BLTZMC02 043-102-hr1 2-11-2009 9:29 Page 43 Set...

----

Organizing and visually representing sets of human blood types is presented in the Blitzer Bonus on page 83. The vital

role that this representation plays in blood transfusions is developed in Exercises 113–117 of Exercise Set 2.4.

–43–

Set TheoryOur bodies are fragile and complex, vulnerable to disease and easily damaged.

The sequencing of the human genome in 2003—all 140,000 genes—should lead

to rapid advances in treating heart disease, cancer, depression, Alzheimer’s, and

AIDS. Neural stem cell research could make it possible to repair brain damage

and even re-create whole parts of the brain. There appears to be no limit to the

parts of our bodies that can be replaced. By contrast, at

the start of the

twentieth century, we

lacked even a basic

understanding of the different

types of human blood. The discovery of blood

types, organized into collections called sets

and illustrated by a special set

diagram,

rescued surgery

patients from random,

often lethal, transfusions. In this

sense, the set diagram for blood types that

you will encounter in this chapter reinforces our

optimism that life does improve and that we are

better off today than we were one hundred

years ago.

2C H A P T E R

BLTZMC02_043-102-hr1 2-11-2009 9:29 Page 43

We tend to place things incategories, which allows us

to order and structurethe world. For example,to which populationsdo you belong? Doyou categorize your-

self as a collegestudent? What about

your gender? Whatabout your academic

major or your ethnicbackground? Our minds

cannot find order andmeaning without creating

collections. Mathematicianscall such collections sets. A set is a collection of objects whose contents can beclearly determined. The objects in a set are called the elements, or members, ofthe set.

A set must be well defined, meaning that its contents can be clearlydetermined. Using this criterion, the collection of actors who have won AcademyAwards is a set. We can always determine whether or not a particular actor is anelement of this collection. By contrast, consider the collection of great actors.Whether or not a person belongs to this collection is a matter of how we interpret theword great. In this text, we will only consider collections that form well-defined sets.

Methods for Representing Sets

An example of a set is the set of the days of the week, whose elements are Monday,Tuesday, Wednesday, Thursday, Friday, Saturday, and Sunday.

Capital letters are generally used to name sets. Let’s use to represent the setof the days of the week.

Three methods are commonly used to designate a set. One method is a worddescription. We can describe set as the set of the days of the week. A secondmethod is the roster method. This involves listing the elements of a set inside a pairof braces, The braces at the beginning and end indicate that we arerepresenting a set. The roster form uses commas to separate the elements of the set.Thus, we can designate the set by listing its elements:

Grouping symbols such as parentheses, and square brackets, are notused to represent sets. Only commas are used to separate the elements of a set.Separators such as colons or semicolons are not used. Finally, the order in which theelements are listed in a set is not important. Thus, another way of expressing the setof the days of the week is

Representing a Set Using a Description

Write a word description of the set

P = 5Washington, Adams, Jefferson, Madison, Monroe6.

EXAMPLE 1

W = 5Saturday, Sunday, Monday, Tuesday, Wednesday, Thursday, Friday6.

3 4,1 2,

W = 5Monday, Tuesday, Wednesday, Thursday, Friday, Saturday, Sunday6.

W

5 6.

W

W

2.1 Basic Set Concepts

44 C H A P T E R 2 Set Theory

O B J E C T I V E S

1 Use three methods to

represent sets.

2 Define and recognize the

empty set.

3 Use the symbols and

4 Apply set notation to sets of

natural numbers.

5 Determine a set’s cardinal

number.

6 Recognize equivalent sets.

7 Distinguish between finite and

infinite sets.

8 Recognize equal sets.

x .H

1Use three methods to

represent sets.

BLTZMC02_043-102-hr1 2-11-2009 9:29 Page 44

S E C T I O N 2 .1 Basic Set Concepts 45

Solution Set is the set of the first five presidents of the United States.

Write a word description of the set

Representing a Set Using the Roster Method

Set is the set of U.S. coins with a value of less than a dollar. Express this set usingthe roster method.

Solution

Set is the set of months beginning with the letter A. Express this setusing the roster method.

The third method for representing a set is with set-builder notation. Using thismethod, the set of the days of the week can be expressed as

We read this notation as “Set is the set of all elements such that is a day of theweek.” Before the vertical line is the variable which represents an element ingeneral. After the vertical line is the condition must meet in order to be anelement of the set.

Table 2.1 contains two examples of sets, each represented with a worddescription, the roster method, and set-builder notation.

xx,

xxW

Set W the set of

allelements x

suchthat

W={x | x is a day of the week}.

is

M2CHEC

KPOIN

T

C = 5penny, nickel, dime, quarter, half-dollar6

C

EXAMPLE 2

L = 5a, b, c, d, e, f6.1C

HEC

KPOIN

T

P

STUDY TIP

Any letter can be used torepresent the variable in set-builder notation. Thus,

andall

represent the same set.5z ƒ z is a day of the week65y ƒ y is a day of the week6,5x ƒ x is a day of the week6,

T A B L E 2 .1 Sets Using Three Designations

Word Description Roster Method Set-Builder Notation

is the set of members ofthe Beatles in 1963.B George Harrison,

John Lennon, PaulMcCartney, Ringo Starr6

B = 5 was a member ofthe Beatles in 19636B = 5x ƒ x

is the set of states whosenames begin with the letter A.

S Alabama, Alaska,Arizona, Arkansas6S = 5 is a U.S. state

whose name begins withthe letter A6

S = 5x ƒ x

Converting from Set-Builder to Roster Notation

Express set

using the roster method.

Solution Set is the set of all elements such that is a month beginning withthe letter M. There are two such months, namely March and May. Thus,

Express the set

using the roster method.

O = 5x ƒ x is a positive odd number less than 1063C

HEC

KPOIN

T

A = 5March, May6.

xxA

A = 5x ƒ x is a month that begins with the letter M6

EXAMPLE 3

The Beatles climbed to the top of the British

music charts in 1963, conquering the United

States a year later.

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Have you ever considered what would happen if we suddenly lost our ability torecall categories and the names that identify them? This is precisely what

happened to Alice, the heroine of Lewis Carroll’s Through the Looking Glass,as she walked with a fawn in “the woods with no names.”

So they walked on together through the woods, Alice with her armsclasped lovingly round the soft neck of the Fawn, till they came outinto another open field, and here the Fawn gave a sudden bound intothe air, and shook itself free from Alice’s arm. “I’m a Fawn!” it criedout in a voice of delight. “And, dear me! you’re a human child!” Asudden look of alarm came into its beautiful brown eyes, and inanother moment it had darted away at full speed.

By realizing that Alice is a member of the set of human beings, which inturn is part of the set of dangerous things, the fawn is overcome by fear.

Thus, the fawn’s experience is determined by the way it structures theworld into sets with various characteristics.

THE LOSS OF SETS

BL

ITZ

ER

BO

NU

SThe representation of some sets by the roster method can be rather long, or

even impossible, if we attempt to list every element. For example, consider the set ofall lowercase letters of the English alphabet. If is chosen as a name for this set, wecan use set-builder notation to represent as follows:

A complete listing using the roster method is rather tedious:

We can shorten the listing in set by writing

The three dots after the element d, called an ellipsis, indicate that the elements inthe set continue in the same manner up to and including the last element z.

L = 5a, b, c, d, Á , z6.

L

L = 5a, b, c, d, e, f, g, h, i, j, k, l, m, n, o, p, q, r, s, t, u, v, w, x, y, z6.

L = 5x ƒ x is a lowercase letter of the English alphabet6.

LL

The Empty Set

Consider the following sets:

Can you see what these sets have in common? They both contain no elements.Thereare no fawns that speak. There are no numbers that are both greater than 10 andalso less than 4. Sets such as these that contain no elements are called the empty set,or the null set.

5x ƒ x is a number greater than 10 and less than 46.

5x ƒ x is a fawn that speaks6

2Define and recognize the

empty set.

Notice that and have the same meaning. However, the empty set is notrepresented by This notation represents a set containing the element �.3�4.

�5 6

THE EMPTY SET

The empty set, also called the null set, is the set that contains no elements. Theempty set is represented by or �.5 6

John Tenniel, colored by Fritz Kredel

BLTZMC02_043-102-hr1 3-11-2009 15:25 Page 46


Recognizing the Empty Set

Which one of the following is the empty set?

a. b. 0

c.

d.

Solutiona. is a set containing one element, 0. Because this set contains an element,

it is not the empty set.

b. 0 is a number, not a set, so it cannot possibly be the empty set. It does,however, represent the number of members of the empty set.

c. contains all numbers thatare either less than 4, such as 3, or greater than 10, such as 11. Because someelements belong to this set, it cannot be the empty set.

d. contains no elements. There are nosquares with exactly three sides. This set is the empty set.

Which one of the following is the empty set?

a.

b.

c. nothing d. 5�6

5x ƒ x is a number less than 3 and greater than 56

5x ƒ x is a number less than 3 or greater than 564C

HEC

KPOIN

T

5x ƒ x is a square with exactly three sides6

5x ƒ x is a number less than 4 or greater than 106

506

5x ƒ x is a square with exactly three sides6


506

EXAMPLE 4

Notations for Set Membership

We now consider two special notations that indicate whether or not a given objectbelongs to a set.

3Use the symbols and x .H

THE NOTATIONS AND

The symbol is used to indicate that an object is an element of a set.The symbolis used to replace the words “is an element of.”

The symbol is used to indicate that an object is not an element of a set. Thesymbol is used to replace the words “is not an element of.”x

x

H

H

xç

Using the Symbols and

Determine whether each statement is true or false:

a. b. c.

Solutiona. Because r is an element of the set the statement

is true.Observe that an element can belong to a set in roster notation when threedots appear even though the element is not listed.

b. Because 7 is not an element of the set the statement

is true.

c. Because is a set and the set is not an element of the set thestatement

is false.5a6 H 5a, b6

5a, b6,5a65a6

7 x 51, 2, 3, 4, 56

51, 2, 3, 4, 56,

r H 5a, b, c, Á , z6

5a, b, c, Á , z6,

5a6 H 5a, b6.7 x 51, 2, 3, 4, 56r H 5a, b, c, Á , z6

xçEXAMPLE 5

BLITZER BONUS

THE MUSICALSOUNDS OF THEEMPTY SETJohn Cage (1912–1992), theAmerican avant-garde composer,translated the empty set into thequietest piece of music everwritten. His piano composition

requires the musician to sitfrozen in silence at a piano stoolfor 4 minutes, 33 seconds, or 273seconds. (The significance of 273is that at approximately all molecular motion stops.) Theset

is the empty set. There are nomusical sounds in thecomposition. MathematicianMartin Gardner wrote, “I havenot heard performed, butfriends who have tell me it isCage’s finest composition.”

4¿33–

from 4¿33–65x ƒ x is a musical sound

-273°C,

4¿33–

STUDY TIP

A set can be an element ofanother set. For example,

is a set with twoelements. One element is the set

and the other element isthe letter Thus,

and

c H 55a, b6, c6.

5a, b6 H 55a, b6, c6c.

5a, b6

55a, b6, c6

BLTZMC02_043-102-hr1 2-11-2009 9:29 Page 47



a.

b.

c.

Sets of Natural Numbers

For much of the remainder of this section, we will focus on the set of numbers used forcounting:

The set of counting numbers is also called the set of natural numbers. Werepresent this set by the bold face letter N.

51, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, Á 6.

5Monday6 H 5x ƒ x is a day of the week6.

r x 5a, b, c, z6

8 H 51, 2, 3, Á , 1065C

HEC

KPOIN

T

4Apply set notation to sets of

natural numbers.

THE SET OF NATURAL NUMBERS

N = 51, 2, 3, 4, 5, Á 6

The three dots, or ellipsis, after the 5 indicate that there is no final element and thatthe listing goes on forever.

Representing Sets of Natural Numbers

Express each of the following sets using the roster method:

a. Set is the set of natural numbers less than 5.

b. Set is the set of natural numbers greater than or equal to 25.

c.

Solution

a. The natural numbers less than 5 are 1, 2, 3, and 4. Thus, set can beexpressed using the roster method as

b. The natural numbers greater than or equal to 25 are 25, 26, 27, 28, and so on.Set in roster form is

The three dots show that the listing goes on forever.

c. The set-builder notation

indicates that we want to list the set of all such that is an element of theset of natural numbers and is even. The set of numbers that meets bothconditions is the set of even natural numbers. The set in roster form is


a. Set is the set of natural numbers less than or equal to 3.

b. Set is the set of natural numbers greater than 14.

c. O = 5x ƒ x H N and x is odd6.

B

A6C

HEC

KPOIN

T

E = 52, 4, 6, 8, Á 6.

xxx

E = 5x ƒ x H N and x is even6

B = 525, 26, 27, 28, Á 6.

B

A = 51, 2, 3, 46.

A

E = 5x ƒ x H N and x is even6.

B

A

EXAMPLE 6

BLTZMC02_043-102-hr1 2-11-2009 9:29 Page 48


STUDY TIP

A page of sheet music, filled withsymbols and notations, representsa piece of music. Similarly, thesymbols and notationsthroughout this book arerepresentations of mathematicalideas. Mathematical notation nomore is mathematics than musicalnotation is music. As you becomefamiliar with variousmathematical notations, the ideasrepresented by the symbols canthen live and breathe in yourmind.

T A B L E 2 . 2 Inequality Notation and Sets

Inequality Symbol Exampleand Meaning Set-Builder Notation Roster Method

x is less than a.

x is less thanor equal to a.

x is greater than a.

x is greater thanor equal to a.

x is greater than aand less than b.

x is greater than orequal to a and lessthan or equal to b.

x is greater than orequal to a andless than b.

x is greater than aand less thanor equal to b.

x is a natural number greaterthan or equal to 4 and less than 8.

x<a

a<x<b

a � x � b

a<x � b

a � x<b

x>a

x � a

x � a

{1, 2, 3}

{5, 6, 7}

{1, 2, 3, 4}

{5, 6, 7, 8, »}

{5, 6, 7, 8}

{4, 5, 6, 7, »}

{4, 5, 6, 7}

{4, 5, 6, 7, 8}

{x | x H N and x<4}

{x | x H N and x � 4}

{x | x H N and x>4}

{x | x H N and x � 4}

{x | x H N and 4<x<8}

{x | x H N and 4 � x � 8}

{x | x H N and 4 � x<8}

{x | x H N and 4<x � 8}

x is a natural numberless than 4.

x is a natural number greater than orequal to 4 and less than or equal to 8.

x is a natural numberless than or equal to 4.

x is a natural numbergreater than 4.

x is a natural numbergreater than or equal to 4.

x is a natural number greaterthan 4 and less than 8.

x is a natural number greater than4 and less than or equal to 8.

Representing Sets of Natural Numbers


a. b.

Solutiona. represents the set of natural numbers less than or

equal to 100. This set can be expressed using the roster method as

b. represents the set of natural numbersgreater than or equal to 70 and less than 100. This set in roster form is570, 71, 72, 73, Á , 996.

5x ƒ x H N and 70 … x 6 1006

51, 2, 3, 4, Á , 1006.

5x ƒ x H N and x … 1006

5x ƒ x H N and 70 … x 6 1006.5x ƒ x H N and x … 1006

EXAMPLE 7

Inequality symbols are frequently used to describe sets of natural numbers.Table 2.2 reviews basic inequality notation.

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a.

b.

Cardinality and Equivalent Sets

The number of elements in a set is called the cardinal number, or cardinality, of theset. For example, the set contains five elements and therefore has thecardinal number 5. We can also say that the set has a cardinality of 5.

5a, e, i, o, u6

5x ƒ x H N and 50 6 x … 2006.

