Howard anton and bernard kolman (auth ) applied finite mathematics elsevier inc, academic press inc (1974)

Figure 1.1 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.. Since A and B overlap, We can define inte

BERNARD KOLMAN

Drexel University

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at eat-os

ACADEMIC PRESS New York San Francisco London

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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 - 1 2 - 0 5 9 5 5 0 - 8

PRINTED IN THE UNITED STATES OF AMERICA

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

7 8.1 8.2 8.3 8.4-8.5 8.6

9

5 6.1 6.6 -6.5

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 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 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

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

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

1

S E T 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

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

A set is a collection of objects; the objects are called the elements

or members of the set

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, 6, c, With this notation,

an arbitrary set with five members might be written

A = {a, 6, c } d, e}

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

a 6 A,

which is read "a is an element of A" or "a belongs to A 11 To indicate that

the element a is not a member of the set A, we shall write

use what is sometimes called set-builder notation To illustrate, consider

the set

A = { 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 } ,

which we have described here by listing its elements Since A consists

precisely of those positive integers that are less than 10, this set can be written in set-builder notation as

A = {z | a; is a positive integer less than 10},

which is read, "A equals the set of all x such that z is a positive integer

less than 10."

In this notation x denotes a typical element of the set, the vertical bar | 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 | x is an IBM stockholder},

which is read "the set of all x such that x is an IBM stockholder." Note

that it would be very inconvenient to list all the members of this set

Example 3 \χ \ χ is 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}

Example 4 If A is the set

( 1 , 2 , 3 , 4 )

and B is the set

{a: | a: is a positive integer and x 2 < 25},

then

A = B

Example 5 Consider the sets of stocks

A = {IBM, 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 deleted

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

exhaust systems of 11 buses, the following percentage decreases in carbon monoxide emissions were observed:

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

in which case we write

A = B

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 f u] and A = {a, o, u)

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

All possible subsets of A are

0 , {a}, {6}, {c}, {a, 6}, {a, c}, {6, c}, {a, 6, c}

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

July 26, 1974 and T is the set of all stocks on the New York Stock Exchange

that traded over 100 shares on July 26, 1974, then

res

Example 10 If A is any set, then

AC A

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

ACB

Figure 1.1

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 dB and B (I 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

2 Let A = {x | x is a real number satisfying x > 5) Answer the following as

true or false

(a) 2 G A (b) 5.5 G A (c) 7 $ A

(d) 5 G A (e) 3 $ A (f) 3 € A

3 Consider the set of water pollutants

A = {sulfur, crude oil, phosphates, mercury}

Answer the following as true or false

(a) sulfurG A (b) phosphates $ A

(c ) arsenic $ A (d ) oil 6 A

4 Let A = {x | x is a real number and x 2 = 9} 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, 5} 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

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

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

(d) [x | x is a positive integer and x 2 < 16}

(e) [x | x is a positive real number and x < 4}

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

9 Consider the set of psychological disorders

A = {schizophrenia, paranoia, depression, megalomania}

Which of the following sets are equal to A?

(a) {schizophrenia, paranoia, depression}

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

10 Which of the following sets are empty?

(a) {x | x is an integer and x 2 = 4}

(b) \x | x is an integer and x 2 = — 4}

(c) [x | x is a real number satisfying x 2 + 1 = 0}

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) {a h 02, a z ) (b) 0

14 Let A = {3, 5, 7, 9} Answer the following as true or false

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:

Figure 1.3

(a) CC T (b) CC S (c) Td C (d) xi C

(e) x 6 T (f) y e C and y G T (g) xÇ CorxG î1

17 In each part find the "smallest" possible set that contains the given sets as subsets

(a) {1,3,7}, {3,5,9,2}, {1,2,3,4,6}, {3};

(b) { α , Μ , 0

18 In each part find the "smallest^ set that contains the given sets as subsets (a) {IBM, Du Pont, Xerox}, {Polaroid, Honeywell, Xerox, IBM, Avco} (b) {1,3,5}, {a, 6,3}, {a}, {a,o}

19 Is it true that 0 € 0 ? Is it true that 0 C 0 ?

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

1.2 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 opera­tions and in later sections we shall illustrate their applications

If A and B are two gi?m sets, then <àe set erf all elements that belong

to both A and Bm a new set, called the interaction of A and B;

it is denoted by the symbol

Àt\B

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 Π B is the

set of points in the shaded region of the figure

■CD-Figure 1.4 Example 14 Let

A = {a, 6, c, d, e}, B = {&, d, e, g} } C = {a, A}

Find A n f i , A n C , a n d ß n C

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(\B

Solution The only elements that belong to both A and B are 6, d, and e Therefore

Af\B = {b,d,e}

Similarly,

AViC = {a}

Since the sets B and C have no elements in common, their intersection is

the empty set; that is,

BnC = 0

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 i 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:

