selection, wc have included a discussion of the structure of the book and a flow chart suggesting possible organizations of the material The prerequisites for each topic are shown in the table below the flow chart Chapter l discusses the elementary set theory needed in later chapters Chapter gives an introduction to cartesian coordinate systems and graphs Equations of straight lines arc discussed and applications arc 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 omitting Chapter Chapter 3, 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 elementary as possible, the material is intrinsically technical, so that some instructors 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 relationship between the model and the corresponding real-world problem Section 6.6 on Bayes' Formula and stochastic processes is somewhat more difficult than the rest of the chapter and is starred Instructors who omit this section should also omit Section 8.1 which applies the material to problems in medical diagnosis Chapter discusses basic concepts in statistics Section 7 introduces hypothesis testing by means of the chi-square test, thereby exposing the student 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 PREFACE / xiv section since it can be omitted from the chapter without loss of con­ tinuity 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 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 the class Chapter covers a number of topics in the mathematics of finance, The chapter is self contained and includes an optional review section on exponents and logarithms Chapter 10 introduces the student to computers and programming While there is no need to have access to any computer facilities, the material is presented in sufficient detail that the student will be able to run pro­ grams on a computer It is not the purpose of this chapter to make the student into a computer expert; rather we are concerned with providing an intelligent understanding of what a computer is and how it works We touch on binary arithmetic and then proceed to some BASIC program­ ming and flow charting We have starred this chapter since we regard it as optional ACKNOWLEDGMENTS We gratefully acknowledge the contributions of the following people whose comments, criticisms and assistance greatly improved the entire manuscript Robert E Beck-Villanova University College Alan I Brooks-Sperry UNIVAC Christopher Newport - College Samuel L Marateck-New York University J A Moreno -S a n Diego City College Donald E Myers University of - Mary W Gray-A merican University Beryl M Indiana - University Elizabeth Berman-Rockhurst Jerry Ferry Daniel P Maki Green- Oregon College of Education Albert J He rr-D rexel University Robert L Higgins-QUANTICS Leo W Lampone-Spring Garden College Arizona John Quigg-D rexel University Ellen Reed-University of Massachusetts at Amherst James Snow-Lane Community College Leon Steinberg-Temple University William H Wheeler-Indiana State University We also thank our typists: Susan R Gcrshuni, Judy A Kummerer, Amelia Maurizio, and Kathleen R McCabe for their skillful work and infinite patience We thank: IBM, Sperry UNIVAC, and Teletype Corporation for providing illustrations for the computer material Finally, we thank the entire staff of Academic Press for their support, encouragement and imaginative contributions 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 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 can be written { 1, 2, 3}; the set of all positive integers can be written 11, 2, 3, }; and the set of all United States Presidents whose last names begin with the 544 / ANSWERS TO SELECTED EXERCISES Exercise Set 8.3, page 364 I (a) 0.973 (a) 0.0352 0.0179 0.933 (b) ( c) 