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

1.1 INTRODUCTION TO SETS / 5 Figure I.I If a set A is not a subset of a set B, we write A Example 11 Let A be the set of points inside the left circle in Figure 1.1 and let B be the s

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PRINTED IN 'fHE UNI1'ED STATES OF AMERICA

To our mothers

• 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

� 2

Coordinate

Systems and Graphs

6.1-6.5 . - Probability

' 6.6 Bayes' Formula and Stochastic Processes

1

8

Applications (See table below.)

7 • 8.1 • 8.2 • 8.3 • 8.4-8.5 • 8.6 •

9

10

r • r-•

�,-

r -r

7 Statistics

10 Computers

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 2 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 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 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 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 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 7 discusses basic concepts in statistics Section 7 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 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 applica­tions 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 9 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 pro­grams 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 program­ming 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

Elizabeth Berman-Rockhurst

College

Alan I Brooks-Sperry UNIVAC

Jerry Ferry-Christopher Newport

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 4 can be written

2 / 1: SET THEORY

letter T can be written

!Taft, Tyler, Taylor, Truman}

We shall denote sets by uppercase letters such as A, B, C, and mem­ bers of a set by lowercase letters such as a, b, c, With this notation,

an arbitrary set with five members might be written

A = I a, b, c, d, e}

To indicate that an element a is a member of the set A, we shall write

a EA, which is read "a is an element of A" or "a belongs to A." To indicate that the clement a is not a member of the set A, we shall write

a i A, which is read "a is not an element of A" or "a does not belong to A."

A = {I, 2, 3, 4, 5, 61 7, 8; 91, which we have described here by listing its elements Since A consists precisely of those positive integers that arc less than 10, this set can be written in set-builder notation as

A = { x I x is a positive integer less than 10}, which is read, "A equals the set of all x such that x is a positive integer less than 10."

In this notation x denotes a typical element of the set, the vertical bar I is read "such that," and following the bar are the conditions that x must satisfy to be a member of the set A

The following examples give further illustrations of this notation

Example 2 The set of all IBM stockholders can be written in set-builder notation as

{ x I x is an IBM stockholder),

which is read "the set of all x such that xis an IBM stockholder." Note that it would be very inconvenient to list all the members of this set

1.1 I NTRODUCTION TO SETS / 3

Example 3 { x I xis a letter in the word stock} is read "the set of all x such that x

is a letter in the word stock." This set can also be described by listing its elements as

{ s, t, o, c, k 1

Two sets A and B are said to be equal if they have the same elements,

in which case we write

A = B

Exam pie 4 If A is the set

and B is the set

Ix I x is a positive integer and x2 < 25 I, then

A = B

Example 5 Consider the sets of stocks

A = IIBM, Du Pont, General Electric}

and

B = {Du Pont, General Electric, IBM}

Even though the members of these sets are listed in different orders, the sets A and B are equal since they have the same members

It is customary to require that all members of a set be distinct; thus when describing a set by listing its members, all duplications should be oP,leted

Example 6 In a study of the effectiveness of antipollution devices attached to the

exhaust systems of 1 1 buses, the following percentage decreases in carbon monoxide emissions were observed:

4 / 1: SET THEORY

has no members since no integer has a square that is negative A set with

no members is called an empty set or sometimes a null set Such a set

is denoted by the symbol 0

Consider the sets

B = {a, e, i, o, u} and A = {a, o, u}

Every member of the set A is also a member of the set B This suggests the following definition

If every member of a set A is also a member of a set B, then we say

A is a subset of B and write

All possible subsets of A are

0, {a}, {bl, {cl, {a, bl, (a, c}, {b, c}, {a, b, c}

Example 9 If S is the set of all stocks listed with the New York Stock Exchange on

July 26, 197 4 and Tis the set of all stocks on the New York Stock Exchange that traded over 100 shares on July 26, 1974, then