5x ƒ x H N and x 6 20067C

HEC

KPOIN

T

Notice that the cardinal number of a set refers to the number of distinct, ordifferent, elements in the set. Repeating elements in a set neither adds newelements to the set nor changes its cardinality. For example, and

represent the same set with three distinct elements, 3, 5,and 7. Thus, and

Determining a Set’s Cardinal Number

Find the cardinal number of each of the following sets:

a. b.

c. d.

Solution The cardinal number for each set is found by determining the numberof elements in the set.

a. contains four distinct elements. Thus, the cardinal numberof set is 4. We also say that set has a cardinality of 4, or

b. contains one element, namely 0. The cardinal number of set is 1.Therefore,

c. Set lists only five elements. However, the threedots indicate that the natural numbers from 16 through 21 are also in the set.Counting the elements in the set, we find that there are 11 natural numbersin set The cardinality of set is 11, and

d. The empty set, contains no elements. Thus,

Find the cardinal number of each of the following sets:

a. b.

c. d.

Sets that contain the same number of elements are said to be equivalent.

D = 5 6.C = 59, 10, 11, Á , 15, 166

B = 58726A = 56, 10, 14, 15, 1668C

HEC

KPOIN

T

n1�2 = 0.�,

n1C2 = 11.CC.

C = 513, 14, 15, Á , 22, 236

n1B2 = 1.BB = 506

n1A2 = 4.AAA = 57, 9, 11, 136

�.C = 513, 14, 15, Á , 22, 236

B = 506A = 57, 9, 11, 136

EXAMPLE 8

n1B2 = 3.n1A2 = 3B = 53, 5, 5, 7, 7, 76

A = 53, 5, 76

5Determine a set’s cardinal

number.

DEFINITION OF A SET’S CARDINAL NUMBER

The cardinal number of set represented by is the number of distinctelements in set The symbol is read “ of ”A.nn1A2A.

n1A2,A,

6Recognize equivalent sets.

DEFINITION OF EQUIVALENT SETS

Set is equivalent to set means that set and set contain the same numberof elements. For equivalent sets, n1A2 = n1B2.

BABA

BLTZMC02_043-102-hr1 2-11-2009 9:29 Page 50


Here is an example of two equivalent sets:

It is not necessary to count elements and arrive at 5 to determine that these sets areequivalent. The lines with arrowheads, indicate that each element of set can bepaired with exactly one element of set and each element of set can be pairedwith exactly one element of set We say that the sets can be placed in a one-to-onecorrespondence.

A.BB

AD ,

n(A) = n(B) = 5

A={x | x is a vowel} = {a, e, i, o, u}

B={x | x H N and 3 � x � 7}={3, 4, 5, 6, 7}.

ONE-TO-ONE CORRESPONDENCES AND EQUIVALENT SETS

1. If set and set can be placed in a one-to-one correspondence, then A isequivalent to

2. If set and set cannot be placed in a one-to-one correspondence, then is not equivalent to B: n1A2 Z n1B2.

ABA

B: n1A2 = n1B2.BA

Determining If Sets Are Equivalent

In most societies, women say they prefer to marry men who are older thanthemselves, whereas men say they prefer women who are younger. Figure 2.1shows the preferred age difference in a mate in five selected countries.

EXAMPLE 9

65432

01

−1−2−3−4

−8−7−6

UnitedStates

2.5

−1.9

Italy

3.3

−2.8

Poland

3.3

−2.8

Colombia

4.5

−4.5

Pre

ferr

ed A

ge D

iffe

renc

e

Yea

rs Y

oung

er T

han

Self

Yea

rs O

lder

Tha

n Se

lf

Preferred Age Difference in a Mate

Country

Zambia

4.2

−7.3

−5MenWomen

F I G U R E 2.1Source: Carole Wade and Carol Tavris,Psychology, Sixth Edition, Prentice Hall, 2000.

Letthe set of the five countries shown in Figure 2.1the set of the average number of years women in each of thesecountries prefer men who are older than themselves.

Are these sets equivalent? Explain.

Solution Let’s begin by expressing each set in roster form.

A={Zambia, Colombia, Poland, Italy, U.S.}

B={ 4.2, 4.5, 3.3, 2.5 }

Do not write 3.3 twice.We are interested in eachset’s distinct elements.

B =

A =

BLTZMC02_043-102-hr1 2-11-2009 9:29 Page 51


7Distinguish between finite and

infinite sets.

8Recognize equal sets.

FINITE SETS AND INFINITE SETS

Set is a finite set if (that is, is the empty set) or is a naturalnumber. A set whose cardinality is not 0 or a natural number is called an infiniteset.

n1A2An1A2 = 0A

DEFINITION OF EQUALITY OF SETS

Set is equal to set means that set and set contain exactly the sameelements, regardless of order or possible repetition of elements.We symbolize theequality of sets and using the statement A = B.BA

BABA

There are two ways to determine that these sets are not equivalent.

Method 1. Trying to Set Up a One-to-One Correspondence

The lines with arrowheads between the sets in roster form indicate that thecorrespondence between the sets is not one-to-one. The elements Poland andItaly from set are both paired with the element 3.3 from set These sets arenot equivalent.

Method 2. Counting Elements

Set contains five distinct elements: Set contains four distinctelements: Because the sets do not contain the same number ofelements, they are not equivalent.

Let

the set of the five countries shown in Figure 2.1 on the previous page

the set of the average number of years men in each of thesecountries prefer women who are younger than themselves.

Are these sets equivalent? Explain.

Finite and Infinite Sets

Example 9 illustrated that to compare the cardinalities of two sets, pair off theirelements. If there is not a one-to-one correspondence, the sets have differentcardinalities and are not equivalent. Although this idea is obvious in the case offinite sets, some unusual conclusions emerge when dealing with infinite sets.

B =

A =

9CHEC

KPOIN

T

n1B2 = 4.Bn1A2 = 5.A

B.A

An example of an infinite set is the set of natural numbers,where the ellipsis indicates that there is no last, or final,

element. Does this set have a cardinality? The answer is yes, albeit one ofthe strangest numbers you’ve ever seen. The set of natural numbers is assigned theinfinite cardinal number (read: “aleph-null,” aleph being the first letter of theHebrew alphabet).What follows is a succession of mind-boggling results, including ahierarchy of different infinite numbers in which is the smallest infinity:

These ideas, which are impossible for our imaginations to grasp, are developed inSection 2.2 and the Blitzer Bonus at the end of that section.

Equal Sets

We conclude this section with another important concept of set theory, equality of sets.

u0 6 u1 6 u2 6 u3 6 u4 6 u5 Á .

u0

u0

N = 51, 2, 3, 4, 5, 6, Á 6,

For example, if and then becausethe two sets contain exactly the same elements.

Because equal sets contain the same elements, they also have the samecardinal number. For example, the equal sets and B = 5z, y, w, x6A = 5w, x, y, z6

A = BB = 5z, y, w, x6,A = 5w, x, y, z6

A={Zambia, Colombia, Poland, Italy, U.S.}

B={ 4.2, 4.5, 3.3, 2.5 }

The sets in roster form (repeated)

BLTZMC02_043-102-hr2 10-11-2009 15:50 Page 52


have four elements each. Thus, both sets have the same cardinal number: 4. Noticethat a possible one-to-one correspondence between the equal sets and can beobtained by pairing each element with itself:

This illustrates an important point: If two sets are equal, then they must be equivalent.

Determining Whether Sets Are Equal


a. b.

Solution

a. The sets and contain exactly the same elements. Therefore,the statement

is true.

b. As we look at the given sets, and we see that 0 is anelement of the second set, but not the first. The sets do not contain exactly thesame elements.Therefore, the sets are not equal.This means that the statement

is false.


a. b. 54, 56 = 55, 4, �6.5O, L, D6 = 5D, O, L610C

HEC

KPOIN

T

51, 3, 56 = 50, 1, 3, 56

50, 1, 3, 56,51, 3, 56

54, 8, 96 = 58, 9, 46

58, 9, 4654, 8, 96

51, 3, 56 = 50, 1, 3, 56.54, 8, 96 = 58, 9, 46

EXAMPLE 10

A={w, x, y, z}

B={z, y, w, x}

BASTUDY TIP

In English, the words equal andequivalent often mean the samething. This is not the case in settheory. Equal sets contain thesame elements. Equivalent setscontain the same number ofelements. If two sets are equal,then they must be equivalent.However, if two sets areequivalent, they are notnecessarily equal.

Practice ExercisesIn Exercises 1–6, determine which collections are not well definedand therefore not sets.

1. The collection of U.S. presidents

2. The collection of part-time and full-time students currentlyenrolled at your college

3. The collection of the five worst U.S. presidents

4. The collection of elderly full-time students currentlyenrolled at your college

5. The collection of natural numbers greater than onemillion

6. The collection of even natural numbers greater than 100

In Exercises 7–14, write a word description of each set. (More thanone correct description may be possible.)

7. Mercury, Venus, Earth, Mars, Jupiter, Saturn, Uranus,Neptune

8. 9. 5January, June, July65Saturday, Sunday6

65

10. 11.

12. 13.

14.

In Exercises 15–32, express each set using the roster method.

15. The set of the four seasons in a year

16. The set of months of the year that have exactly 30 days

17.

18. is a lowercase letter of the alphabet that follows dand comes before

19. The set of natural numbers less than 4

20. The set of natural numbers less than or equal to 6

21. The set of odd natural numbers less than 13

22. The set of even natural numbers less than 10

23.

24.

25. 5x ƒ x H N and x 7 56

5x ƒ x H N and x … 46

5x ƒ x H N and x … 56

j65x ƒ x

5x ƒ x is a month that ends with the letters b-e-r6

59, 10, 11, 12, Á , 256

56, 7, 8, 9, Á , 20659, 10, 11, 12, Á 6

56, 7, 8, 9, Á 65April, August6

Exercise Set 2.1

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26.

27.

28.

29.

30.

31. 32.

In Exercises 33–46, determine which sets are the empty set.

33. 34.

35. is a woman who served as U.S. president before2000

36.

37. is the number of women who served as U.S. presidentbefore 2000

38. is the number of living U.S. presidents born before1200

39.

40. is a month of the year whose name begins with theletter

41.

42.

43.

44.

45.

46.

In Exercises 47–66, determine whether each statement is true or false.

47. 48.

49. 50.

51. 52.

53. 54.

55. 56.

57.

58.

59.

60.

61.

62.

63. 64.

65. 66.

In Exercises 67–80, find the cardinal number for each set.

67.

68.

69.

70.

71. is a day of the week that begins with the letter

72. is a month of the year that begins with the letter

73. 74. D = 5six6D = 5five6

W6C = 5x ƒ x

A6C = 5x ƒ x

B = 51, 3, 5, Á , 216

B = 52, 4, 6, Á , 306

A = 516, 18, 20, 22, 24, 266

A = 517, 19, 21, 23, 256

-2 x N-1 x N

576 H 57, 86536 H 53, 46

19 x 5x ƒ x H N and 16 … x 6 216

16 x 5x ƒ x H N and 15 … x 6 206

20 x 5x ƒ x H N and x 6 206

13 x 5x ƒ x H N and x 6 136

2 H 5x ƒ x H N and x is odd6

4 x 5x ƒ x H N and x is even6

26 x 51, 2, 3, Á , 50637 x 51, 2, 3, Á , 406

17 x 51, 2, 3, Á , 16611 x 51, 2, 3, Á , 96

8 H 51, 3, 5, Á 1965 H 52, 4, 6, Á , 206

10 H 51, 2, 3, Á , 16612 H 51, 2, 3, Á , 146

6 H 52, 4, 6, 8, 1063 H 51, 3, 5, 76



5x ƒ x H N and 3 6 x 6 76

5x ƒ x H N and 2 6 x 6 56

5x ƒ x 6 3 and x 7 76

5x ƒ x 6 2 and x 7 56

X65x ƒ x

5x ƒ x is a U.S. state whose name begins with the letter X6

65x ƒ x

65x ƒ x

5x ƒ x is a living U.S. president born before 12006

65x ƒ x

50, �65�, 06

5x ƒ x + 3 = 965x ƒ x + 5 = 76

5x ƒ x H N and 15 … x 6 606

5x ƒ x H N and 10 … x 6 806

5x ƒ x H N and 7 6 x … 116

5x ƒ x H N and 6 6 x … 106

5x ƒ x H N and x 7 46 75.

76.

77.

78.

79.

80.

In Exercises 81–90,

a. Are the sets equivalent? Explain.

b. Are the sets equal? Explain.

81. is the set of students at your college. is the set ofstudents majoring in business at your college.

82. is the set of states in the United States. is the set ofpeople who are now governors of the states in the UnitedStates.

83.

84.

85.

86.

87.

88.

89.

90.

In Exercises 91–96, determine whether each set is finite or infinite.

91.

92.

93.

94.



Practice PlusIn Exercises 97–100, express each set using set-builder notation.Use inequality notation to express the condition x must meet inorder to be a member of the set. (More than one correct inequalitymay be possible.)

97.

98.

99.

100. 536, 37, 38, 39, Á , 596

561, 62, 63, 64, Á , 896

536, 37, 38, 39, Á 6

561, 62, 63, 64, Á 6

5x ƒ x H N and x … 2,000,0006

5x ƒ x H N and x … 1,000,0006

5x ƒ x H N and x Ú 506

5x ƒ x H N and x Ú 1006

B = 5x ƒ x H N and 199 6 x 6 2076

A = 5x ƒ x H N and 200 … x … 2066

B = 5x ƒ x H N and 99 6 x 6 1066

A = 5x ƒ x H N and 100 … x … 1056

B = 5x ƒ x H N and 20 … x 6 256

A = 5x ƒ x H N and 12 6 x … 176

B = 5x ƒ x H N and 9 6 x … 136

A = 5x ƒ x H N and 6 … x 6 106

B = 53, 2, 1, 06

A = 50, 1, 1, 2, 2, 2, 3, 3, 3, 36

B = 54, 3, 2, 16

A = 51, 1, 1, 2, 2, 3, 46

B = 52, 4, 6, 8, 106

A = 51, 3, 5, 7, 96

B = 50, 1, 2, 3, 46

A = 51, 2, 3, 4, 56

BA

BA

C = 5x ƒ x 6 5 and x Ú 156

C = 5x ƒ x 6 4 and x Ú 126

B = 5x ƒ x H N and 3 … x 6 106

B = 5x ƒ x H N and 2 … x 6 76

A = 5x ƒ x is a letter in the word six6

A = 5x ƒ x is a letter in the word five6

BLTZMC02_043-102-hr1 2-11-2009 9:29 Page 54


In Exercises 101–104, give examples of two sets that meet the givenconditions. If the conditions are impossible to satisfy, explain why.

101. The two sets are equivalent but not equal.

102. The two sets are equivalent and equal.

103. The two sets are equal but not equivalent.

104. The two sets are neither equivalent nor equal.

Application ExercisesThe bar graph shows the countries with the greatest percentage oftheir population having used marijuana. In Exercises 105–112, usethe information given by the graph to represent each set by theroster method, or use the appropriate notation to indicate that theset is the empty set.

105. The set of countries in which the percentage having usedmarijuana exceeds 12%

106. The set of countries in which the percentage having usedmarijuana exceeds 9%

107. The set of countries in which the percentage having usedmarijuana is at least 8% and at most 18%

108. The set of countries in which the percentage having usedmarijuana is at least 8.5% and at most 20%

109. is a country in which percentage having usedmarijuana

110. is a country in which percentage havingused marijuana

111. is a country in which the percentage having usedmarijuana

112. is a country in which the percentage having usedmarijuana

A study of 900 working women in Texas showed that theirfeelings changed throughout the day. The line graph in the nextcolumn shows 15 different times in a day and the average levelof happiness for the women at each time. Based on theinformation given by the graph, represent each of the sets inExercises 113–116 using the roster method.