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

A n£n<7

is the shaded region in the figure

The intersection of any collection of sets is the set of elements that

belong to every one of the sets in the collection

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 OR

Solution The members of S Π R belong to both S and Ä, so that S Π 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

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

1.7a, then A U B is the set of points in the shaded region shown in Figure

1.7b

Figure 1.7

Example 21 Let A be any set Find A U 0

Solution The members of A U 0 are those elements that lie either in A or 0 or

both Since 0 has no elements, we obtain

A U 0 = A

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

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

Figure 1.8a, then the union of these sets, denoted by

A OBUC

is the shaded region in Figure 1.8b

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

Shade the sets

(a) AüB (b) Cn(AüB) (c) ( C f l A ) U ( C n £ )

Solution (a) See Figure 1.10

c

Figure 1.9

Figure 1.10

Solution (b) To find C f i ( i U ß ) we intersect C with the shaded set A UB

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

Solution (c) We begin by shading the sets C Π A and C Γ\Β; this gives the

diagrams in Figure 1.12 To find (C fi A) U ( C n B ) , we form the union of the shaded sets C Γ\Α and C Γ\Β; this yields the shaded set in Figure 1.13

Observe that the sets C n ( A u B ) and ( C n A ) Ό (C 0 B) obtained in

parts (b) and (c) are identical We have thus established the following basic law of sets:

C n(A u5) = (CnA) u ( C n B )

This is called the first distributive law for sets We leave it as an exercise

Figure 1.11

are called Venn diagrams In the exercises we have indicated some other

useful set relationships that can be established using Venn diagrams

Figure 1.13

f John Venn (1834-1923)—Venn was the son of a minister He graduated from

Gon-ville and Caius College in Cambridge, England in 1853, after which he pursued theo­ logical interests 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

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

sets as follows:

Q = quick-growing R = rust resistant

W = all white kernels Y = all yellow kernels

Describe the characteristics of the plants in the following sets:

(a) Q n Y (b) R U W

(c) ( Ä n Q ) u ( Ä n F ) (d) Qu(TFnF)

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

quick-growing, yellow-kerneled plants

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

of plants that are either rust resistant or white-kerneled

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

plants in R fi Y are rust resistant and yellow-kerneled Thus (R 0 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 Π Y is empty since the kernels cannot be both all

white and all yellow Therefore,

(e) AOBOC (f) DO0

2 Let A, By C, and D be the sets in Exercise 1 Compute

( a ) i U ß (b) i U C

(c) ß U C (d) A U ß U D ;

(e) A U 0 (f) DUD

G

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:

(a) SOT (b) s oc

(e) TOC (d) SO TOC

4 Let S, T, and C be the sets in Exercise 3 Shade the following sets: (a) SUT (b) SUC

(c) T U C (a) SU TU C

5 Let S, T, and C be the sets in Exercise 3 Shade the following sets: (a) CU (SOT) (b) CO (SUT)

(c) (COS) UT (d) (CUS)O(COT)

6 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) 0 , {1,2}

(d) {book, 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,zbe the indicated points

9 Answer the following as true or false

(a) we CU (SOT) (b) w£ CCl(SUT)

11 Use Venn diagrams to establish the second distributive law

CU(A (\B) = (CU A) n(CUB)

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

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

(a) ( A n B) U C (b) A U (B fi C) (c) A U E U C

(d) AflßfiC (e) Afl(£UC) (f) ( A U £ ) n C

13 Explain why the following are true

(a) A U B = B U A (b) A Π £ = £ Π A

14 Use Venn diagrams to establish the following

(a) (Af)B) 0C= An(BOC) (b) (A UB) U C = A U (B U C)

15 Use Venn diagrams to establish the following

(a) A C A U B and B C A U B

(b) A fi B C A and i n ß C 5

16 Use Venn diagrams to establish the following

(a) If A C C and B C C, then ( A U B ) C C

(b) If C C A and C C 5, then C C (AflB)

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

(a) M fi A (b) M U F (c) A D T fl F

(d) i u r u F ( e ) i n i n Ä

18 Let M, F, A, and S 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

(a) x € M U F U A (b) x £ M Π S

(c) x e F n S (d) s 6 (M n S) U F

19 An automobile insurance company classifies its policy holders into the fol­lowing categories:

A = the set of all policy holders who drive cars with engines

that are more than 200 horsepower

B = the set of all policy holders who drive cars with engines

that are more than 250 horsepower

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

Describe the policy holders in the following sets:

(a) i n ß (b) A ÜB (c)incnjii

(d) AU DU F (e) BO (DO 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 are 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 x is

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

con-cerned with sets of integers, U would be the set of all integers; in problems concerned with sets of stocks on the New York Stock Exchange, U would