Example 10 If A is any set, then

TCS

1.1 INTRODUCTION TO SETS / 5

Figure I.I

If a set A is not a subset of a set B, we write

A<! B

Example 11 Let A be the set of points inside the left circle in Figure 1.1 and let B

be the set of points inside the right circle Then

A <!B and B<!A

Example 12 Let A be the set of points inside the larger circle in Figure 1.2 and let B

be the set of points inside the smaller circle Then

(c) a E A (f) A E {al

(c) 7 (j: A (f) 3 E A

3 Consider the set of water pollutants

A = {sulfur, crude oil, phosphates, mercuryj

Answer the following as true or false

(a) sulfur E A

(c) arsenic (j: A

(b) phosphates (j: A

(d ) oil E A

4 Let A = Ix J x is a real number and x2 = 9 j List the elements of A

5 In each part form a set from the letters in the given words:

(a) AARDVARK (b) MISSISSIPPI (c) TABLE

6 Write A = { 1, 2, 3, 4, 5j in set-builder notation

7 Write the following in set-builder notation:

(a) the set of U.S citizens;

( b) the set of U.S citizens over 40 years of age

3 Let A = { 1, 2, 3, 4} Which of the following sets are equal to A?

(a) {3,2,1,4j (b) {1,2,3} (c) {1,2,3,4,0)

(d) {x J xis a positive integer and x2 � 16)

(e) Ix J xis a positive real number and x � 4)

( f) { x J x is a positive integer and x < 4)

9 Consider the set of psychological disorders

A = I schizophrenia, paranoia, depression, megalomania)

Which of the following sets are equal to A?

(a) I schizophrenia, paranoia, depression I

(b) (schizophrenia, paranoia, megalomania, depression)

IO Which of the following sets are empty?

(a) {x J xis an integer and x2 = 4j

(b) Ix Ix is an integer and x2 = -4)

(c) lxJxis a real number satisfying x2+ 1 = Oj

11 List all subsets of the set { 2, 5}

12 List all subsets of the set (Roosevelt, Truman, Kennedy)

13 List all subsets of

(a) {a1,a.z,a.a} (b) [25

S = the set of points inside the square

T = the set of points inside the triangle

C = the set of points inside the circle and let x and y be the indicated points Answer the following as true or false:

8 / 1: SET THEORY

19 Is it true that f2f E f2f? Is it true that f2f C f2f?

20 Is the set of letters in the word latter the same as the set of letters in the word later?

l.Z UNION AND INTERSECTION OF SETS

We all know that the operations of addition, subtraction, multiplication, and division of real numbers can be used to solve a variety of problems Analogously, we can introduce on sets operations that can be used to solve many important problems In this section we shall discuss two such opera­ tions and in later sections we shall illustrate their applications

If A and B are two given sets, then the set of all elements that belong

to both A and B is a new set, called the intersection of A and B;

it is denoted by the symbol

A nB

Example 13 If, as in Figure 1.4, A is the set of points inside the circle on the left

and B is the set of points inside the circle on the right, then A n B is the set of points in the shaded region of the figure

A

Example 14 Let

A = {a, b, c, d, e}, Find A n B, A n C, and B n C

Figure 1.4

B = { b, d, e, g}, C = {a, h}

1.2 UNION AND INTERSECTION OF SETS / 9

Solution The only elements that belong to both A and Bare b, d, and e Therefore

Two sets that have no common elements, like B and C in Example 14,

are called disjoint sets

Example 15 In Figure 1.5, let A, B, C, and D be the sets of points inside the indi­

cated circles Since A and B overlap,

We can define intersections of more than two sets as follows:

The intersection of any collection of sets is the set of elements that belong to every one of the sets in the collection

Example 16 If A, B, and C are the sets of points inside the circles indicated in

Figure 1.6, then the intersection of these sets, denoted by

AnBnC

is the shaded region in the figure

Example 18 Let S be the set of stocks on the New York Stock Exchange that have

paid a dividend for each of the past 40 years, and let R be the set of railroad stocks listed on the New York Stock Exchange Describe the set

S nR

Solution The members of S n R belong to both S and R, so that S n R consists of

all railroad stocks on the New York Stock Exchange that have paid a dividend for each of the past 40 years