Ú 22.2%65x ƒ x

7 22.2%65x ƒ x

6 9%67.6% …5x ƒ x

6 12.3%68% …5x ƒ x

Per

cent

age

Hav

ing

Use

d M

ariju

ana

Reefer Madness: Countries withthe Greatest Marijuana Use

Country

24%21%18%15%12%9%6%3%

New

Zea

land

22.2

Aus

tral

ia

18

Uni

ted

Stat

es

12.3

Uni

ted

Kin

gdom

9

Swit

zerl

and

8.5

Irel

and

8

Spai

n

7.6

Can

ada

7.4

Source: Organization for Economic Cooperation and Development

113. is a time of the day when the average level ofhappiness was

114. is a time of the day when the average level ofhappiness was

115. is a time of the day when

3 average level of happiness

116. is a time of the day when

average level of happiness

117. Do the results of Exercise 113 or 114 indicate a one-to-one correspondence between the set representing thetime of day and the set representing average level ofhappiness? Are these sets equivalent?

Writing in Mathematics118. What is a set?

119. Describe the three methods used to represent a set. Givean example of a set represented by each method.

120. What is the empty set?

121. Explain what is meant by equivalent sets.

122. Explain what is meant by equal sets.

123. Use cardinality to describe the difference between a finiteset and an infinite set.

Critical Thinking ExercisesMake Sense? In Exercises 124–127, determine whether eachstatement makes sense or does not make sense, and explainyour reasoning.

124. I used the roster method to express the set of countries thatI have visited.

125. I used the roster method and natural numbers to express theset of average daily Fahrenheit temperatures throughoutthe month of July in Vostok Station, Antarctica, the coldestmonth in one of the coldest locations in the world.

… 463 6

5x ƒ x

6 466

5x ƒ x

165x ƒ x

365x ƒ x

5

4

3

2

222120191817161514131211109

Ave

rage

Lev

elof

Hap

pine

ss

Time of Day

Average Level of Happinessat Different Times of Day

8

1 Noon 10P.M.

Source: D. Kahneman et al. “A Survey Method for Characterizing Daily LifeExperience,” Science.

BLTZMC02_043-102-hr1 2-11-2009 9:29 Page 55

Math tattoos. Who knew? Emerging fromtheir often unsavory reputation of

the recent past, tattoos havegained increasing prominence

as a form of body artand self-expression. A

Harris poll conductedin 2008 estimated that

32 million Americans, or14% of the adult population, have at

least one tattoo.Table 2.3 shows the percentage of

Americans, by age group, with tattoos. Thecategories in the table divide the set of

tattooed Americans into smaller sets, calledsubsets, based on age. The age subsets can be broken into still-smaller subsets. Forexample, tattooed Americans ages 25–29 can be categorized by gender, politicalparty affiliation, race/ethnicity, or any other area of interest.This suggests numerouspossible subsets of the set of Americans with tattoos. Every American in each ofthese subsets is also a member of the set of tattooed Americans.

Subsets

Situations in which all the elements of one set are also elements of another set aredescribed by the following definition:

2.2 Subsets


127. Using the bar graph in Exercise 126, I can see that there isa one-to-one correspondence between the set of theaverage number of hours that men sleep per day and theset of the average number of hours that women sleep perday.

In Exercises 128–135, determine whether each statement is true orfalse. If the statement is false, make the necessary change(s) toproduce a true statement.

128. Two sets can be equal but not equivalent.129. Any set in roster notation that contains three dots must be

an infinite set.130.131. Some sets that can be written in set-builder notation

cannot be written in roster form.132. The set of fractions between 0 and 1 is an infinite set.133. The set of multiples of 4 between 0 and 4,000,000,000 is an

infinite set.134. If the elements in a set cannot be counted in a trillion years,

the set is an infinite set.135. Because 0 is not a natural number, it can be deleted from

any set without changing the set’s cardinality.136. In a certain town, a barber shaves all those men and only

those men who do not shave themselves. Consider each ofthe following sets:

is a man of the town who shaves himself

is a man of the town who does not shave himself

The one and only barber in the town is Sweeney Todd. If represents Sweeney Todd,

a. is b. is s H B?s H A?

s

6. B = 5x ƒ x

6 A = 5x ƒ x

n(�) = 1

Hou

rs S

lept

per

Day

Hours Slept per Day, by Age

Age17

8.0

9.7

9.3

22

8.7

9.1

30

8.4

8.8

40

8.38.5

50

8.28.4

60

8.38.5

Men Women

8.4

8.8

9.2

9.6

10.0

Source: ATUS, Bureau of Labor Statistics

T A B L E 2 . 3 Percentage of Tattooed

Americans, by Age Group

Age Group Percent Tattooed

18–24 9%

25–29 32%

30–39 25%

40–49 12%

50–64 8%

65+ 9%

Source: Harris Interactive

126. Using this bar graph that shows the average number ofhours that Americans sleep per day, I can see that there is aone-to-one correspondence between the set of six ages onthe horizontal axis and the set of the average number ofhours that men sleep per day.

O B J E C T I V E S

1 Recognize subsets and use the

notation

2 Recognize proper subsets and

use the notation

3 Determine the number of

subsets of a set.

4 Apply concepts of subsets and

equivalent sets to infinite sets.

( .

8 .

BLTZMC02_043-102-hr1 2-11-2009 9:29 Page 56

S E C T I O N 2 . 2 Subsets 57

DEFINITION OF A SUBSET OF A SET

Set is a subset of set expressed as

if every element in set is also an element in set B.A

A 8 B,

B,A1Recognize subsets and use the

notation 8 .

Let’s apply this definition to the set of people ages 25–29 in Table 2.3.

Observe that a subset is itself a set.The notation means that is not a subset of Set is not a subset of

set if there is at least one element of set that is not an element of set Forexample, consider the following sets:

Can you see that 3 is an element of set that is not in set Thus, set is not asubset of set

We can show that by showing that every element of set also occurs asan element of set We can show that by finding one element of set that isnot in set


Write or in each blank to form a true statement:

a.

b.

c.

Solutiona. All the elements of are also contained in

Therefore, set is a subset of set

b. Let’s write the set of letters in the word proof and the set of letters in theword roof in roster form. In each case, we consider only the distinct elements,so there is no need to repeat the o.

Because there is an element in set that is not in set set is not a subsetof set

A h B.B:

AB,A

A={p, r, o, f} B={r, o, f}

The element p is in set A but not in set B.

A 8 B.

B:AB = 51, 3, 5, 7, 9, 116.A = 51, 3, 5, 76

A B. B = 5Mercury, Venus, Earth, Mars, Jupiter, Saturn, Uranus, Neptune6

A = 5x ƒ x is a planet of Earth’s solar system6

A B B = 5y ƒ y is a letter in the word roof6

A = 5x ƒ x is a letter in the word proof6

A B B = 51, 3, 5, 7, 9, 116 A = 51, 3, 5, 76

h8

h8EXAMPLE 1

B.AA h BB.

AA 8 BA h B.B:

AB?A

A = 51, 2, 36 and B = 51, 26.

B.ABAB.AA h B

The set of tattooed Americansin the 25-29 age group is a subset of the set of all tattooed Americans.

{x | x is a tattooed Americanand 25 � x's age � 29}

{x | x is a tattooed American}�

Every person in this set, tothe left of the subset symbol,

is also a member of this set,to the right of the subset symbol.

Earth

VenusMercury

Mars

Jupiter

Saturn

Uranus

Neptune

The eight planets in Earth’s solar system

No, we did not forget Pluto. In 2006, basedon the requirement that a planet mustdominate its own orbit (Pluto is slave toNeptune’s orbit), the InternationalAstronomical Union removed Pluto fromthe list of planets and decreed that itbelongs to a new category of heavenlybody, a “dwarf planet.”

BLTZMC02_043-102-hr1 2-11-2009 9:29 Page 57


c. All the elements of

are contained in

Because all elements in set are also in set set is a subset of set

Furthermore, the sets are equal

Write or in each blank to form a true statement:

a.

b.

c.Monday, Tuesday, Wednesday, Thursday, Friday, Saturday,Sunday

Proper Subsets

In Example 1(c) and Check Point 1(c), the given sets are equal and illustrate thatevery set is a subset of itself. If is any set, then because it is obvious thateach element of is a member of

If we know that set is a subset of set and we exclude the possibility of thesets being equal, then set is called a proper subset of set written A ( B.B,A

BAA.A

A 8 AA

A B6

B = 5A = 5x ƒ x is a day of the week6

A B B = 5y ƒ y is a letter in the word proof6

A = 5x ƒ x is a letter in the word roof6

A B B = 51, 3, 5, 76 A = 51, 3, 5, 6, 9, 116

h81CHEC

KPOIN

T

1A = B2.

A 8 B.

B:AB,A

B = 5Mercury, Venus, Earth, Mars, Jupiter, Saturn, Uranus, Neptune6.

A = 5x ƒ x is a planet of Earth’s solar system6

2Recognize proper subsets and

use the notation ( .

DEFINITION OF A PROPER SUBSET OF A SET

Set is a proper subset of set expressed as if set is a subset of set and sets and are not equal 1A Z B2.BA

BAA ( B,B,A

Try not to confuse the symbols for subset, and proper subset, In somesubset examples, both symbols can be placed between sets:

By contrast, there are subset examples where only the symbol can be placedbetween sets:

Because the lower part of the subset symbol in suggests an equal sign, itis possible that sets and are equal, although they do not have to be. By contrast,the missing lower line for the proper subset symbol in indicates that sets and cannot be equal.B

AA ( BBA

A 8 B

{1, 3, 5} � {1, 3, 5}.

Set A Set B

A is a subset of B. Every element in A is alsoan element in B. A is not a proper subset

of B because A = B. The symbol � should not beplaced between the sets.

8

{1, 3} � {1, 3, 5} and

Set A Set B

A is a subset of B. Every elementin A is also an element in B.

{1, 3} � {1, 3, 5}.

Set A Set B

A is a proper subset of Bbecause A and B are not equal sets.

( .8 ,

BLTZMC02_043-102-hr1 2-11-2009 9:29 Page 58



Write or both in each blank to form a true statement:

a.

b.

Solutiona. We begin with is a person and lives in San Francisco and

a person and lives in California . Every person living in SanFrancisco is also a person living in California. Because each person in set iscontained in set set is a subset of set

Can you see that the two sets do not contain exactly the same elements and,consequently, are not equal? A person living in California outside SanFrancisco is in set but not in set Because the sets are not equal, set isa proper subset of set

The symbols and can both be placed in the blank to form a true statement.

b. Every number in is contained in so set isa subset of set

Because the sets contain exactly the same elements and are equal, set isnot a proper subset of set The symbol cannot be placed in the blank ifwe want to form a true statement. (Because set is not a proper subset of set

it is correct to write )

Write or both in each blank to form a true statement:

a.

b.

A B.

B = 5x ƒ x is a person and x lives in Georgia6

A = 5x ƒ x is a person and x lives in Atlanta6

A B

B = 52, 8, 4, 6, 106

A = 52, 4, 6, 86

8 , ( ,2CHEC

KPOIN

T

A X B.B,A

(B.A

A 8 B.

B:AB = 52, 8, 4, 66,A = 52, 4, 6, 86

(8

A ( B.

B:AA.B,

A 8 B.

B:AB,A

6xB = 5x ƒx6xA = 5x ƒx

A B. B = 52, 8, 4, 66 A = 52, 4, 6, 86

A B

B = 5x ƒ x is a person and x lives in California6

A = 5x ƒ x is a person and x lives in San Francisco6

8 , ( ,

(8EXAMPLE 2

STUDY TIP

Do not confuse the symbols and The symbol means “is an element of” and thesymbol means “is a subset of.” Notice the differences among the following truestatements:

The set containing 4 is not anelement of {4, 8}, although

{4} H {{4}, {8}}.

{4} x {4, 8}.

4 is an elementof the set {4, 8}.

The set containing 4 is asubset of the set {4, 8}.

4 H {4, 8} {4} 8 {4, 8}

8

H8 .H

STUDY TIP

• The notation for “is a subsetof,” is similar to thenotation for “is less than orequal to,” Because thenotations share similar ideas,

applies to finite setsonly if the cardinal number ofset is less than or equal tothe cardinal number of set

• The notation for “is a propersubset of,” is similar to thenotation for “is less than,”Because the notations sharesimilar ideas, applies tofinite sets only if the cardinalnumber of set is less thanthe cardinal number of set B.

A

A ( B

6 .( ,

B.A

A 8 B

… .

8 ,

BLTZMC02_043-102-hr1 2-11-2009 9:29 Page 59


The Number of Subsets of a Given Set

If a set contains elements, how many subsets can be formed? Let’s observe somespecial cases, namely sets with 0, 1, 2, and 3 elements.We can use inductive reasoningto arrive at a general conclusion. We begin by listing subsets and counting thenumber of subsets in our list. This is shown in Table 2.4.

n3Determine the number of

subsets of a set.

T A B L E 2 . 4 The Number of Subsets: Some Special Cases

SetNumber of Elements

List of All SubsetsNumber of

Subsets

5 6 0 5 6 1

5a6 1 5a6, 5 6 2

5a, b6 2 5a, b6, 5a6, 5b6, 5 6 4

5a, b, c6 3 5a, b, c6, 8

5a, b6, 5a, c6, 5b, c6,

5a6, 5b6, 5c6, 5 6

THE EMPTY SET AS A SUBSET

1. For any set

2. For any set other than the empty set, � ( B.B

� 8 B.B,

Subsets and the Empty Set

The meaning of leads to some interesting properties of the empty set.

The Empty Set as a Subset

Let and Is

Solution is not a subset of if there is at least one element of set that is not an element of set Because represents the empty set, there are noelements in set period, much less elements in that do not belong to Becausewe cannot find an element in that is not contained in this means that Equivalently,

Let and Is

Example 3 illustrates the principle that the empty set is a subset of every set.Furthermore, the empty set is a proper subset of every set except itself.

A 8 B?B = 56, 7, 86.A = 5 63CHEC

KPOIN

T

� 8 B.A 8 B.B = 51, 2, 3, 4, 56,A = 5 6

B.AA,AB.

AB 1A h B2A

A 8 B?B = 51, 2, 3, 4, 56.A = 5 6

EXAMPLE 3

A 8 B

Number of elements 0 1 2 3

Number of subsets 1 = 20 2 = 21 4 = 2 * 2 = 22 8 = 2 * 2 * 2 = 23

Table 2.4 suggests that when we increase the number of elements in the setby one, the number of subsets doubles. The number of subsets appears to be apower of 2.

BLTZMC02_043-102-hr1 2-11-2009 9:29 Page 60

STUDY TIP

If powers of 2 have you in anexponentially increasing state ofconfusion, here’s a list of valuesthat should be helpful. Observehow rapidly these values areincreasing.

Powers of 2

230= 1,073,741,824

225= 33,554,432

220= 1,048,576

215= 32,768

212= 4096

211= 2048

210= 1024

29= 512

28= 256

27= 128

26= 64

25= 2 * 2 * 2 * 2 * 2 = 32

24= 2 * 2 * 2 * 2 = 16

23= 2 * 2 * 2 = 8

22= 2 * 2 = 4

21= 2

20= 1


The power of 2 is the same as the number of elements in the set. Using inductivereasoning, if the set contains elements, then the number of subsets that can beformed is 2n.

n

NUMBER OF SUBSETS

The number of subsets of a set with elements is 2n.n

NUMBER OF PROPER SUBSETS

The number of proper subsets of a set with elements is 2n- 1.n

For a given set, we know that every subset except the set itself is a propersubset. In Table 2.4, we included the set itself when counting the number of subsets.If we want to find the number of proper subsets, we must exclude counting the givenset, thereby decreasing the number by 1.