0.0999 (b) (d) 0.0273 0.280 0.00671 at least $346.51 Exercise Set 8.4, page 375 I (b) , (c) , (f) (a) row 1, colwnn 2; - (b) ( c) row 1, colwnn 2; - (d ) row 1, column 2; row 1, column 2; Player II Player I [ -7 •00•• Player I scissors paper - l :J Player II - stone scissors paper -1 -1 n player R shows one finger, player C shows two fingers 13 firms A and B should each use television Exercise Set 8.5, page 390 I (a) Pi H = t, P2 = H (b) -J, qi (c) t = -h;_, qz = H, E = l.,(l t, P2 = t, q1 = !, qz = t, E = t P1 = t, P2 = t, Pa = 0, qi = l, qz = t, qa = 0, E = t II P1 = 0, Pz = t, Pa = �' qi = 0, q2 = -.h, qa H, E = t P1 = = CHAPTER / 545 13 Columbus should keep going with probability 627 15 i male, ! female Exercise Set 8.6, page 408 Next state [i 1 Present state (a) !J i If the system is in state 1, the probability that at the next observation the system will be in state is l (b) [t l] [ff tt] (c) [t !J [.722 278] 11 13 [.320 ( a) 258 422] No power of P has all positive entries (b) p'l has all positive entries (a) 167 (b) 15 279 spaces at Kennedy 115 spaces at La.Guardia 107 spaces at Newark CHAPTER Exercise Set 9.1, page 417 (a) 243 (f) -h\ (a) (d) 59 a� rtlr (b) (g) (c) (h) 16 (b) 2s (c) t2 (f) (e) (a) 7 (a) (d) 7781 2386 (a) -3.4655 (b) 1f (b) (e) (c) (d) (i) - 1f 64 a2 y3 xa - 6t -.1761 -2.699 (b) 87.664 (d) (c) 9542 (f) J 7323 (c) 4.4817 (e) - -h 546 / ANSWERS TO SELECTED 11 (a) EXERCISES y I -I -2 (b) -3 I '-" ' I\) ! ""' I\) '-" (c) I '-" ' I\) ! (J1 I\) ct> ,, ""' CHAPTER 10 J 547 13 (a) 172 TI1i 17 (a) (b) 3,000 units 3,738 units 15 No (c) Hff (b) * (d) ! Exercise Set 9.2, page 437 (a) $300 (b) $308.39 (a) % (b) 6.167781 % (a) $3150.85 $6346.66 (c) $306.82 6.136355% (c) % per year compounded quarterly (b) 11 (a) $5309 13 10.52% 19 (a) $57,507.39 21 $1855.30 23 (a) $41,001.97 33 (b) $5471 15 25 $4769.05 $3108.61 27 (b) (b) $4100 29 10% (d) $2790.80 (c) $83,721.17 (d) $2069.02 $522.50 (c) 17 (d) 6.18% $175,610.81 (c) (b) $13,677.74 (b) $6450.24 (a) $6181.92 1.551666 years $30,200.99 $520.83 (d) 6.09% (c) $3122.99 (c) (a) $900 (d) $304.50 31 $1 16,263.17 $781.92 (d) $1050.24 CHAPTER 10 Exercise Set 10.2, page 450 (b) 8796 I (a) 5627 (a) (e) 0 0 I 101 (b) (f) 10 110 (c) 3579 (c) 1 ( g) 1 (d) 100 Exercise Set 10.3, page 461 I LET M = 48 (a) Increase the number stored in X by and store the new result back into X (b) Replace the number stored in X by its square (c) Replace the number stored in I by K minus I 548 / ANSWERS TO SELECTED EXERCISES a, c, and d (a) (a) ll (c) * C/ ( D (b) or D) (d) (e) (a) 10 10 * ( A + B ) /2 12.5 C/ ( D + E ) C/ ( D j 2) D) or (f) ( 4/3 ) (C + E ) / ( D * 14 * j (R 2) j 3) One possible solution is: 10 15 ( A + B ) /C (d) (c) 7.5 15.625 ( C + E ) / (D (b) 13 (b) LET A = 7.2 20 LET B 30 LET c 40 LET D 50 LET x 60 PRINT 70 END = 3.8 = 1.6 = 2.7 = ( A + B + C + D ) /4 x One possible solution is: 10 LET A 20 LET B = 30 LET C = 40 LET D = 50 LET E = 60 LET X = (A 70 PRINT X 80 END = l j ) + (B j 2) + (C Another possibility is: j j 10 LET A = l 20 LET B == 30 LET c = i 40 LET D = j 50 LET E 60 LET F = A + B + C 70 PRINT 80 END == F j + D + E j ) + (D j ) + (E j 2) CHAPTER Exercise Set 10.4, I PRINT (a) page 10 I 549 467 "INVENTORY" BACTERIA COUNT K = 600000 ( b) BACTERIA COUNT 600000 (c) BACTERIA COUNT K 600000 BACTERIA COUNT (d) = = = (a) TEMP 98 l, TEMP 98 (b) TEMP Tl T2 98 TEMP 98 T3 2, TEMP 99 l 3 The computer asks the operator to type in a value for X ; the computer calculates Y X2 and prints the values of X and Y = ll 13 10 READ Cl, C2 , C3 20 DATA , 20, 37 30 LET Fl = 40 LET F2 = (9/5) 50 LET F3 = (9/5) 60 PRINT Fl 70 PRINT F2 80 PRINT F3 90 END ( 9/5) * Cl + 32 * * C2 + 32 C3 + 32 One possible solution is : 10 READ X l , Yl , Zl 20 DATA , 