If A and B are two given sets, then the set of all elements that belong

to either A or B or both is a new set, called the union of A and B;

it is denoted by the symbol

A UB

1.2 UNION AND INTERSECTION OF SETS / 1 1

Example 20 If A and Bare the sets of points inside the indicated circles in Figure

1.7a, then A u Bis the set of points in the shaded region shown in Figure l.7b

Figure 1.7 Example 21 Let A be any set Find A u f2f

Solution The members of A U f2f are those elements that lie either in A or f2f or

both Since f2f has no elements, we obtain

A u f2f = A

We can define unions of more than two sets as follows:

Given any collection of sets, their union is the set of elements that belong to one or more of the sets in the collection

Example 22 If A, B, and C are the sets of points inside the circles indicated in

Figure 1.Sa, then the union of these sets, denoted by

A uBu C

is the shaded region in Figure 1.8b

Example 24 Let A, B, and C be the points inside the circles indicated in Figure 1.9

Shade the sets

1.2 UNION AND INTERSECTION OF SETS / 13

c

A UB

Figure 1.10

Solution (b) To find C n (A U B) we intersect C with the shaded set A U B

in Figure 1.10; this yields the shaded set in Figure 1.11

Solution (c) We begin by shading the sets C n A and C n B; thiB gives the

diagrams in Figure 1.12 To find (C n A) u (C n B), we form the union of the shaded sets C n A and C n B; this yields the shaded set in Figure 1.13 Observe that the sets C n (A u B) and (C n A) u (C n B) obtained in parts (b) and ( c) are identical We have thus established the following ha.sic law of sets:

{C n Al u {C n Bl

}'igure 1.13

t John Venn (1834-1923)-Venn was the son of a minister He graduated from Gon­ ville and Caius College in Cambridge, England in 18.53, after which he pursued theo­ logical interest.s as a curate in the parishes of London As a result of his contact with intellectual agnostics and the works of Augustus DeMorgan, George Boole, and John Stuart Mill, Venn became interested in logic In addition to his work in logic, he made important contributions to the mathematics of probability He was an accomplished linguist, a botanist, and a noted mountaineer

1.2 UNION AND INTERSECTION OF SETS / 15

Example 25 In an experiment with hybrid corn, the corn plants were classified into

sets as follows :

Q = quick-growing

W = all white kernels

R = rust resistant

Y = all yellow kernels

Describe the characteristics of the plants in the following sets:

(a) Q n Y (c) (R n Q) u (Rn Y)

(b) R u W

(d) Q u (W n Y)

Solution (a) The plants in Q n Y are in both Q and Y Thus Q n Y consists of

quick-growing, yellow-kerneled plants

Solution (b) The plants in R U W arc either in R or W Thus R U W consists

of plants that arc either rust resistant or white-kerneled

Solution (c) The plants in R n Q are rust resistant and quick-growing The

plants in R n Y are rust resistant and yellow-kerneled Thus (R n Q) u

(R n Y) consists of plants that are either rust resistant and quick-growing

or rust resistant and yellow-kerneled

Solution ( d) The set W n Y is empty since the kernels cannot be both all

white and all yellow Therefore,

16 I 1: SET THEORY

Figure 1.14

3 In Figure 1.14, let

S = the set of points inside the square

T = the set of points inside the triangle

C = the set of points inside the circle

Shade the following sets:

6 In each part determine if the given sets are disjoint

(a) {a,b, dj, {e, f, g}

(b) {l,2, 31, {3,7,9}

(c) f2f, { 1, 2}

(d) !hook, candle, bell}, {page, fire, ring)

For Problems 7-9, refer to Figure 1.15 and let

S = the set of points inside the square

T = the set of points inside the triangle

C = the set of points inside the circle and let v, w, x, y, z be the indicated points

1.2 UNION AND INTERSECTION OF SETS / 17

8 Answer the following as true or false

(a) x E CUT

(c)xECUSUT

(e) v E CU SU T

(b) x E Cu S (d) y E Cu C

9 Answer the following as true or false

(b) CU(AnB) =(CUA) n(CUB)

11 Use Venn diagrams to establish the second distributive law

CU(A nB) =(CUA) n(CUB)