Finding the Number of Subsets and Proper Subsets

Find the number of subsets and the number of proper subsets for each set:

a.

b.

Solution

a. A set with elements has subsets. Because the set contains5 elements, there are subsets. Of these, wemust exclude counting the given set as a proper subset, so there are

proper subsets.

b. We can write in roster form asBecause this set contains 7 elements, there are

subsets. Of these, there areproper subsets.

Find the number of subsets and the number of proper subsets for each set:

a.

b.

The Number of Subsets of Infinite Sets

In Section 2.1, we mentioned that the infinite set of natural numbers,is assigned the cardinal number (read “aleph-null”), called a

transfinite cardinal number. Equivalently, there are natural numbers.Once we accept the cardinality of sets with infinitely many elements, a

surreal world emerges in which there is no end to an ascending hierarchy ofinfinities. Because the set of natural numbers contains elements, it has subsets, where Denoting by we have Because the setof subsets of the natural numbers contains elements, it has subsets, where

Denoting by we now have Continuing in thismanner, is the “smallest” transfinite cardinal number in an infinite hierarchyof different infinities!

u0

u2 7 u1 7 u0 .u2 ,2u12u1 7 u1 .2u1u1

u1 7 u0 .u1 ,2u02u0 7 u0 .2u0u0

u0

u051, 2, 3, 4, 5, 6, Á 6,

5x ƒ x H N and 3 … x … 86.

5a, b, c, d64C

HEC

KPOIN

T

27- 1 = 128 - 1 = 127

27= 2 * 2 * 2 * 2 * 2 * 2 * 2 = 128

59, 10, 11, 12, 13, 14, 156.5x ƒ x H N and 9 … x … 156

25- 1 = 32 - 1 = 31

25= 2 * 2 * 2 * 2 * 2 = 32

5a, b, c, d, e62nn

5x ƒ x H N and 9 … x … 156.

5a, b, c, d, e6

EXAMPLE 4

4Apply concepts of subsets and

equivalent sets to infinite sets.

“Infinity is where things happen thatdon’t.”

W. W. Sawyer, Prelude toMathematics, Penguin Books, 1960

BLTZMC02_043-102-hr1 2-11-2009 9:29 Page 61


Practice Exercises

In Exercises 1–18, write or in each blank so that the resultingstatement is true.

1. _____

2. _____

3. _____

4. _____ 5-4, -3, -1, 1, 3, 465-4, 0, 46

5-3, -1, 1, 365-3, 0, 36

51, 2, 3, 4, 5, 6, 7652, 3, 76

51, 2, 3, 4, 5, 6, 7651, 2, 56

h8

5. _____

6. _____

7. _____

8. _____

9. _____5v, o, i, c, e, s, r, a, n, t, o, n65c, o, n, v, e, r, s, a, t, i, o, n6

5x ƒ x is a pure-bred dog65x ƒ x is a dog6

5x ƒ x is a black cat65x ƒ x is a cat6

5Venus, Earth, Mars, Jupiter65Mercury, Venus, Earth6

5Saturday, Sunday, Monday, Tuesday, Wednesday65Monday, Friday6

The mirrors in the painting Time and Time Againhave the effect of repeating the image infinitelymany times, creating an endless tunnel of mirrorimages. There is something quite fascinating aboutthe idea of endless infinity. Did you know that forthousands of years religious leaders warned thathuman beings should not examine the nature of theinfinite? Religious teaching often equated infinitywith the concept of a Supreme Being. One of the lastvictims of the Inquisition, Giordano Bruno, wasburned at the stake for his explorations into thecharacteristics of infinity. It was not until the 1870sthat the German mathematician Georg Cantor(1845–1918) began a careful analysis of themathematics of infinity.

It was Cantor who assigned the transfinite cardinalnumber to the set of natural numbers. He used one-to-one correspondences to establish somesurprising equivalences between the set of naturalnumbers and its proper subsets. Here are two examples:

These one-to-one correspondences indicate that the set of even natural numbers and the set of odd natural numbersare equivalent to the set of all natural numbers. In fact, an infinite set, such as the natural numbers, can be defined as anyset that can be placed in a one-to-one correspondence with a proper subset of itself. This definition boggles the mindbecause it implies that part of a set has the same number of objects as the entire set. There are even natural numbers,

odd natural numbers, and natural numbers. Because the even and odd natural numbers combined make up theentire set of natural numbers, we are confronted with an unusual statement of transfinite arithmetic:

As Cantor continued studying infinite sets, his observations grew stranger and stranger. It was Cantor who showedthat some infinite sets contain more elements than others.This was too much for his colleagues, who considered this workridiculous. Cantor’s mentor, Leopold Kronecker, told him,“Look at the crazy ideas that are now surfacing with your workwith infinite sets. How can one infinity be greater than another? Best to ignore such inconsistencies. By considering thesemonsters and infinite numbers mathematics, I will make sure that you never gain a faculty position at the University ofBerlin.” Although Cantor was not burned at the stake, universal condemnation of his work resulted in numerous nervousbreakdowns. His final days, sadly, were spent in a psychiatric hospital. However, Cantor’s work later regained the respectof mathematicians. Today, he is seen as a great mathematician who demystified infinity.

u0 + u0 = u0 .

u0u0

u0

Natural Numbers: {1, 2, 3, 4, 5 , 6 , », n, »}

Even Natural Numbers: {2, 4, 6, 8, 10, 12, », 2n, »}

Natural Numbers: {1, 2, 3, 4, 5, 6, », n, »}

Odd Natural Numbers: {1, 3, 5, 7, 9, 11, », 2n-1, »}

Each natural number, n, is paired with its double, 2n,in the set of even natural numbers.

Each natural number, n, is paired with 1 less thanits double, 2n - 1, in the set of odd natural numbers.

u0

CARDINAL NUMBERS OF INFINITE SETS

BL

ITZ

ER

BO

NU

S

PJ Crook “Time and Time Again” 1981. Courtesy Loch Gallery, Toronto.

Exercise Set 2.2

BLTZMC02_043-102-hr2 11-11-2009 12:38 Page 62


10. _____

11. _____ 12. _____

13. _____ 14. _____

15. _____ 16. _____

17. _____ 18. _____

In Exercises 19–40, determine whether both, or neither canbe placed in each blank to form a true statement.

19. _____

20. _____

21. _____

22. _____

23. _____

24. _____

25. _____

26. _____

27. _____

28. _____

29.set of natural numbers between 5 and 12




33.

34.

35.

36.

37. _____

38. _____

39. _____

40. _____

In Exercises 41–54,determine whether each statement is true or false. Ifthe statement is false, explain why.

41.

42.

43.

44.

45.

46.

47. � H 5Archie, Edith, Mike, Gloria6

5Canada6 8 5Mexico, United States, Canada6

5Ralph6 8 5Ralph, Alice, Trixie, Norton6

Canada 8 5Mexico, United States, Canada6

Ralph 8 5Ralph, Alice, Trixie, Norton6

Canada H 5Mexico, United States, Canada6

Ralph H 5Ralph, Alice, Trixie, Norton6

�5 65 6�

�5101, 102, 103, Á 6

�57, 8, 9, Á 6

� 5101, 102, 103, Á , 2006

� 57, 8, 9, Á , 1006

A BB = 5x ƒ x H N and 2 … x … 86

A = 5x ƒ x H N and 3 6 x 6 106

A BB = 5x ƒ x H N and 2 … x … 116

A = 5x ƒ x H N and 5 6 x 6 126

A BB = theA = 5x ƒ x H N and 3 6 x 6 106




5x ƒ x is a person65x ƒ x is a woman or a man6

5x ƒ x is a person65x ƒ x is a man or a woman6

5x ƒ x is a person65x ƒ x is a woman6

5x ƒ x is a person65x ƒ x is a man6

5x ƒ x is a man65x ƒ x is a woman6

5x ƒ x is a woman65x ƒ x is a man6

51, 3, 4, 7, 9659, 1, 7, 3, 46

58, 0, 6, 2, 4650, 2, 4, 6, 86

5F, I, N, K65F, I, N6

5V, C, R, S65V, C, R6

8 , ( ,

5 6��5 6�51, 3, 56�52, 4, 66

51, 3, 56�52, 4, 66�

52, 3, 56E12 , 13 FE74 , 139 FE47 , 9

13 F

5t, o, l, o, v, e, r, u, i, n65r, e, v, o, l, u, t, i, o, n6 48.

49. 50.

51. 52.

53. 54.

In Exercises 55–60, list all the subsets of the given set.

55. 56.

57. 58.

59. 60.

In Exercises 61–68, calculate the number of subsets and thenumber of proper subsets for each set.

61. 62.

63. 64.

65.

66.

67.

68.

Practice PlusIn Exercises 69–82, determine whether each statement is true orfalse. If the statement is false, make the necessary change(s) toproduce a true statement.

69. The set has proper subsets.

70. The set has proper subsets.

71.

72.

73. 74.

75. 76.

77. If

78. If

79. If set is equivalent to the set of natural numbers, then

80. If set is equivalent to the set of even natural numbers,then

81. The set of subsets of contains 64 elements.

82. The set of subsets of contains 128 elements.

Application ExercisesSets and subsets allow us to order and structure data. In the datashown below, the set of tattooed Americans is divided into subsetscategorized by party affiliation. These subsets are further brokendown into subsets categorized by gender. All numbers in thebranching tree diagram are based on the number of people per10,000 American adults.

5a, b, c, d, e, f6

5a, e, i, o, u6

n1A2 = u0 .A

n1A2 = u0 .A

A 8 B and B 8 C, then A 8 C.

A 8 B and d H A, then d H B.

5�6 H 5�, 5�66� H 5�, 5�66

5�6 h 5�, 5�66� h 5�, 5�66

5x ƒ x H N and 20 … x … 606h 5x ƒx H N and 20 6 x 6 606

5x ƒ x H N and 30 6 x 6 5068 5x ƒx H N and 30 … x … 506

210,00051, 2, 3, Á , 10,0006

2100051, 2, 3, Á , 10006

5x ƒ x H N and 2 … x … 66

5x ƒ x H N and 2 6 x 6 66

5x ƒ x is a U.S. coin worth less than a dollar6

5x ƒ x is a day of the week6

5a, b, c, d, e, f652, 4, 6, 8, 10, 126

E12 , 13 , 14 , 15 F52, 4, 6, 86

�506

5I, II, III65t, a, b6

5Romeo, Juliet65border collie, poodle6

506 h �0 x �

51, 46 X 54, 1651, 46 h 54, 16

516 H 5516, 5366556 H 5556, 5966

� 8 5Charlie Chaplin, Groucho Marx, Woody Allen6

Breakdown of Tattooed Americansby Party Affiliation and Gender

1400 TattooedAmericans

per 10,000 Adults

572 Democrats

446 Republicans

382 Independents

295 Men

277 Women

230 Men

216 Women

197 Men

185 Women

Source: Harris Interactive

BLTZMC02_043-102-hr1 2-11-2009 9:29 Page 63


Based on the tree diagram on the bottom of the previous page, let


83. 84.

85. 86.

87. If then 88. If then

89. If then 90. If then

91. The set of elements in and combined is equal to set

92. The set of elements in and combined is equivalent toset

93. Houses in Euclid Estates are all identical. However, a personcan purchase a new house with some, all, or none of a set ofoptions.This set includes pool, screened-in balcony, lake view,alarm system, upgraded landscaping How many options arethere for purchasing a house in this community?

94. A cheese pizza can be ordered with some, all, or none of thefollowing set of toppings: beef, ham, mushrooms, sausage,peppers, pepperoni, olives, prosciutto, onion How manydifferent variations are available for ordering a pizza?

95. Based on more than 1500 ballots sent to film notables, theAmerican Film Institute rated the top U.S. movies. TheInstitute selected Citizen Kane (1941), Casablanca (1942),The Godfather (1972), Gone With the Wind (1939), Lawrenceof Arabia (1962), and The Wizard of Oz (1939) as the top sixfilms. Suppose that you have all six films on DVD and decideto view some, all, or none of these films. How many viewingoptions do you have?

96. A small town has four police cars. If a radio dispatcherreceives a call, depending on the nature of the situation, nocars, one car, two cars, three cars, or all four cars can be sent.How many options does the dispatcher have for sending thepolice cars to the scene of the caller?

97. According to the U.S. Census Bureau, the most ethnicallydiverse U.S. cities are New York City, Los Angeles, Miami,Chicago,Washington, D.C., Houston, San Diego, and Seattle.If you decide to visit some, all, or none of these cities, howmany travel options do you have?

98. Some of the movies with all-time high box office grossesinclude

Titanic ($601 million), Star Wars: Episode IV—A New

Hope ($461 million), Shrek 2 ($441 million),

E. T. the Extra-Terrestrial ($435 million),

Star Wars: Episode I—The Phantom Menace ($431 million),

Pirates of the Carribean: Dead Man’s Chest ($421 million), and

Spider-Man ($404 million).

Suppose that you have all seven films on DVD and decide,over the course of a week, to view some, all, or none of thesefilms. How many viewing options do you have?

6.5

6.5

D.WM

D.WM

x x R.x H D,x x D.x H R,

x H M.x H D,x H W.x H D,

W ( TM ( T

R H TD H T

W = the set of tattooed Democratic women. M = the set of tattooed Democratic men D = the set of tattooed Democrats R = the set of tattooed Republicans T = the set of tattooed Americans

Writing in Mathematics99. Explain what is meant by a subset.

100. What is the difference between a subset and a proper subset?101. Explain why the empty set is a subset of every set.102. Describe the difference between the symbols and

Explain how each symbol is used.103. Describe the formula for finding the number of subsets for

a given set. Give an example.104. Describe how to find the number of proper subsets for a

given set. Give an example.

Critical Thinking ExercisesMake Sense? In Exercises 105–108, determine whether eachstatement makes sense or does not make sense, and explain yourreasoning.

105. The set of my six rent payments from January through Juneis a subset of the set of my 12 cable television payments fromJanuary through December.

106. Every time I increase the number of elements in a set byone, I double the number of subsets.

107. Because Exercises 93–98 involve different situations, Icannot solve them by the same method.

108. I recently purchased a set of books and am deciding whichbooks, if any, to take on vacation. The number of subsets ofmy set of books gives me the number of differentcombinations of the books that I can take.

In Exercises 109–112, determine whether each statement is true orfalse. If the statement is false,make the necessary change(s) to producea true statement

109. The set has or eight, subsets.110. All sets have subsets.111. Every set has a proper subset.112. The set has eight subsets.113. Suppose that a nickel, a dime, and a quarter are on a table.

You may select some, all, or none of the coins. Specify all ofthe different amounts of money that can be selected.

114. If a set has 127 proper subsets, how many elements arethere in the set?

Group Exercises115. This activity is a group research project and should result in a

presentation made by group members to the entire class.Georg Cantor was certainly not the only genius in historywho faced criticism during his lifetime, only to have his workacclaimed as a masterpiece after his death. Describe the lifeand work of three other people, including at least onemathematician, who faced similar circumstances.

116. Research useful Web sites and present a report on infinitesets and their cardinalities. Explain why the sets of wholenumbers, integers, and rational numbers each have cardinalnumber Be sure to define these sets and show the one-to-one correspondences between each set and the set ofnatural numbers. Then explain why the set of real numbersdoes not have cardinal number by describing how a realnumber can always be left out in a pairing with the naturalnumbers. Spice up the more technical aspects of yourreport with ideas you discovered about infinity that youfind particularly intriguing.

u0

u0 .