7, 30 LET X2 = Xl 40 LET X3 = Xl 50 LET Y2 = Yl 60 LET Y3 = Yl i i i i i i 3 70 LET Z2 = Zl 80 LET Z3 = Zl 90 PRINT "NUMBER" , "SQUARE" , "CUBE" 100 PRINT Xl , X2, X3 110 PRINT Yl, Y2, Y3 120 PRINT Z l , Z2, Z3 130 END 550 I ANSWERS TO SELECTED EXERCISES Exercise Set 10.5, page 477 I It prints the larger of two input values It reads in seven pairs of numbers; then computes and prints out the average for each pair REM PROGRAM OUTPUTS FIRST 200 EVEN INTEGERS 10 FOR I = l TO 200 20 LET J = * I J 30 PRINT 40 NEXT I 50 END 10 REM INTEREST PROGRAM 20 PRINT 30 FOR I = l TO 25 40 LET s 50 PRINT 60 NEXT I 70 END "YEAR" , "AMOUNT OWNED" = 1000 * ( l 05 I, S j I) A possible solution is: 10 REM TOTAL RECEIPTS COMPUTATION 20 REM S = STOCK NO , N = NO SOLD, P 30 REM ENTER S 40 PRINT "S, N, P" 50 INPUT S , N, P 60 IF S THEN 110 70 LET = = UNIT PRICE TO STOP T = N * P 80 PRINT "STOCK NO " , "TOTAL RECEIPTS" 90 PRINT s 100 GO TO 40 110 END T Exercise 10.6, page 483 I It prints all odd integers from to 85 inclusive For an integer K entered by the operator, the program prints a l if K is odd and a if K is even The program prints out the tax to be paid, based on the salary entered by the operator 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 Otherwise the tax is 25 % of the salary CHAPTER 10 I 551 START READ EMPLOYEE CODE AND SALARY YES STOP A1 = SALARY - 10,000.00 DUES - 50.00 A2 = SALARY - 5,000.00 DUES = 50.00 + 07 DUES = 150.00 + 06 • At t �����-; EMPLOYEE CODE PRINT • A1 AND DUES 552 I ANSWERS TO SELECTED EXERCISES START READ CUSTOMER ACCOUNT NUMBER NUMBER OF SHARES PRICE YES STOP SET K1 - o K1 = K1 = K1 + PERCENT = 3.25 PERCENT = 2.50 PERCENT = 2.75 COMMISSION = (NUMBER OF SHARESI (PRICEI (PERCENT) • • INDEX Abscissa, 45 Babbage, Charles, 443 Addition principle, 212 Addresses, 449 BASIC, 452 Basic feasible solution, 165 Basic variables, 169 ALGOL, 487 Alleles, 350 Bayes, Thomas, 262 Addition of matrices, 124 Bayes' formula, 262, 267 Amount, 420, 424, 432 Bayes tree, 267, 268 Analog computer, 443 Beginning variables, 113 Bernoulli Annuity, 429 amount, 432 future value, 432 experiments, 309 trials, 309 Bernoulli, Jacob, 309 ordinary, 430 payment interval, 430 Binary number system, 447 payment period, 430 Binomial experiment, 314 Binomial probabilities table, 489-490 payment size, 432 Binomial random variable, 314 present value, 436 Bit, 448 rent, 432 sum, 432 Borel, Emil, 365 term, 430 APL, 487 Ari:>as under standard normal table, 488 Arithmetic average, 282 Arit hmetic unit, 446 curve, Cnr, 282 Axis Chebyshev's Theorem, 303 Chi-square (x'), 336 run·c, 338 coordinate, 45 x, 45 y, 45 Chebyshev, Pafnuti Liwowich, 303 test, 339 COBOL, 487 Coefficient matrix, 139 553 554 / INDEX Column maximum, 369 Combination, 237 Common logarithm, 415 Commutative law for matrix addition, 125 Difference of matrices, 127 Digital computer, 443 Discount, 428 formula, 429 simple, 428 Compiler, 451 Discounting, 428 Compilation, 460 Disjoint sets, Distributive Jaw Complement of a set, 20 Compound interest, 421 Compound interest formula, 422 for matrices, 136 for sets, 13, Compounding, 421 Dominant trait, 352 Computer (s) analog, 443 digital, 443 Dual problem, 195 gcncrnl purpose, 443 program, 444 programmer, 444 programming, 444 special purpose, 443 Conditional probability, 247, 248 Consistent, 105 Constant, 454 Constant sum games, 367 Constraints, 89 Continuous compounding, 427 formula, 427 Continuous random variable, 277 Conversion period, 421 Convex set, 93 bounded, 93 corner point in, 94 unbounded , 93 Coordinate, 44 x, 45 y, 45 Coordinate axes, 45 Coordinate system Cartesian, 45 rectangular, 45 Corner point, 94 Counter, 472 Critical level, 338 Dantzig, George B., 175 Data, 442 Degeneracy, 189 Degrees of