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

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

A = {x | 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 1 is the set of real numbers that do

not satisfy x > 0; that is,

A' = {# | z 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 $10,000 taxable income in 1974 Find the complement of W

If U is the universal set and A is a subset of (7, then the set of all elements in U that are not members of A is called the complement

of A It is denoted by the symbol

A'

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 y so that

U' = 0

Solution (b) The set 0' consists of all elements that are not in 0 Since 0

has no elements, all elements are outside of 0; thus

0' = 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'nBnC (b) A n B ' n c "

Figure 1.17

Figure 1.18

Solution (a) The members of A' n B (1 C belong to both B and C but not A

This yields the shaded region in Figure 1.18

Solution (b) The members of A n B' Π C" belong to A but not to 5 and not

to C This yields the shaded region in Figure 1.19

There are 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'sf laws

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 as well as several successful elementary textbooks

If A and B are any two sets, then

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

(A uJB)'

consists of all elements that are not in

A US

This is given by the shaded region in Figure 1.20a

In Figures 1.20b and 1.20c we have shaded the regions A' and B' and in

Figure 1.20d we formed their intersection to obtain

A 1 n B'

Comparing Figures 1.20a and 1.20d we see that

(A [)BY = 4 ' n ß ' ,

which is the first law of DeMorgan

The next example illustrates DeMorgan's laws

as guaranteed by the first DeMorgan law

(A U B)' = A' n B' first DeMorgan law

(A n B)' = A' U 5 ' second DeMorgan law

Figure 1.20

The verifications of the following results are left as exercises

In many problems we are interested in paired data; for example, the height and weight of an individual, the wind speed and wind direction at a certain time, the total assets and total liabilities of a business firm, and so

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

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

(A')' = A

A UA' = U

A nA' = 0

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

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

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

An ordered pair (a, 6) 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

If A and B are two sets, the set of all ordered pairs (a, 6), where

a £ A and b £ B y is called the Cartesian] product of A and B; it is

denoted by the symbol

AXB

emanating from each of these dots correspond to the possible choices for the second entry in the ordered pair, namely 1, 2, 3, or 4 The various ordered pairs can be listed by tracing out all possible paths or "branches" from the top of the tree to the bottom of the tree For example, the path in color corresponds to the pair (s, 2) and the heavy black path to the pair

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

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

cording to three characteristics

S (sex) : m = male / = female

I (income) : A = high a = average I = low

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

registration) For example, a person filed under ( /, A, r) would be a female with high income who is registered as a Republican

The Cartesian product

SXIXP

contains all possible classifications From the tree in Figure 1.22 the different possible classifications are

(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) ( /, Z, r) ( /, Z, i)

Figure 1.22

A = the set of stocks traded on the New York Stock Exchange

that have paid a dividend for the past 10 years without any interruption

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 sets

D = the set of all Democratic U.S senators

Specify a universal set and compute D'

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

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

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

(A n £ ) ' = A'Uß'

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

A, By and C:

( A U 5 u c ) ' = i ' n s ' n c ' ( i n ß 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 Ai, A2, , An

14 Explain why the following are true

(a) (A f )'= A (b) AUA'=U (e) A fi A ' = 0

In Exercises 15 and 16, let A, Z?, and C be the indicated sets and let x, y, z, and w be

the indicated points in Figure 1.23

15 In each case determine which of the points x, y, z, w belong to the indicated set (a) c't\B' (b) An#nc"

(c) i n s ' n c (d) i ' n s ' n c

16 Follow the directions of Exercise 15 for the sets

(a) A'VB (b) J5'UC"

(c) A ' U t f ' U C " (d) A'UBUC'

17 In each part, find the 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, 14) (d) (x2 , 9) = (16, 9)

18 Let A = {a, b, c, d) and B = {0, 1, 2}

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

19 Let A = {a, b}, B= {1,2}, and C= {x, y, z} List the elements of

AXBXC

20 An air traffic control station supplies the following data to airline pilots:

Traffic: crowded (c), average (a), light (I) Visibility: poor (p), good (g)

Windspeed: negligible (n), medium (m), high (h)

Flying conditions are described by an ordered triple; for example (c, p, n)

means crowded traffic, poor visibility, negligible windspeed List all the possible flying conditions

21 In a psychology experiment, a rat is placed in a cage with three doors, a, 6,

and c (see Figure 1.24) The rat leaves the cage through one of the doors Upon reaching the intersection he turns either left or right and at the next inter­ section he turns either left or right again The rat then proceeds to the reward section

Figure 1.24

(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, ò, c}, T = {l y r), I = left, r = right

(b) List all possible paths

1.4 COUNTING ELEMENTS 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 UB and A X 5 ? In this section we investigate problems like these and

illustrate some of their applications

If S is a set with a finite number of elements, then we shall denote the

number of elements in S by the symbol

together; in other words

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