12 Let A= {ATT, IBM, GE}, B = !Du Pont, Burroughs, GE, Kodak), C =

{Avco, Sun Oil, IBM, GE, Du Pont} Compute

18 / 1 : SET THEORY

14 Use Venn diagrams to establish the following

(a) (A n B) n c = A n (B n C) (b) (AU B) UC= AU (BU C)

15 Use Venn diagrams to establish the following

(a) AC A U B

(b) AnBCA

and and

BCAUB AnBCB

16 Use Venn diagrams to establish the following

(a) If AC C and BC C, then (AU B) C C

(b) If CC A and CC B, then CC (A n B)

17 The personnel department of a major company classifies its employees into the following categories:

M = the set of all male employees

F = the set of all female employees

A = the set of all administrative employees

T = the set of all technical employees

S = the set of all employees working for the company at least 5 years

Describe the members of the following sets

(d) AU TU F (e) Mn An 8

18 Let M, F, A, and 8 be the sets in Exercise 17 Let x designate a male employee who has worked for the company at least 5 years Answer the fol­lowing as true or false

C = the set of all policy holders who are over 25 years of age

D = the set of all policy holders who are over 20 years of age

M = the set of all male policy holders

F = the set of all female policy holders

1.3 COMPLEMENTATION AND CARTESIAN PRODUCT OF SETS I 19

Describe the policy holders in the following sets:

(a) A n B (h) A u B (c) A n C n M

(d) .4 U D U F (e) B n (D U F)

20 Let A , B, C, D, M, and F be the sets in Exercise 19 Write the following sets using unions and intersections of A, B, C, D, M, and F:

(a) the set of all female policy holders who are over 25 years of age;

(b) the set of all policy holders who are either male or drive cars with engines that are over 200 horsepower;

( c) the set of all female policy holders over 20 years of age who drive cars with engines that are over 250 horsepower;

(d) the set of all male policy holders who are either over 25 years of age or drive cars with engines that are over 200 horsepower

1.3 COMPLEMENTATION AND CARTESIAN PRODUCT OF SETS

In this section we introduce some other set operations that will be useful

in our later work

Given a set A, we may want to consider those elements that arc not in A

Usually, however, there are elements not in A that are extraneous to the problem being studied For example, suppose A is the set of all positive numbers and we talk about "an element x that is not in A " Clearly, if xis

a negative number, then x is not in A However, if x is a cabbage or a walrus, then x is also not in A since cabbages and walruses are not positive numbers If we are concerned with a problem about real numbers, then what we really mean when we talk about

"an element x that is not in A"

is

"a real number x that is not in A"

-cabbages and walruses are extraneous to the problem

One way of eliminating extraneous elements is to specify in advance that all elements in the problem under consideration come from some fixed set U, called the universal set for the problem For example, in a problem concerned with real numbers, we might agree to take the universal set U to

be the set of all real numbers Then, if we speak of an "element x that is not

in A," we must mean

"a real number x that is not in A"

since we have agreed that all elements come from the universal set U of real numbers

In many problems, the universal set is intuitively clear from the nature

of the problem and is not explicitly stated For example, in problems

be the set of all stocks on the New York Stock Exchange, and so on

In Figure 1.16, we describe the above ideas using a Venn diagram The points inside the rectangle form the universal set U Since all the sets under study must have their members in U, they lie inside the rectangle In particular, if A is the set of points inside the circle, then the shaded region

in Figure 1.16 forms the set of all elements not in A

If U is the universal set and A is a subset of U, then the set of all elements in U that are not members of A is called the complement

of A It is deMted by the symbol

A'

Example 26 Let A be the set of positive real numbers; that is,

A = { x I x is a real number satisfying x > 0}

Find A'