freedom, 339 De Morgan, Augustus, 22 De Morgan's laws, 22 Departing variable, 179 selection of, 183 Depreciation, 66 Descartes, Rene, 25, 43 Deviation, 293 Effective rate of interest, 424 formula, 425, 427 Elementary events, 1 Elements o f a set, Empty set, Entering variable, 178 selection of, 179 Entries, 122 Equal matrices, 124 Equal sets, Equation(s) linear, 59 system of, 60, 103 Event (s), 205 certain, 205 elementary, 1 impossible, 205 independent, 256, 258 probability of, 202 mutually exclusive, 207 Exact interest year, 421 Execution, 460 Expectation of a random variable, 287 Expected frequencies, 336 Expected value of a random variable, 287 Expected winnings, 379 Explicit variables, 169 Exponential notation, 454 Feasible solution, 92 Federalist papers, 271 Fields, 463 Finite sample space, 1 Finite discrete random variable, 276 5% critical level (s), 338 table of, 490 Flow chart(s), 478 FORTRAN, 487 Frequencies, 284 expected, 336 observed, 336 INDEX / 555 Future value, 420, 424, 432 formula, 432 Games constant sum, 367 matrix, 367 saddle point for, 371 strictly determined, 371 two-person, 367 zero sum, 367 Gauss, Carl Friedrich, 1 Gaussian curves, 321 Gauss-Jordan elimination, 1 General purpose computer, 443 Genes, 350 Genetics, 335, 349-357, 406 Genotype, 350 Graph, 48 of a probability function, 280 Half plane, 75 Hardy, G H., 353 Hardy-Weinberg Stability Principle, 356 Heredity, 349 Histogram, 320 Key punch, 444 Language (s) , 487 (See Lethal gene, 358 Line number, 459 Linear depreciation, 67 Linear cquation (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 Logarithms common, 415 natural, 415 Logic unit, 446 Loop, 471 Impossible event, 205 Inconsistent, 105 Magnetic tape, 4'-16 Index, 476 Infinite discrete random variable' 276 Inflection points, 323 Initial basic feasible solution, 176 Initial state, 396 Integers, 1 Interactive mode, 452 Interest compound, 421 continuous compounding, 427 effective rate, 424 formulas, 420, 422, 425, 427 nominal rate, 424 rate, 420 simple, 420 Interest rate, 420 Interpreter, 447 Intersection of sets, 8, Inverse of a matrix, 143 Invertible matrix, 143 Irrational number, 414 Jordan, Camille, 1 Program- Life insurance, 358-365 Identity matrix, 138 Implicit variables, 169 Independent events, 256, 258 al,so ming languages) Least squares, 70, 71 Machine language, 451 Main diagonal, 138 Markov, Andrei Andreevich, 396 Markov chains, 396 Markov processes, 396 Matrices difference of, 127 distributive laws equal, 124 product of, 132 for, subtraction, 127 sum of, 124 Matrix, 107, 122 addition, 124 augmented, 107 coefficient, 139 entries of, 122 games, 367 identity, 138 inverse of, 143 invertible, 143 main diagonal of, 138 multiplication, 132 negative of, 127 nonsingular, 143 136 556 / INDEX payoff, 367 reduced row echelon form' 1 row operations, size of, 122 square, 122 state, :399 steady state, 403 transition, 393 transpose of, 140 zero, 126 l\Iatrix games, 367 (See also Games) Matrix multiplication, 132 associative law for' 136 Maximality test, 177 l\Iean, 282 of a random variable 287 Medical diagnosis, 34i-349 Members of a set, Memory, 444 Mendel, Gregor, Johann 335 349 Mixed strategies, 379 Ylorgrnstern, Oskar, 365 Mortality tnblc(s), 358 Commissioners Standard Ordinary· ' 359 Mosteller, F., 271 Multiplication principle, 229, 230 Mutually exclusive events, 207 ' Natural logarithm, 415 Negative direction, 44 Negative of a matrix, 127 �ominal interest rate 424 Nonbasic variables i69 Konsingular matri�, 143 Normal curves, 321 standard, 324 n(S), 31 Null set, Number of elements in a set, , 38, 39 Numbers irrntional, 414 rational, 412 Objective function 89 Observed frequencies, 336 One-to-one correspondence, 44 Optimal solution, 92 Optimal strategy, 