53, 51, 466

23,536

8 .H

BLTZMC02_043-102-hr1 2-11-2009 9:29 Page 64

S E C T I O N 2 . 3 Venn Diagrams and Set Operations 65

2.3 Venn Diagrams and SetOperationsSí TV, a 24-hour cable channel targetedto young U.S. Latinos, was launched in2004 and is now in more than 18 millionhouseholds. Its motto: “Speak English.Live Latin.” As Latino spending powersteadily rises, corporate America hasdiscovered that Hispanic Americans,particularly young spenders between theages of 14 and 34, want to be spoken to inEnglish, even as they stay true to their Latinoidentity.

What is the primary language spokenat home by U.S. Hispanics? In this section,we use sets to analyze the answer to thisquestion. By doing so, you will see how sets andtheir visual representations provide precise ways oforganizing, classifying, and describing a wide varietyof data.

Universal Sets and Venn Diagrams

The circle graph in Figure 2.2 categorizes America’s 46 million Hispanics by theprimary language spoken at home. The graph’s sectors define four sets:

• the set of U.S. Hispanics who speak Spanish at home

• the set of U.S. Hispanics who speak English at home

• the set of U.S. Hispanics who speak both Spanish and English at home

• the set of U.S. Hispanics who speak neither Spanish nor English at home.

In discussing sets, it is convenient to refer to a general set that contains allelements under discussion.This general set is called the universal set.A universal set,symbolized by is a set that contains all the elements being considered in a givendiscussion or problem.Thus, a convenient universal set for the sets described above is

Notice how this universal set restricts our attention so that we can divide it into thefour subsets shown by the circle graph in Figure 2.2.

We can obtain a more thorough understanding of sets and their relationship toa universal set by considering diagrams that allow visual analysis. Venn diagrams,named for the British logician John Venn (1834–1923), are used to show the visualrelationship among sets.

Figure 2.3 is a Venn diagram. The universal set is represented by a regioninside a rectangle. Subsets within the universal set aredepicted by circles, or sometimes by ovals or other shapes.In this Venn diagram, set is represented by the light blueregion inside the circle.

The dark blue region in Figure 2.3 represents the setof elements in the universal set that are not in set Bycombining the regions shown by the light blue shadingand the dark blue shading, we obtain the universal set, U.

A.U

A

U = the set of U.S. Hispanics.

U,

English32%

Both23%

Spanish43%

Other2%

Languages Spoken at Homeby U.S. Hispanics

F I G U R E 2.2Source: Time

1Understand the meaning of a

universal set.

O B J E C T I V E S

1 Understand the meaning of a

universal set.

2 Understand the basic ideas of

a Venn diagram.

3 Use Venn diagrams to visualize

relationships between two

sets.

4 Find the complement of a set.

5 Find the intersection of two

sets.

6 Find the union of two sets.

7 Perform operations with sets.

8 Determine sets involving set

operations from a Venn

diagram.

9 Understand the meaning of

and and or.

10 Use the formula for n1A ´ B2.

STUDY TIP

The size of the circle representingset in a Venn diagram hasnothing to do with the number ofelements in set A.

A

AU

F I G U R E 2.3

2Understand the basic ideas of

a Venn diagram.

BLTZMC02_043-102-hr1 2-11-2009 9:29 Page 65


A156

79

U

F I G U R E 2.5

A

$M5

U

F I G U R E 2.4

Determining Sets from a Venn Diagram

Use the Venn diagram in Figure 2.4 to determine each of the following sets:

a. b. c. the set of elements in that are not in

Solutiona. Set the universal set, consists of all the elements within the rectangle.

Thus,

b. Set consists of all the elements within the circle. Thus,

c. The set of elements in that are not in shown by the set of all theelements outside the circle, is


a.

b.

c. the set of elements in that are not in

Representing Two Sets in a Venn Diagram

There are a number of different ways to represent two subsets of a universal setin a Venn diagram. To help understand these representations, consider thefollowing scenario:

You need to determine whether there is sufficient support on campus to havea blood drive. You take a survey to obtain information, asking students

Would you be willing to donate blood?

Would you be willing to help serve a free breakfast to blood donors?

Set represents the set of students willing to donate blood. Set represents the setof students willing to help serve breakfast to donors. Possible survey results includethe following:

• No students willing to donate blood are willing to serve breakfast, and vice versa.• All students willing to donate blood are willing to serve breakfast.• The same students who are willing to donate blood are willing to serve breakfast.• Some of the students willing to donate blood are willing to serve breakfast.

We begin by using Venn diagrams to visualize these results. To do so, we considerfour basic relationships and their visualizations.

Relationship 1: Disjoint Sets Two sets that have no elements in common arecalled disjoint sets.Two disjoint sets, and are shown in the Venn diagram inFigure 2.6. Disjoint sets are represented as circles that do not overlap. Noelements of set are elements of set and vice versa.

Since set represents the set of students willing to donate blood and set represents the set of students willing to serve breakfast to donors, the set diagramillustrates

No students willing to donate blood are willing to serve breakfast, andvice versa.

Relationship 2: Proper Subsets If set is a proper subset of set therelationship is shown in the Venn diagram in Figure 2.7. All elements of set are elements of set If an representing an element is placed inside circle it automatically falls inside circle

Since set represents the set of students willing to donate blood and set represents the set of students willing to serve breakfast to donors, the setdiagram illustrates

All students willing to donate blood are willing to serve breakfast.

BAB.

A,xB.A

B 1A ( B2,A

BAB,A

B,A

BA

A.U

A

U1C

HEC

KPOIN

T

5$, M, 56.A,U

A = 5n, ^6.A

U = 5n, ^, $, M, 56.U,

A.UAU

EXAMPLE 1

3Use Venn diagrams to visualize

relationships between two

sets.

B

A

U

F I G U R E 2.7

A BU

F I G U R E 2.6

BLTZMC02_043-102-hr1 2-11-2009 9:29 Page 66


F I G U R E 2.11

Neither blooddonors norbreakfast servers

U

A B

A: Set of blood donorsB: Set of breakfast servers

IV

I II III

Blood donors andbreakfast servers

Blood donors, butnot breakfastservers

Breakfast servers,but not blooddonors

F I G U R E 2.10

U A B

Common elements arein this region.

F I G U R E 2.9

A = BU

F I G U R E 2.8

Relationship 3: Equal Sets If then set contains exactly the sameelements as set This relationship is shown in the Venn diagram in Figure 2.8.Because all elements in set are in set and vice versa, this diagramillustrates that when then and

Since set represents the set of students willing to donate blood and set represents the set of students willing to serve breakfast to donors, the setdiagram illustrates

The same students who are willing to donate blood are willing to servebreakfast.

Relationship 4: Sets with Some Common Elements In mathematics, the wordsome means there exists at least one. If set and set have at least one elementin common, then the circles representing the sets must overlap. This isillustrated in the Venn diagram in Figure 2.9.

Since set represents the set of students willing to donate blood and set represents the set of students willing to serve breakfast to donors, the presenceof at least one student in the dark blue region in Figure 2.9 illustrates

Some students willing to donate blood are willing to serve breakfast.

In Figure 2.10, we’ve numbered each of the regions in the Venn diagram inFigure 2.9. Let’s make sure we understand what these regions represent in terms ofthe campus blood drive scenario. Remember that is the set of blood donors and is the set of breakfast servers.

In Figure 2.10, we’ll start with the innermost region, region II, and workoutward to region IV.

Region II This region represents the set of students willing to donateblood and serve breakfast. The elements that belong to bothset and set are in this region.

Region I This region represents the set of students willing to donateblood but not serve breakfast.The elements that belong to set but not to set are in this region.

Region III This region represents the set of students willing to servebreakfast but not donate blood. The elements that belong toset but not to set are in this region.

Region IV This region represents the set of students surveyed who are notwilling to donate blood and are not willing to serve breakfast.The elements that belong to the universal set that are not insets or are in this region.

Determining Sets from a Venn Diagram

Use the Venn diagram in Figure 2.11 to determineeach of the following sets:

a. b.

c. the set of elements in but not

d. the set of elements in that are not in

e. the set of elements in both and

Solutiona. Set the universal set, consists of all elements within the rectangle. Taking

the elements in regions I, II, III, and IV, we obtain

b. Set consists of the elements in regions II and III. Thus, B = 5d, e6.B

U = 5a, b, c, d, e, f, g6.U,

B.A

BU

BA

BU

EXAMPLE 2

BAU

AB

BA

BA

BA

BA

BA

BAB 8 A.A 8 BA = B,B,A

B.AA = B,

II

AU

B

III

IVfg

Iabc

d e

BLTZMC02_043-102-hr1 2-11-2009 9:29 Page 67

A

A′

U

F I G U R E 2.12


c. The set of elements in but not found in region I, is d. The set of elements in that are not in found in regions I and IV, is

e. The set of elements in both and found in region II, is


a.b. the set of elements in but not c. the set of elements in that are not in d. the set of elements in that are not in or

The Complement of a Set

In arithmetic, we use operations such as addition and multiplication to combinenumbers. We now turn to three set operations, called complement, intersection, andunion. We begin by defining a set’s complement.

B.AU

AU

AB

A2C

HEC

KPOIN

T

5d6.B,A5a, b, c, f, g6.

B,U5a, b, c6.B,A

The shaded region in Figure 2.12 represents the complement of set or This region lies outside circle but within the rectangular universal set.

In order to find a universal set must be given. A fast way to find is tocross out the elements in that are given to be in set is the set that remains.

Finding a Set’s Complement

Let and Find

Solution Set contains all the elements of set that are not in set Becauseset contains the elements 1, 3, 4, and 7, these elements cannot be members of set

Thus, set contains 2, 5, 6, 8, and 9:

A Venn diagram illustrating and is shown in Figure 2.13.

Let and Find

The Intersection of Sets

If and are sets, we can form a new set consisting of all elements that are in bothand This set is called the intersection of the two sets.B.A

BA

A¿.A = 5a, d6.U = 5a, b, c, d, e63CHEC

KPOIN

T

A¿A

A¿ = 52, 5, 6, 8, 96.

A¿

5 1 , 2, 3 , 4 , 5, 6, 7 , 8, 96.

A¿:AA.UA¿

A¿.A = 51, 3, 4, 76.U = 51, 2, 3, 4, 5, 6, 7, 8, 96

EXAMPLE 3

A¿A.UA¿UA¿,

A,A¿.A,

4Find the complement of a set.

DEFINITION OF THE COMPLEMENT OF A SET

The complement of set symbolized by is the set of all elements in theuniversal set that are not in This idea can be expressed in set-builder notationas follows:

A¿ = 5x ƒ x H U and x x A6.

A.A¿,A,

DEFINITION OF THE INTERSECTION OF SETS

The intersection of sets and written is the set of elements common toboth set and set This definition can be expressed in set-builder notation asfollows:

A ¨ B = 5x ƒ x H A and x H B6.

B.AA ¨ B,B,A

A1347

25689A′

U

F I G U R E 2.13

5Find the intersection of two

sets.

II

AU

B

III

IVfg

Iabc

d e

F I G U R E 2.11 (repeated)

BLTZMC02_043-102-hr1 2-11-2009 9:29 Page 68


U

13579

2468

F I G U R E 2.15 These disjoint setshave no common elements.

U

7911

810

612

F I G U R E 2.14 The numbers 8 and10 belong to both sets.

DEFINITION OF THE UNION OF SETS

The union of sets and written is the set of elements that aremembers of set or of set or of both sets. This definition can be expressed inset-builder notation as follows:

A ´ B = 5x ƒ x H A or x H B6.

BAA ´ B,B,A

In Example 4, we are asked to find the intersection of two sets. This is done bylisting the common elements of both sets. Because the intersection of two sets is alsoa set, we enclose these elements with braces.

Finding the Intersection of Two Sets

Find each of the following intersections:

a.

b.

c.

Solutiona. The elements common to and are and Thus,

The Venn diagram in Figure 2.14 illustrates this situation.

b. The sets and have no elements in common. Thus,

The Venn diagram in Figure 2.15 illustrates this situation.The sets are disjoint.

c. There are no elements in the empty set. This means that there can be noelements belonging to both and Therefore,

Find each of the following intersections:

a.

b.

c.

The Union of Sets

Another set that we can form from sets and consists of elements that are in or or in both sets. This set is called the union of the two sets.BA

BA

51, 2, 36 ¨ �.

51, 2, 36 ¨ 54, 5, 6, 76

51, 3, 5, 7, 106 ¨ 56, 7, 10, 1164C

HEC

KPOIN

T

51, 3, 5, 7, 96 ¨ � = �.

�.51, 3, 5, 7, 96�,

51, 3, 5, 7, 96 ¨ 52, 4, 6, 86 = �.

52, 4, 6, 8651, 3, 5, 7, 96

57, 8, 9, 10, 116 ¨ 56, 8, 10, 126 = 58, 106.

10.856, 8, 10, 12657, 8, 9, 10, 116

51, 3, 5, 7, 96 ¨ �.

51, 3, 5, 7, 96 ¨ 52, 4, 6, 86

57, 8, 9, 10, 116 ¨ 56, 8, 10, 126

EXAMPLE 4

We can find the union of set and set by listing the elements of set Then,we include any elements of set that have not already been listed. Enclose allelements that are listed with braces. This shows that the union of two sets is also a set.

Finding the Union of Two Sets

Find each of the following unions:

a.

b.

c. 51, 3, 5, 7, 96 ´ �.

51, 3, 5, 7, 96 ´ 52, 4, 6, 86

57, 8, 9, 10, 116 ´ 56, 8, 10, 126

EXAMPLE 5

BA.BA

6Find the union of two sets.

BLTZMC02_043-102-hr1 2-11-2009 9:29 Page 69


Solution This example uses the same sets as in Example 4. However, this timewe are finding the unions of the sets, rather than their intersections.

a. To find start by listing all the elements fromthe first set, namely 7, 8, 9, 10, and 11. Now list all the elements from thesecond set that are not in the first set, namely 6 and 12. The union is the setconsisting of all these elements. Thus,

b. To find list the elements from the first set,namely 1, 3, 5, 7, and 9. Now add to the list the elements in the second set thatare not in the first set. This includes every element in the second set, namely2, 4, 6, and 8. The union is the set consisting of all these elements, so

c. To find list the elements from the first set, namely 1, 3, 5, 7,and 9. Because there are no elements in the empty set, there are noadditional elements to add to the list. Thus,

Examples 4 and 5 illustrate the role that the empty set plays in intersectionand union.

51, 3, 5, 7, 96 ´ � = 51, 3, 5, 7, 96.

�,51, 3, 5, 7, 96 ´ �,

51, 3, 5, 7, 96 ´ 52, 4, 6, 86 = 51, 2, 3, 4, 5, 6, 7, 8, 96.

51, 3, 5, 7, 96 ´ 52, 4, 6, 86,

57, 8, 9, 10, 116 ´ 56, 8, 10, 126 = 56, 7, 8, 9, 10, 11, 126.

57, 8, 9, 10, 116 ´ 56, 8, 10, 126,

Find each of the following unions:

a.

b.

c.

Performing Set Operations

Some problems involve more than one set operation. The set notation specifiesthe order in which we perform these operations. Always begin by performing anyoperations inside parentheses. Here are two examples involving sets we will findin Example 6.

• Finding (A � B)'

• Finding A' � B'

Step 1. Parentheses indicate tofirst find the union of A and B.

Step 2. Find thecomplement of A � B.

Step 1. Find thecomplement of A.