Solution Since A is a set of real numbers, it is natural to take the universal set U

to be the set of all real numbers Thus A' is the set of real numbers that do

not satisfy x > 0; that is,

A' = { x I x is a real number satisfying x ::; 0}

Example 27 Let W be the set of United States citizens who file tax returns listing

less than Sl0,000 taxable income in 1974 Find the complement of W

1.3 COMPLEMENTATION AND CARTESIAN PRODUCT OF SETS / 2 1

Solution Since W is a set of United States citizens filing tax returns in 1974,

it is reasonable to let the universal set U be the set of all U.S citizens filing tax returns in 1974 Thus W' is the set of all U.S citizens filing tax returns in 1974 listing at least $10,000 taxable income

Solution (a) By definition of complement, U' consists of all elements that are

not in U But every element lies in U, so that

U' = fZ)

Solution (b) The set fZ)' consists of all elements that are not in fZ) Since fZ)

has no elements, all elements are outside of fZ); thus

fZJ' = u

Example 30 Let A, B, and C be the sets of points inside the circles indicated in

Figure 1.17 Shade the sets:

(a) A' n B n C

c (b) A n B' n C'

Figure 1.17

u

22 / 1: SET THEORY

c Figure 1.18

Solution (a) The members of A' n B n C belong to both Band C but not A

This yields the shaded region in Figure 1.18

Solution (b) The members of A n B' n C' belong to A but not to B and not

to C This yields the shaded region in Figure 1.19

There arc a number of useful properties of the complementation opera­tion We shall discuss a few of the more important ones here We begin with the following two important results, called DeMorgan' st laws

c Figure 1.19

t Augustus DeMorgan (1806-1871), British mathematician and logician-DeMorgan, the son of a British army officer, was born in Madura, India He graduated from Trinity College in Cambridge, England in 1827, but was denied a teaching position there for refusing to subscribe to religious tests He was, however, appointed to a mathematics professorship at the newly opened University of London

He was a man of firm principles who was described as "indifferent to politics and society and hostile to the animal and vegetable kingdoms." He twice resigned his teaching position on matters of principle (but later returned), refused to accept honorary degrees, and declined memberships in many learned societies

He is best known for his work Formal Logic which appeared in 1847; but he also wrote papers on the foundations of algebra, philosophy of mathematical methods, and prob­ ability well several successful elementary textbooks

1.3 COMPLEMENTATION AND CARTESIAN PRODUCT OF SETS / 23

If A and B are any two sets, then

(A UB)' = A' nB' (A nB)' =A' uB'

first DeMorgan law second DeMorgan law

We shall verify the first DeMorgan law using Venn diagrams and leave the second as an exercise The set

(A u B)' consists of all elements that are not in

A UB

This is given by the shaded region in Figure l.20a

In Figures l.20b and l.20c we have shaded the regions A' and B' and in Figure l.20d we formed their intersection to obtain

A' n B'

Comparing Figures l.20a and 1.20d we see that

(A UB)' =A' n B', which is the first law of DeMorgan

The next example illustrates DeMorgan's laws

(Au B)' = {c, h}

On the other hand,

A'={c, f,g,h} and B' = {a, c, e, h},

24 / 1: SET THEORY

(b) A'

(al

(AUBl'

(d) A'ne' Figure 1.20

(cl

B'

The verifications of the following results are left as exercises

If U is the universal set, and A is a subset of U, then

on It is often the case that the order in which the information is listed is

1.3 COMPLEMENTATION AND CARTESIAN PRODUCT OF SETS / 25

important For example, suppose we want to describe the financial status

of a business firm by listing a pair of numbers, the first number indicating its total assets in millions of dollars and the second indicating its total liabilities in millions of dollars

If the pair for Company A is

(9, 0)

and the pair for Company B is

(0, 9) , then Company A is in excellent financial position, while Company B is likely on the verge of bankruptcy Thus, even though the pairs

(9,0) and (0, 9)

involve the same numbers, they convey very different information because

of the order

This notion of ordered data gives rise to the following concept

An ordered pair (a, b) is a listing of two objects a and b in definite order The element a is called the.first entry in the ordered pair and the element b is called the second entry in the ordered pair

Two ordered pairs are called equal if they list the same objects in the same order Thus, for (a, b) and (c, d) to be equal ordered pairs, we must have

a = c and b = d

We are now in a position to define the last set operation we shall need

If A and B are two sets, the set of all ordered pairs (a, b) , where

a E A and b E B, is called the Cartesiant product of A and B ; it is denoted by the symbol