381 Ordered pair, 25 Ordinary annuity, 430 formula, 432 Ordinary interest year, 421 Ordinate, 45 Origin, 44, 45 Output, 446 Paper tape, 446 Path probabilities, 267 Payment interval, 430 period, 430 size, 432 Payoff matrix, 367 Permutation, 230 i· at a time, 234 Pivot column, 181 entry, 180 row, 181 Pivotal elimination, 181 PL/I, 487 Point-Slope form, 57 Positive direction, 44 Premiums, 358 Present value, 420, 424, 436 formula, 426, 436 Primal problem, 195 Principal, 66, 420, 424 Probability, 202 conditional, 247, 248 product principle, 250 Probability function, 279 graph of, 280 Probability model, 211 uniform, 214 Proceeds, 428 Product of matrices, 132 Product principle for probabilities, 250 Programming languages, 451 ALGOL, 487 BASIC, 452 COBOL, 487 FORTRAN, 487 PL/I, 487 Punch card (s), 444 Punched paper tape, 446 Pure strategies, 379 Quadrant, 46 Quotient for a row, 183 Handom variable, 275 binomial, 314 continuous, 277 expectation of, 287 expected value of, 287 finite discrete, 276 infinite discrete, 276 INDEX / 557 mean of, 287 standard deviation of, 296 variance of, 296 Rational numbers, 412 Real number of line, 44 Recessive trait, 353 Rectangular coordinate system, 45 Reduced row echelon form, 1 Hegular transition matrix, 403 Relative freciuency, 202 Remote terminal, 446 Rent, 432 Row minimum, 369 Row operations, 107 · Saddle point, 371 Sample points, 203 Sample space, 203 finite, 211 Scalar(s), 122 Scientific notation, 454 Sentinel, 475 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, member of, null, number of elements in, 31, :18, 39 subset of, union of, 10, 1 universal, 19 Venn diagrams for, 14 u-units, 305 Simple interest, 65, 420 Simple interest formula, 420 Simplex method, 175 steps in, 176-183 Sinking fund, 433 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 program, 460 Sriecial purpose computer, 443 Square matrix, 122 Square roots, table, 491 Standard deviation, 296 Standard linear programming problem, 156 Standard normal curve, 324 State matrix, 399 State vector, 399 States, 392 Steady state matrix, 403 Stochastic process, 251, 396 Strategies mixed, 379 optimal, 381 pure, 381 Strictly determined matrix games, 371 Subset, Subtraction of matrices, 127 Sum, 420, 432 Sum of matrices, 124 System(s), 392 of inequalities, 79 of linear eciuations, 60, 103 solutions of, 60, 103 Tableau, 169 initial, 176 Tables areas under standard normal curve, 488 binomial probabilities, 489-490 exponentials, 493 5% critical levels for X' curves, 490 interest, 494-523 logarithms, 492 square roots, 491 Term, 430 Time sharing, 446 Trailer, 475 Trait dominant, 352 recessive, 353 Transition matrix, 393 regular, 403 Transpose of a matrix, 140 Tree diagram, 26 Two-person games, 367 558 / INDEX Uniform probability model, 214 Venn diagrams, Union of sets, 10, 1 Von Neumann, John, 365 Universal set, Wallace, D L , 271 Weinberg, Wilhelm, 353 Variable (s) basic, 169 departing, 179 entering, 178 explicit, 169 implicit, 169 nonbasic, 169 random, 275 Variance, 2!J6 Venn, John, Word, 449 x x axis, 45 coordinate, 45 y axis, 45 y coordinate, 45 Zero matrix, 126 Zero sum games, 367 A B c D E f G H I J ... IN WRITING FROM THE WITHOUT PUBLISHER ACADEMIC PRESS, INC 111 FIFTH AVENUE, NEW YORK, NEW UNITED YORK 10003 KINGDOM EDITION PUBLISHED BY ACADEMIC PRESS, INC. , 24/28 ( LONDON) LTD OVAL llOAD,... fZf Since A has four elements, we have n(A) = Since B contains infinitely many elements and since we defined n ( S) only when S has finitely many elements, n (B) is undefined Finally, since C... 40 included high performance engine and air conditioning 30 included power steering and air conditioning 20 included power steering and high performance engine 80 included power steering 60 included