Step 2. Find thecomplement of B.

Step 3. Find the intersectionof A' and B'.

51, 2, 36 ´ �.

51, 2, 36 ´ 54, 5, 6, 76

51, 3, 5, 7, 106 ´ 56, 7, 10, 1165C

HEC

KPOIN

T

7Perform operations with sets.

THE EMPTY SET IN INTERSECTION AND UNION

For any set

1.

2. A ´ � = A.

A ¨ � = �

A,

STUDY TIP

The words union and intersectionare helpful in distinguishing thesetwo operations. Union, as in amarriage union, suggests joiningthings, or uniting them.Intersection, as in the intersectionof two crossing streets, brings tomind the area common to both,suggesting things that overlap.

STUDY TIP

When finding the union of twosets, some elements may appearin both sets. List these commonelements only once, not twice, inthe union of the sets.

BLTZMC02_043-102-hr1 2-11-2009 9:29 Page 70


F I G U R E 2.16

Performing Set Operations

Given

find each of the following sets:

a. b.

Solutiona. To find we will first work inside the parentheses and determine

Then we’ll find the complement of namely

These are the given sets.

Join (unite) the elements, listing thecommon elements (3 and 7) only once.

Now find the complement of

List the elements in the universal set thatare not listed in

b. To find we must first identify the elements in and Set isthe set of elements of that are not in set

List the elements in the universal set that are notlisted in

Set is the set of elements of that are not in set

List the elements in the universal set that are not listedin

Now we can find the set of elements belonging to both and to

The numbers 2, 4, 5, and 6are common to both sets.

Given and find each ofthe following sets:

a. b.

Determining Sets froma Venn Diagram

The Venn diagram in Figure 2.16 percolates withinteresting numbers. Use the diagram todetermine each of the following sets:

a. b. c.

d. e. f. A ´ B¿.A¿ ¨ B1A ¨ B2¿

A ¨ B1A ´ B2¿A ´ B

EXAMPLE 7

A¿ ¨ B¿.1A ´ B2¿

B = 5b, c, e6,U = 5a, b, c, d, e6, A = 5b, c6,6CHEC

KPOIN

T

= 52, 4, 5, 66.

A¿ ¨ B¿ = 52, 4, 5, 6, 8, 106 ¨ 51, 2, 4, 5, 6, 96

B¿:A¿A¿ ¨ B¿,

B = 53, 7, 8, 106: 51, 2, 3 , 4, 5, 6, 7 , 8 , 9, 10 6.B¿ = 51, 2, 4, 5, 6, 96.

B:UB¿

A = 51, 3, 7, 96: 5 1 , 2, 3 , 4, 5, 6, 7 , 8, 9 , 106.A¿ = 52, 4, 5, 6, 8, 106.

A:UA¿B¿.A¿A¿ ¨ B¿,

5 1 , 2, 3 , 4, 5, 6, 7 , 8 , 9 , 10 6.51, 3, 7, 8, 9, 106:

= 52, 4, 5, 66

1A ´ B2¿ = 51, 3, 7, 8, 9, 106¿

A ´ B.1A ´ B2¿,

= 51, 3, 7, 8, 9, 106

A ´ B = 51, 3, 7, 96 ´ 53, 7, 8, 106

1A ´ B2¿.A ´ B,A ´ B.1A ´ B2¿

A¿ ¨ B¿.1A ´ B2¿

B = 53, 7, 8, 106,

A = 51, 3, 7, 96

U = 51, 2, 3, 4, 5, 6, 7, 8, 9, 106

EXAMPLE 6

8Determine sets involving set

operations from a Venn

diagram.

IIA

UB

III

IV666

Ipe

epi

10100

2u0

�2�−1

BLTZMC02_043-102-hr1 2-11-2009 9:29 Page 71

F I G U R E 2.17


Set to Determine Description of SetRegions in Venn

Diagram in Figure 2.16 Set in Roster Form

a. A ´ B set of elements in or or bothBA I, II, III Ep, e, 22, 2-1, epi, 10100, 2u0F

b. 1A ´ B2¿ set of elements in U that are not in A ´ B IV 56666

c. A ¨ B set of elements in both and BA II E22, 2-1F

d. 1A ¨ B2¿ set of elements in U that are not in A ¨ B I, III, IV 5p, e, epi, 10100, 2u0, 6666

e. A¿ ¨ B set of elements that are not in and are in BA III 5epi, 10100, 2u06

f. A ´ B¿ set of elements that are in or not in or bothBA I, II, IV Ep, e, 22, 2-1, 666F

Use the Venn diagram in Figure 2.17 todetermine each of the following sets:

a.b.c.d.e.f.

Sets and Precise Use of Everyday English

Set operations and Venn diagrams provide precise ways of organizing, classifying,and describing the vast array of sets and subsets we encounter every day. Let’s seehow this applies to the sets from the beginning of this section:

When describing collections in everyday English, the word or refers to theunion of sets. Thus, U.S. Hispanics who speak Spanish or English at home meansthose who speak Spanish or English or both.The word and refers to the intersectionof sets. Thus, U.S. Hispanics who speak Spanish and English at home means thosewho speak both languages.

In Figure 2.18, we revisit the circle graph showing languages spoken at homeby U.S. Hispanics. To the right of the circle graph, we’ve organized the data using aVenn diagram. The voice balloons indicate how the Venn diagram provides a moreaccurate understanding of the subsets and their data.

E = the set of U.S. Hispanics who speak English at home. S = the set of U.S. Hispanics who speak Spanish at home U = the set of U.S. Hispanics

A ¨ B¿.A¿ ´ B1A ´ B2¿A ´ B

1A ¨ B2¿A ¨ B

7CHEC

KPOIN

T

Solution Refer to Figure 2.16, repeated below.

Languages Spoken at Home by U.S. Hispanics

English32%

Both23%

Spanish43%

Other2%

II

U

III

2%IV

I

43% 23% 32%

Spanish English43% + 23%

+ 32% = 98% speakSpanish or English:

S � E.

23% speakSpanish and English:

S � E.

23% + 32% = 55% speak English.This includes people who speak only Englishor people who speak Spanish and English.

43% + 23% = 66% speak Spanish.This includes people who speak only Spanish

or people who speak both Spanish and English.

43% speak Spanishand not English:

S � E '.

32% speak Englishand not Spanish:

E � S '.

F I G U R E 2.18 Comparing a circle graph and a VenndiagramSource: Time

9Understand the meaning of

and and or.

IIA

UB

III

IV1719

I23

5 71113

IIA

UB

III

IV666

Ipe

epi

10100

2u0

�2�−1


BLTZMC02_043-102-hr1 2-11-2009 9:29 Page 72


The Cardinal Number of the Union of Two Finite Sets

Can the number of elements in or be determined by adding the number

of elements in and the number of elements inThe answer is no. Figure 2.19

illustrates that by doing this, we are countingelements in both sets, or region II, twice.

To find the number of elements in theunion of finite sets and add the number ofelements in and the number of elements in Then subtract the number of elements commonto both sets. We perform this subtraction so thatwe do not count the number of elements in theintersection twice, once for and again for n1B2.n1A2,

B.AB,A

A ¨ B,

n1A2 + n1B2?B,A

n1A ´ B2,B,A10

Use the formula for

n1A ´ B2.

FORMULA FOR THE CARDINAL NUMBER OF THE UNION OFTWO FINITE SETS

is

n(A � B)=n(A)+n(B)-n(A � B)

The number ofelements in A or B

the number of elements in Aplus the number of elements in B

minus the number ofelements in A and B.

UA B

I II

IV

III

n(A) is the numberof elements in

regions I and II.

n(B) is the numberof elements in

regions II and III.

F I G U R E 2.19

Using the Formula for

Some of the results of the campus blood drive survey indicated that 490 studentswere willing to donate blood, 340 students were willing to help serve a freebreakfast to blood donors, and 120 students were willing to donate blood and servebreakfast. How many students were willing to donate blood or serve breakfast?

Solution Let set of students willing to donate blood and set ofstudents willing to serve breakfast. We are interested in how many students werewilling to donate blood or serve breakfast. Thus, we need to determine

We see that 710 students were willing to donate blood or serve a free breakfast.

The admissions department at a college looked at the registration of500 of its students and found the following results: 244 students wereregistered in a mathematics class, 230 students were registered in anEnglish class, and 89 students were registered in a math class and anEnglish class. How many students were registered in a math class or anEnglish class?

8CHEC

KPOIN

T

n(A � B)=n(A)+n(B)-n(A � B)

=490+340-120

=830-120

=710

number of blood donorsor breakfast servers

number ofblood donors

number ofbreakfast servers

number of blood donorsand breakfast servers

n1A ´ B2.

B = theA = the

n1A ´ B2EXAMPLE 8

BLTZMC02_043-102-hr1 2-11-2009 9:29 Page 73


Practice Exercises

In Exercises 1–4, describe a universal set U that includes allelements in the given sets. Answers may vary.

1.

2.

3.

4.

In Exercises 5–8, let and

Use the roster method to write each of the following sets.

5. 6. 7. 8.

In Exercises 9–12, let and

Use the roster method to write each of the following sets.

9. 10. 11. 12.

In Exercises 13–16, let

and Use the roster method to write each of the following sets.

13. 14. 15. 16.


Find each of the following sets.

17. 18. 19.

20. 21. 22.

23. 24. 25.

26. 27. 28.

29. 30. 31.

32. 33. 34.

35. 36. 37.

38. 39. 40.


C = 5b, c, d, e, f6.

B = 5b, g, h6

A = 5a, g, h6

U = 5a, b, c, d, e, f, g, h6

B ¨ UA ¨ UB ´ U

A ´ UC ¨ �A ¨ �

C ´ �A ´ �1A ´ C2¿

1A ´ B2¿A¿ ´ B¿A¿ ´ C¿

1A ¨ B2¿1A ¨ C2¿B ´ C¿

A ´ C¿B¿ ¨ CA¿ ¨ B¿

B¿A¿B ´ C

A ´ BB ¨ CA ¨ B

C = 52, 3, 4, 5, 66.

B = 51, 2, 36

A = 51, 3, 5, 76

U = 51, 2, 3, 4, 5, 6, 76

D¿C¿B¿A¿

D = 51, 3, 5, 7, Á 6.C = 52, 4, 6, 8, Á 6,A = 51, 2, 3, 4, Á , 206, B = 51, 2, 3, 4, Á , 506,

U = 51, 2, 3, 4, Á 6,

D¿C¿B¿A¿

D = 52, 4, 6, 8, Á , 206.C = 51, 3, 5, 7, Á , 196,B = 56, 7, 8, 96,A = 51, 2, 3, 4, 56,

U = 51, 2, 3, 4, Á , 206,

D¿C¿B¿A¿

D = 5a, b, c, d, e, f6.B = 5c, d, e6, C = 5a, g6,A = 5a, b, f, g6,U = 5a, b, c, d, e, f, g6,

B = 5Dodge Ram, Chevrolet Impala6

A = 5Acura RSX, Toyota Camry, Mitsubishi Lancer6

B = 5Coca Cola, Seven-Up6

A = 5Pepsi, Sprite6

B = 5Mark Twain, Robert Louis Stevenson6

A = 5William Shakespeare, Charles Dickens6

B = 5Brahms, Schubert6

A = 5Bach, Mozart, Beethoven6


41. 42. 43.

44. 45. 46.

47. 48. 49.

50. 51. 52.

53. 54. 55.

56. 57. 58.

59. 60. 61.

62. 63. 64.

65. 66.

In Exercises 67–78, use the Venn diagram to represent each set inroster form.

67. 68. 69.

70. 71. 72.

73. 74. 75.

76. 77. 78.

In Exercises 79–92, use the Venn diagram to determine each set orcardinality.

79. 80. 81.

82. 83. 84.

85. 86. 87.

88. 89. 90.

91. 92.

Use the formula for the cardinal number of the union of two sets tosolve Exercises 93–96.

93. Set contains 17 elements, set contains 20 elements, and6 elements are common to sets and How manyelements are in


95. Set contains 8 letters and 9 numbers. Set contains 7 letters and 10 numbers. Four letters and 3 numbers arecommon to both sets and Find the number ofelements in set or set B.A

B.A

BA

A ´ B?B.A

BA

A ´ B?B.A

BA

n1U2 - n1A2n1U2 - n1B2

A ¨ B¿A¿ ¨ B1A ´ B2¿

1A ¨ B2¿n1B¿2n1A¿2

n1A ¨ B2n1A ´ B2A ¨ B

A ´ BAB

AU

B

1001

#$ ∆

twosix

four

A ´ B¿A ¨ B¿A¿ ¨ B

1A ´ B2¿1A ¨ B2¿B¿

A¿A ¨ BA ´ B

UBA

UA B

14

37

89

256

1A ´ B2 ¨ B¿1A ¨ B2 ´ B¿

B ¨ UA ¨ UB ´ U

A ´ UC ¨ �A ¨ �

C ´ �A ´ �1A ´ C2¿

1A ´ B2¿A¿ ´ B¿A¿ ´ C¿

1A ¨ B2¿1A ¨ C2¿B ´ C¿

A ´ C¿B¿ ¨ CA¿ ¨ B¿

B¿A¿B ´ C

A ´ BB ¨ CA ¨ B

Exercise Set 2.3

BLTZMC02_043-102-hr1 2-11-2009 9:29 Page 74


96. Set contains 12 numbers and 18 letters. Set contains14 numbers and 10 letters. One number and 6 letters arecommon to both sets and Find the number ofelements in set or set

Practice PlusIn Exercises 97–104, let


97. 98.

99. 100.

101. 102.

103. 104.

In Exercises 105–108, use the Venn diagram to determine each setor cardinality.

105.

106.

107.

108.

Application ExercisesA math tutor working with a small group of students asked eachstudent when he or she had studied for class the previous weekend.Their responses are shown in the Venn diagram.

In Exercises 109–116, use the Venn diagram to list the elements ofeach set in roster form.

109. The set of students who studied Saturday

110. The set of students who studied Sunday

111. The set of students who studied Saturday or Sunday

112. The set of students who studied Saturday and Sunday

113. The set of students who studied Saturday and not Sunday114. The set of students who studied Sunday and not Saturday

115. The set of students who studied neither Saturday norSunday

116. The set of students surveyed by the math tutor

Jacob

StudiedSaturday

StudiedSunday

Ashley MikeJosh

EmilyHanna

Ethan

U

n1A ¨ B23n1A ´ B2 - n1A¿24

n1U23n1A ´ B2 - n1A ¨ B24

1A¿ ¨ B2 ´ 1A ¨ B2

A ´ 1A ´ B2¿

AU

B

5359616771

23293137

4143 47

1A ¨ C2¿1B ¨ C2¿

A ¨ B¿A ¨ C¿

A ´ UA ¨ U

B ´ CA ´ B

C = 5x ƒ x H N and 1 6 x 6 66.

B = 5x ƒ x is an even natural number and x 6 96

A = 5x ƒ x is an odd natural number and x 6 96

U = 5x ƒ x H N and x 6 96

B.AB.A

BA The bar graph shows the percentage of Americans with genderpreferences for various jobs.

70%

60%

50%

40%

30%

20%

Per

cent

age

of A

mer

ican

s

Gender and Jobs: Percentage of Americans Who PreferMen or Women in Various Jobs

LawyerFamilyDoctor

AirlinePilot

SurgeonBankerPoliceOfficer

ElementarySchoolTeacher

10%

Prefer men Prefer women No Preference

Source: Pew Research Center

In Exercises 117–122, use the information in the graph to place theindicated job in the correct region of the following Venn diagram.