A X B

t Rene Descartes (1596-1650)-Descartes wa.s the son of a government official He graduated from the University of Poitiers with a law degree at age 20, after which he went to Paris, where he lived a dissipative life a.s a man of fashion In 1618 he joined the

army of the Prince of Orange, where he worked as a military engineer Descartes was a genius of the first magnitude having made major contributions in philosophy, mathe-

mat ic s, physiology, and science in general His work in mathematics gave new direction

to science; along with William Harvey, he is considered founder of modern physiology

(t, 3) The reader will find it instructive to compute the set A X B by this method

It is possible to consider Cartesian products of more than two sets For example, the Cartesian product

1.3 COMPLEMENTATION AND CARTESIAN PRODUCT OF SETS / 27

where

a E A, b E B, and c E C;

and more generally, the Cartesian product

of n sets, where n is a positive integer, consists of all ordered n-tuples

(a1, <22, • • • , a,.) ,

where

,

Example 33 A firm that conducts political polls classifies people for its files ac­

cording to three characteristics

S (sex) : m = male f = female

P (political : d = Democrat r = Republican i = independent

registration)

income who is registered as a Republican

The Cartesian product

f

Figure l.22

B = the set of stocks traded on the New York Stock Exchange that have a price-to-earnings ratio of no more than 12

Describe the following set�

D = the set of all Democratic U.S senators

Specify a universal set and compute D'

1.3 COMPLEMENTATION AND CARTESIAN PRODUCT OF SETS / 29

7 Let C be the set of all consonants in the English alphabet Specify a universal set and compute C'

8 Let A, B, and C be the sets of points in Figure 1.17 Shade the following sets

(a) A n B' n C (b) A n B nC' (c) A ' n B n C' (d) A'n B' n C

9 Let A, B, and C be the sets of points in Figure 1 17 Shade the following sets (a) A1 nB1 n c� (b) (A U B UC)' (c) (A n B nO)' (d) [(A U B) n O]'

10 Use a Venn diagram to verify that if A C B, then B' C A'

11 Use a Venn diagram to verify the second DeMorgan law:

(A n B) ' = A' U B'

12 Use a Venn diagram to verify the following DeMorgan laws for three sets

A, B, and C:

( A u B u C) ' = A' n B' n C' (A n B n C) ' = A' u B' u C'

13 (a) Conjecture DeMorgan laws for four sets A, B, C, and D

(b) Conjecture DeMorgan laws for n sets A1, A2, • • • , A,

14 Explain why the following are true

18 Let A = {a, b, c, dl and B = {0, l, 2 l

(a) List the elements i n A X B

(c) List the elements in A X A

( b) List the elements in B X A ( d) List the elements in B X B

19 Let A = {a, b l , B = ( 1, 2 1 , and C = {x, y, zl List the elements of

(a) Show that the various paths that the rat can take can be represented as the elements of the Cartesian product

E X T X T,

where

E = {a, b, c\, T = {l, r), l = left, r = right

(b) List all possible paths

Reword section

1.4 COUNTING ELEMENTS IN SETS / 31

22 A menu lists a choice of soup, salad, or juice for an appetizer; a choice of beef, chicken, or fish for the entree; and a choice of pie or fruit for dessert

A complete dinner consists of one choice for each course Draw a tree for the possible complete dinners

(a) How many different complete dinners are possible?

(b) If a man refuses to eat chicken or pie, how many different complete dinners can he choose?

(c) A certain customer eats pie for dessert if he has soup as an appetizer; otherwise he chooses fruit for dessert How many different complete meals are available to him?

1.4 COUN TING EL EMEN T S IN SETS

Suppose we know the number of elements in a set A and the number of elements in a set B What can we say about the number of elements in

A U B and A X B? In this section we investigate problems like these and illustrate some of their applications

If S is a set with a finit,e number of elements, then we shall denote the number of elements in S by the symbol

n ( S) Example 34 Consider the sets

n(A u B) = n(A) + n(B) if A and B are disjoint sets