117. elementary school teacher 118. police officer

119. surgeon 120. banker

121. family doctor 122. lawyer

palindromic number is a natural number whose value does notchange if its digits are reversed. Examples of palindromic numbersare 11, 454, and 261,162. In Exercises 123–132, use this definitionto place the indicated natural number in the correct region of thefollowing Venn diagram.

123. 11 124. 22

125. 15 126. 17

127. 454 128. 101

129. 9558 130. 9778

131. 9559 132. 9779

UA B

I II

IV

III

U = the setof naturalnumbers

A = the set ofpalindromic numbers

B = the set ofeven numbers

A

UA B

I II

IV

III

U = theset of jobs

A = the set of jobsfor which more than

20% prefer men

B = the set of jobsfor which more than20% prefer women

BLTZMC02_043-102-hr1 2-11-2009 9:29 Page 75


Toys Requested by Children

100 20 30 40 50

Toy carsand trucks

Dollhouses

Domesticaccessories

Dolls

Spacial-temporal toys(e.g., construction sets)

Sports equipment

Percentage Requesting the Toy

Boys

Girls

“Mas

culin

e” T

oys

“Fem

inin

e” T

oys

Source: Richard, J. G., & Simpson, C. H. (1982). Children, gender and socialstructure: An analysis of the contents of letters to Santa Claus. Child Development,53, 429–436.

133.

134.

135.

136.

137. The set of toys requested by more than 40% of the boysand more than 10% of the girls

138. The set of toys requested by more than 40% of the boys ormore than 10% of the girls

139. A winter resort took a poll of its 350 visitors to see whichwinter activities people enjoyed.The results were as follows:178 people liked to ski, 154 people liked to snowboard, and49 people liked to ski and snowboard. How many people inthe poll liked to ski or snowboard?

140. A pet store surveyed 200 pet owners and obtained thefollowing results: 96 people owned cats, 97 people owneddogs, and 29 people owned cats and dogs. How manypeople in the survey owned cats or dogs?

Writing in Mathematics141. Describe what is meant by a universal set. Provide an

example.

142. What is a Venn diagram and how is it used?

143. Describe the Venn diagram for two disjoint sets. How doesthis diagram illustrate that the sets have no commonelements?

5x ƒ x is a toy requested by fewer than 20% of the girls65x ƒ x is a toy requested by fewer than 5% of the boys6 ´

5x ƒ x is a toy requested by less than 20% of the girls65x ƒ x is a toy requested by more than 10% of the boys6 ´

5x ƒ x is a toy requested by fewer than 20% of the girls65x ƒ x is a toy requested by fewer than 5% of the boys6 ¨

5x ƒ x is a toy requested by less than 20% of the girls65x ƒ x is a toy requested by more than 10% of the boys6 ¨

144. Describe the Venn diagram for proper subsets. How doesthis diagram illustrate that the elements of one set are alsoin the second set?

145. Describe the Venn diagram for two equal sets. How does thisdiagram illustrate that the sets are equal?

146. Describe the Venn diagram for two sets with commonelements. How does the diagram illustrate this relationship?

147. Describe what is meant by the complement of a set.

148. Is it possible to find a set’s complement if a universal set isnot given? Explain your answer.

149. Describe what is meant by the intersection of two sets. Givean example.

150. Describe what is meant by the union of two sets. Give anexample.

151. Describe how to find the cardinal number of the union oftwo finite sets.


152. Set and set share only one element, so I don’t need touse overlapping circles to visualize their relationship.

153. Even if I’m not sure how mathematicians define irrationaland complex numbers, telling me how these sets arerelated, I can construct a Venn diagram illustrating theirrelationship.

154. If I am given sets and the set indicates Ishould take the union of the complement of and thecomplement of

155. I suspect that at least 90% of college students have nopreference whether their professor is a man or a woman, soI should place college professors in region IV of the Venndiagram that precedes Exercises 117–122.


156.

157.

158.

159. If then

160.

161.

162. If then

163. If then

In Exercises 164–167, assume Draw a Venn diagram thatcorrectly illustrates the relationship between the sets.

164. 165.

166. 167. A ´ B = BA ´ B = A

A ¨ B = BA ¨ B = A

A Z B.

A ¨ B = B.B 8 A,

A ¨ B = �.A 8 B,

A ´ � = �

A ¨ U = U

A ¨ B = B.A 8 B,

1A ´ B2 8 A

A ¨ A¿ = �

n1A ´ B2 = n1A2 + n1B2

B.A

1A ´ B2¿B,A

BA

As a result of cultural expectations about what is appropriatebehavior for each gender, boys and girls differ substantially in theirtoy preferences. The graph shows the percentage of boys and girlsasking for various types of toys in letters to Santa Claus. Use theinformation in the graph to write each set in Exercises 133–138 inroster form or express the set as �.

BLTZMC02_043-102-hr1 2-11-2009 9:29 Page 76

Should your blood type determine what you eat? The blood-type diet, developed bynaturopathic physician Peter D’Adamo, is based on the theory that people withdifferent blood types require different diets for optimal health.D’Adamo gives very detailed recommendations for whatpeople with each type should and shouldn’t eat. Forexample, he says shitake mushrooms aregreat for type B’s, but bad for typeO’s.Type B? Type O? In this section,we present a Venn diagram withthree sets that will give you a uniqueperspective on the different types ofhuman blood. Despite this perspective,we’ll have nothing to say about shitakes,avoiding the question as to whether or not theblood-type diet really works.

Set Operations with Three SetsWe now know how to find the union and intersection of two sets.We also know howto find a set’s complement. In Example 1, we apply set operations to situations con-taining three sets.

Set Operations with Three Sets

Given

find each of the following sets:

a. b. c.

Solution Before determining each set, let’s be sure we perform the operations inthe correct order. Remember that we begin by performing any set operations insideparentheses.

• Finding A � (B � C)

• Finding (A � B) � (A � C)

• Finding A � (B � C ')

Step 2. Find the union ofA and (B � C).

Step 1. Find theunion of A and B.

Step 3. Find the intersection of (A � B) and (A � C).

Step 2. Find the union of A and C.

Step 1. Find the intersection of B and C.

Step 1. Find thecomplement of C.

Step 2. Find the union of B and C '.

Step 3. Find the intersection of A and (B � C ').

A ¨ 1B ´ C¿2.1A ´ B2 ¨ 1A ´ C2A ´ 1B ¨ C2

C = 52, 3, 4, 6, 76,

B = 51, 2, 3, 6, 86

A = 51, 2, 3, 4, 56

U = 51, 2, 3, 4, 5, 6, 7, 8, 96

EXAMPLE 1

S E C T I O N 2 .4 Set Operations and Venn Diagrams with Three Sets 77

2.4 Set Operations and VennDiagrams with Three Sets

1Perform set operations with

three sets.

O B J E C T I V E S

1 Perform set operations with

three sets.

2 Use Venn diagrams with three

sets.

3 Use Venn diagrams to prove

equality of sets.

BLTZMC02_043-102-hr1 2-11-2009 9:29 Page 77


a. To find first find the set within the parentheses,

Now finish the problem by finding

b. To find first find the sets within parentheses. Start with

Now find


c. As in parts (a) and (b), to find begin with the set in parentheses.First we must find the set of elements in that are not in

List the elements in that are not in

Now we can identify elements of


Given andfind each of the following sets:

a.

b.

c. A ¨ 1B ´ C¿2.

1A ´ B2 ¨ 1A ´ C2

A ´ 1B ¨ C2

C = 5b, c, f6,U = 5a, b, c, d, e, f6, A = 5a, b, c, d6, B = 5a, b, d, f6,1C

HEC

KPOIN

T

A � (B � C ')={1, 2, 3, 4, 5} � {1, 2, 3, 5, 6, 8, 9}={1, 2, 3, 5}.

Common elements are 1, 2, 3, and 5.

A ¨ 1B ´ C¿2:

B � C '={1, 2, 3, 6, 8} � {1, 5, 8, 9}={1, 2, 3, 5, 6, 8, 9}.

List all elements in B and then add theunlisted elements in C ′, namely 5 and 9.

B ´ C¿:

C = 52, 3, 4, 6, 76: 51, 2 , 3 , 4 , 5, 6 , 7 , 8, 96.UC¿ = 51, 5, 8, 96.

C:UC¿,A ¨ 1B ´ C¿2,

(A � B) � (A � C)={1, 2, 3, 4, 5, 6, 8} � {1, 2, 3, 4, 5, 6, 7}={1, 2, 3, 4, 5, 6}.

Common elements are 1, 2, 3, 4, 5, and 6.

1A ´ B2 ¨ 1A ´ C2:

A � C={1, 2, 3, 4, 5} � {2, 3, 4, 6, 7}={1, 2, 3, 4, 5, 6, 7}.

List all elements in A and then add theunlisted elements in C, namely 6 and 7.

A ´ C:

A � B={1, 2, 3, 4, 5} � {1, 2, 3, 6, 8}={1, 2, 3, 4, 5, 6, 8}.

List all elements in A and then add theunlisted elements in B, namely 6 and 8.

A ´ B:1A ´ B2 ¨ 1A ´ C2,

A � (B � C)={1, 2, 3, 4, 5} � {2, 3, 6}={1, 2, 3, 4, 5, 6}.

List all elements in A and then add theonly unlisted element in B � C, namely 6.

A ´ 1B ¨ C2:

B � C={1, 2, 3, 6, 8} � {2, 3, 4, 6, 7}={2, 3, 6}.

Common elements are 2, 3, and 6.

B ¨ C:A ´ 1B ¨ C2,

BLTZMC02_043-102-hr1 2-11-2009 9:29 Page 78


F I G U R E 2.21

I

AU

B

C

II III

IV

V

VI

VII

VIII

FIGURE 2.20 Three intersectingsets separate the universal set intoeight regions.

Venn Diagrams with Three Sets

Venn diagrams can contain three or more sets, such as the diagram in Figure 2.20.The three sets in the figure separate the universal set, into eight regions. Thenumbering of these regions is arbitrary—that is, we can number any region as I, anyregion as II, and so on. Here is a description of each region, starting with theinnermost region, region V, and working outward to region VIII.

U,2

Use Venn diagrams with three

sets.

The Region Shown in Dark Blue

Region V This region represents elements that are common to setsand

The Regions Shown in Light Blue

Region II This region represents elements in both sets and thatare not in set

Region IV This region represents elements in both sets and thatare not in set

Region VI This region represents elements in both sets and thatare not in set

The Regions Shown in White

Region I This region represents elements in set that are in neithersets nor

Region III This region represents elements in set that are in neithersets nor

Region VII This region represents elements in set that are in neithersets nor

Region VIII This region represents elements in the universal set thatare not in sets or A¿ ¨ B¿ ¨ C¿.C:A, B,

U

C ¨ (A¿ ¨ B¿).B:AC

B ¨ (A¿ ¨ C¿).C:AB

A ¨ (B¿ ¨ C¿).C:BA

1B ¨ C2 ¨ A¿.A:CB

1A ¨ C2 ¨ B¿.B:CA

1A ¨ B2 ¨ C¿.C:BA

A ¨ B ¨ C.C:A, B,

Determining Sets from a Venn Diagram with ThreeIntersecting Sets

Use the Venn diagram in Figure 2.21 to determineeach of the following sets:

a. b. c.

d. e. A ¨ B ¨ C.C¿

B ¨ CA ´ BA

EXAMPLE 2

I113

AU

B

C

II12

III1 2 10

IV6

V5 7

VI9

VII8

4VIII

Set to Determine Description of SetRegions in Venn

Diagram Set in Roster Form

a. A set of elements in A I, II, IV, V 511, 3, 12, 6, 5, 76

b. A ´ B set of elements in or or bothBA I, II, III, IV, V, VI 511, 3, 12, 1, 2, 10, 6, 5, 7, 96

c. B ¨ C set of elements in both and CB V, VI 55, 7, 96

d. C¿ set of elements in that are not in CU I, II, III, VIII 511, 3, 12, 1, 2, 10, 46

e. A ¨ B ¨ C set of elements in and and CBA V 55, 76

Solution

BLTZMC02_043-102-hr1 2-11-2009 9:29 Page 79



a. b.

c. d.

e.

In Example 2, we used a Venn diagram showing elements in the regions todetermine various sets. Now we are going to reverse directions. We’ll use sets

and to determine the elements in each region of a Venn diagram.To construct a Venn diagram illustrating the elements in and start

by placing elements into the innermost region and work outward. Because the fourinner regions represent various intersections, find and

Then use these intersections and the given sets to place the variouselements into regions. This procedure is illustrated in Example 3.

Determining a Venn Diagram from Sets

Construct a Venn diagram illustrating the following sets:

Solution We begin by finding four intersections. In each case, common elementsare shown in red.

•

•

•

•

Now we can place elements into regions, starting with the innermost region, regionV, and working outward.

This is A � B from above.

A ¨ B ¨ C = 5e, g, h6 ¨ 5a, c, e, h6 = 5e, h6

B ¨ C = 5b, e, g, h, l6 ¨ 5a, c, e, h6 = 5e, h6

A ¨ C = 5a, d, e, g, h, i, j6 ¨ 5a, c, e, h6 = 5a, e, h6

A ¨ B = 5a, d, e, g, h, i, j6 ¨ 5b, e, g, h, l6 = 5e, g, h6

U = 5a, b, c, d, e, f, g, h, i, j, k, l6.

C = 5a, c, e, h6

B = 5b, e, g, h, l6

A = 5a, d, e, g, h, i, j6

EXAMPLE 3

A ¨ B ¨ C.A ¨ B, A ¨ C, B ¨ C,

U,A, B, C,UA, B, C,

A ´ B ´ C.

B¿A ¨ C

B ´ CC2C

HEC

KPOIN

T

I

AU

B

C

II III

IV

Veh VI

VII

VIII

I

AU

B

C

IIIII

IV

Veh

g

VI

VII

VIII

I

AU

B

C

IIIII

IV

Veh

g

a VI

VII

VIII

I

AU

B

C

IIIII

IV

Veh

g

a VI

VII

VIII

A � B � C :Region V

A � B � C = {e, h}Place e and h into V.

A � B:Regions ΙΙ and V

A � B = {e, g, h}With e and h in V,place g into II.

A � C:Regions IV and V

A � C = {a, e, h}With e and h in V,place a into IV.

B � C:Regions V and VI

B � C = {e, h}With e and h in V,

place no letters into VI.

STEP 1 STEP 2 STEP 3 STEP 4

I113

AU

B

C

II12

III1 2 10

IV6

V5 7

VI9

VII8

4VIII


BLTZMC02_043-102-hr1 2-11-2009 9:29 Page 80


UA B

I II

IV

III

F I G U R E 2.22

STUDY TIP

In summary, here are the twoforms of reasoning discussed inChapter 1.

• Inductive Reasoning: Startswith individual observationsand works to a generalconjecture (or educated guess)

• Deductive Reasoning: Startswith general cases and worksto the proof of a specificstatement (or theorem)

The completed Venn diagram in step 8 illustrates the given sets.

Construct a Venn diagram illustrating the following sets:

Proving the Equality of Sets

Throughout Section 2.3, you were given two sets and and their universl set and asked to find and In each example, and resulted in the same set. This occurs regardless of which sets we choose for and in a universal set Examining these individual cases and applying inductivereasoning, a conjecture (or educated guess) is that

We can apply deductive reasoning to prove the statement for all sets and in any universal set To prove that and areequal, we use a Venn diagram. If both sets are represented by the same regions inthis general diagram, then this proves that they are equal. Example 4 shows how thisis done.


Use the Venn diagram in Figure 2.22 to prove that

Solution Begin by identifying the regions representing 1A ¨ B2¿.

1A ¨ B2¿ = A¿ ´ B¿.

EXAMPLE 4

A¿ ´ B¿1A ¨ B2¿U.BA1A ¨ B2¿ = A¿ ´ B¿

1A ¨ B2¿ = A¿ ´ B¿.U.

BAA¿ ´ B¿1A ¨ B2¿A¿ ´ B¿.1A ¨ B2¿

UBA

U = 51, 2, 3, 4, 5, 6, 7, 8, 9, 106.

C = 53, 4, 5, 8, 9, 106

B = 54, 7, 9, 106

A = 51, 3, 6, 1063C

HEC

KPOIN

T

3Use Venn diagrams to prove

equality of sets.

Set Regions in the Venn Diagram

A I, II

B II, III

A ¨ B II (This is the region common to and )B.A

1A ¨ B2¿ I, III, IV (These are the regions in that are not in )A ¨ B.U

Id i j

AU

B

C

IIIII

IV

Veh

g

a VI

VII

VIII

Id i j

AU

B

C

II III

IV

Veh

g b l

a VI

VII

VIII

Id i j

AU

B

C

II III

IV

Veh

g

c

b l

a VI

VII

VIII

Id i j

AU

B

C

II III

IV

Veh

fk

g

c

b l

a VI

VII

VIII

STEP 5 STEP 6 STEP 7 STEP 8

C:Regions IV, V, VI, VII

C = {a, c, e, h}With a, e, and h already

placed in C, place c into VII.

A:Regions I, II, IV, V

A = {a, d, e, g, h, i, j}With a, e, g, and h already placed

in A, place d, i, and j into I.

B:Regions II, III, V, VI

B = {b, e, g, h, l}With e, g, and h already placedin B, place b and l into III.

U:Regions I–VIII

U = {a, b, c, d, e, f, g, h, i, j, k, l}With all letters, except f and k, alreadyplaced in U, place f and k into VIII.

BLTZMC02_043-102-hr1 2-11-2009 9:29 Page 81

STUDY TIP

Learn the pattern of De Morgan’slaws.They are extremelyimportant in logic and will bediscussed in more detail inChapter 3, Logic.



A¿ III, IV (These are the regions not in )A.

B¿ I, IV (These are the regions not in )B.

A¿ ´ B¿ I, III, IV (These are the regions obtained by uniting the regionsrepresenting and )B¿.A¿

Both and are represented by the same regions, I, III, and IV, ofthe Venn diagram. This result proves that

for all sets and in any universal set

Can you see how we applied deductive reasoning in Example 4? We startedwith the two general sets in the Venn diagram in Figure 2.22 and worked to thespecific conclusion that and represent the same regions in thediagram. Thus, the statement is a theorem.

Use the Venn diagram in Figure 2.22 to solve this exercise.

a. Which region represents

b. Which region represents

c. Based on parts (a) and (b), what can you conclude?

The statements proved in Example 4 and Check Point 4 are known as De Morgan’s laws, named for the British logician Augustus De Morgan (1806–1871).

A¿ ¨ B¿?

1A ´ B2¿?4C

HEC

KPOIN

T

1A ¨ B2¿ = A¿ ´ B¿

A¿ ´ B¿1A ¨ B2¿

U.BA

1A ¨ B2¿ = A¿ ´ B¿

A¿ ´ B¿1A ¨ B2¿

Next, find the regions in Figure 2.22 representing A¿ ´ B¿.

DE MORGAN’S LAWS

The complement of the intersection of two sets isthe union of the complements of those sets.

The complement of the union of two sets is theintersection of the complements of those sets.

1A ´ B2¿ = A¿ ¨ B¿:

1A ¨ B2¿ = A¿ ´ B¿:


Use a Venn diagram to prove that

Solution Use a Venn diagram with three sets and as shown in Figure 2.23. Begin by identifying the regions representing A ´ 1B ¨ C2.

C,A, B,

A ´ 1B ¨ C2 = 1A ´ B2 ¨ 1A ´ C2.

EXAMPLE 5

I

AU

B

C

II III

IV

V

VI

VII

VIII

F I G U R E 2.23


A I, II, IV, V

B ¨ C V, VI (These are the regions common to and )C.B

A ´ 1B ¨ C2 I, II, IV, V, VI (These are the regions obtained by uniting theregions representing and )B ¨ C.A

UA B

I II

IV

III


BLTZMC02_043-102-hr1 2-11-2009 9:30 Page 82


Next, find the regions representing 1A ´ B2 ¨ 1A ´ C2.

Practice ExercisesIn Exercises 1–12, let

C = 52, 3, 4, 5, 66.

B = 51, 2, 36

A = 51, 3, 5, 76

U = 51, 2, 3, 4, 5, 6, 76


1. 2.

3. 4.

5. 6.

7. 8.

9. 10.

11. 12. 1B ´ C2¿ ¨ A1A ´ B2¿ ¨ C

1A ¨ B ¨ C2¿1A ´ B ´ C2¿

1C¿ ¨ A2 ´ 1C¿ ¨ B¿21A¿ ¨ B2 ´ 1A¿ ¨ C¿2

C¿ ¨ 1A ´ B¿2A¿ ¨ 1B ´ C¿2

1A ¨ B2 ´ 1A ¨ C21A ´ B2 ¨ 1A ´ C2

A ¨ 1B ´ C2A ´ 1B ¨ C2


A I, II, IV, V

B II, III, V, VI

C IV, V, VI, VII

A ´ B I, II, III, IV, V, VI (Unite the regions representing and )B.A

A ´ C I, II, IV, V, VI, VII (Unite the regions representing and )C.A

1A ´ B2 ¨ 1A ´ C2 I, II, IV, V, VI (These are the regions common to and)A ´ C.

A ´ B

Both and are represented by the same regions, I,II, IV, V, and VI, of the Venn diagram. This result proves that

for all sets and in any universal set Thus, the statement is a theorem.

Use the Venn diagram in Figure 2.23 to solve this exercise.

a. Which regions represent

b. Which regions represent


1A ¨ B2 ´ 1A ¨ C2?

A ¨ 1B ´ C2?5C

HEC

KPOIN

T

U.CA, B,

A ´ 1B ¨ C2 = 1A ´ B2 ¨ 1A ´ C2

1A ´ B2 ¨ 1A ´ C2A ´ 1B ¨ C2

In the early 1900s, the Austrian immunologist Karl Landsteiner discovered that allblood is not the same. Blood serum drawn from one person often clumped whenmixed with the blood cells of another. The clumping was caused by differentantigens, proteins, and carbohydrates that trigger antibodies and fight infection.Landsteiner classified blood types based on the presence or absence of the antigensA, B, and Rh in red blood cells. The Venn diagram in Figure 2.24 contains eightregions representing the eight common blood groups.

In the Venn diagram, blood with the Rh antigen is labeled positive and bloodlacking the Rh antigen is labeled negative. The region where the three circlesintersect represents type indicating that a person with this blood type has theantigens A, B, and Rh. Observe that type O blood (both positive and negative) lacksA and B antigens. Type lacks all three antigens, A, B, and Rh.

In blood transfusions, the recipient must have all or more of the antigens presentin the donor’s blood. This discovery rescued surgery patients from random, oftenlethal, transfusions. This knowledge made the massive blood drives during WorldWar I possible. Eventually, it made the modern blood bank possible as well.

O-

AB+,

BLOOD TYPES AND VENN DIAGRAMS

BL

ITZ

ER

BO

NU

S

Exercise Set 2.4

A−

U

Rh

AB− B−

A+AB+

B+

O−O+

A B

F I G U R E 2.24 Human blood types

BLTZMC02_043-102-hr1 2-11-2009 9:30 Page 83




13. 14.

15. 16.

17. 18.

19. 20.

21. 22.

23. 24.

In Exercises 25–32, use the Venn diagram shown to answer eachquestion.

25. Which regions represent set

26. Which regions represent set

27. Which regions represent







33. 34.

35. 36.

37. 38. 1B ´ C2¿1A ´ B2¿

B ´ CA ´ B

BA

AU

B

C13

45

6

12

78

9

12

3

1011

C¿?

B¿?

A ¨ C?

A ¨ B?

B ´ C?

A ´ C?

C?

B?

I

AU

B

C

II III

IV

V

VI

VII

VIII

1B ´ C2¿ ¨ A1A ´ B2¿ ¨ C

1A ¨ B ¨ C2¿1A ´ B ´ C2¿

1C¿ ¨ A2 ´ 1C¿ ¨ B¿21A¿ ¨ B2 ´ 1A¿ ¨ C¿2

C¿ ¨ 1A ´ B¿2A¿ ¨ 1B ´ C¿2

1A ¨ B2 ´ 1A ¨ C21A ´ B2 ¨ 1A ´ C2

A ¨ 1B ´ C2A ´ 1B ¨ C2

C = 5b, c, d, e, f6.

B = 5b, g, h6

A = 5a, g, h6

U = 5a, b, c, d, e, f, g, h6

39. 40.

41. 42.

43. 44.

In Exercises 45–48, construct a Venn diagram illustrating thegiven sets.

45.

46.

47.

48.

Use the Venn diagram shown to solve Exercises 49–52.

49. a. Which region represents b. Which region represents c. Based on parts (a) and (b), what can you conclude?

50. a. Which regions represents b. Which regions represents c. Based on parts (a) and (b), what can you conclude?

51. a. Which region(s) represents b. Which region(s) represents c. Based on parts (a) and (b), are and

equal for all sets and Explain youranswer.

52. a. Which region(s) represents b. Which region(s) represents

c. Based on parts (a) and (b), are and equal for all sets and Explain your

answer.

In Exercises 53–58, use the Venn diagram for Exercises 49–52to determine whether the given sets are equal for all sets and

53. 54.

55. 56.

57. 58. 1A ´ B¿2¿, A¿ ¨ B1A¿ ¨ B2¿, A ´ B¿

1A ´ B2¿, A¿ ¨ B1A ´ B2¿, 1A ¨ B2¿

A¿ ¨ B, A ´ B¿A¿ ´ B, A ¨ B¿

B.A

B?AA¿ ´ B¿

1A ´ B2¿

A¿ ´ B¿?1A ´ B2¿?

B?AA¿ ¨ B¿

1A ¨ B2¿A¿ ¨ B¿?1A ¨ B2¿?

B ´ A?A ´ B?

B ¨ A?A ¨ B?

UA B

I II

IV

III

U = 5x1 , x2 , x3 , x4 , x5 , x6 , x7 , x8 , x96

C = 5x3 , x4 , x5 , x6 , x96

B = 5x1 , x2 , x3 , x5 , x66

A = 5x3 , x96

U = 5+ , - , * , , , ¿ , ¡ , : , 4 , '6

C = 5¿ , ¡ , : , 4 6 B = 5* , , , : 6 A = 5+ , - , * , , , : , 4 6

C = 5e, f, g6, U = 5a, b, c, d, e, f, g, h, i6

A = 5a, e, h, i6, B = 5b, c, e, f, h, i6,

C = 53, 4, 76, U = 51, 2, 3, 4, 5, 6, 7, 8, 96

A = 54, 5, 6, 86, B = 51, 2, 4, 5, 6, 76,

1A ´ B ´ C2¿1A ¨ B ¨ C2¿

A ´ B ´ CA ¨ B ¨ C

A ¨ CA ¨ B

BLTZMC02_043-102-hr1 2-11-2009 9:30 Page 84


Use the Venn diagram shown to solve Exercises 59–62.

59. a. Which regions represent








c. Based on parts (a) and (b), are andequal for all sets and Explain

your answer.



c. Based on parts (a) and (b), are andequal for all sets and Explain

your answer.

In Exercises 63–68, use the Venn diagram shown above todetermine which statements are true for all sets and and,consequently, are theorems.

63.64.65.66.67.68.

Practice Plus69. a. Let and

Find and

b. Let and Find and

c. Based on your results in parts (a) and (b), use inductivereasoning to write a conjecture that relates and

d. Use deductive reasoning to determine whether yourconjecture in part (c) is a theorem.

70. a. Let andFind and

b. Let and Find andA¿ ¨ 1B¿ ¨ C2.

1A ´ B2¿ ¨ CU = 5a, b, c, Á , h6.A = 5d, f, g, h6, B = 5a, c, f, h6, C = 5c, e, g, h6,

A¿ ¨ 1B¿ ¨ C2.1A ´ B2¿ ¨ CU = 51, 2, 3, 4, 5, 66.

A = 536, B = 51, 26, C = 52, 46,

1A ´ B¿2 ¨ 1A ´ C¿2.A ´ 1B¿ ¨ C¿2

1A ´ B¿2 ¨ 1A ´ C¿2.A ´ 1B¿ ¨ C¿2U = 51, 2, 3, Á , 86.

A = 51, 3, 7, 86, B = 52, 3, 6, 76, C = 54, 6, 7, 86,1A ´ B¿2¨ 1A ´ C¿2.

A ´ 1B¿ ¨ C¿2c, d, e, f6.U = 5a, b, A = 5c6, B = 5a, b6, C = 5b, d6,

A ´ 1B ¨ C2¿ = A ´ 1B¿ ´ C¿2

A ¨ 1B ´ C2¿ = A ¨ 1B¿ ¨ C¿2

B ¨ 1A ´ C2 = 1A ¨ B2 ´ 1B ¨ C2

B ´ 1A ¨ C2 = 1A ´ B2 ¨ 1B ´ C2

A ´ 1B ¨ C2 = 1A ´ B2 ¨ C

A ¨ 1B ´ C2 = 1A ¨ B2 ´ C

C,A, B,

C?A, B,C ¨ 1B ´ A2C ´ 1B ¨ A2

C ¨ 1B ´ A2?

C ´ 1B ¨ A2?

C?A, B,A ´ 1B ¨ C2A ¨ 1B ´ C2

A ´ 1B ¨ C2?

A ¨ 1B ´ C2?

1A ¨ C2 ´ 1B ¨ C2?

1A ´ B2 ¨ C?

1A ´ C2 ¨ 1B ´ C2?

1A ¨ B2 ´ C?

I

AU

B

C

II III

IV

V

VI

VII

VIII

c. Based on your results in parts (a) and (b), use inductivereasoning to write a conjecture that relates and

d. Use deductive reasoning to determine whether yourconjecture in part (c) is a theorem.

In Exercises 71–78, use the symbols and asnecessary, to describe each shaded region. More than one correctsymbolic description may be possible.

71. U A B

¿,A, B, C, ¨ , ´ ,

A¿ ¨ 1B¿ ¨ C2.

72.

73. U A B

U A B

74.

75.A

UB

C

U A B

76.

77.A

UB

C

AU

B

C

78.

Application ExercisesA math tutor working with a small study group has classifiedstudents in the group by whether or not they scored 90% or aboveon each of three tests. The results are shown in the Venn diagram.

LilyEmma

Ann Al Gavin

AmyJose

RonGrace

LeeMaria

FredBen

SheilaEllenGary

Exam 1≥ 90%

Exam 2≥ 90%

Exam 3≥ 90%

AU

B

C

BLTZMC02_043-102-hr1 2-11-2009 9:30 Page 85


In Exercises 79–90, use the Venn diagram at the bottom of theprevious page to represent each set in roster form.

79. The set of students who scored 90% or above on exam 280. The set of students who scored 90% or above on exam 381. The set of students who scored 90% or above on exam 1

and exam 382. The set of students who scored 90% or above on exam 1

and exam 283. The set of students who scored 90% or above on exam 1 and

not on exam 284. The set of students who scored 90% or above on exam 3 and

not on exam 185. The set of students who scored 90% or above on exam 1 or

not on exam 286. The set of students who scored 90% or above on exam 3 or

not on exam 187. The set of students who scored 90% or above on exactly

one test88. The set of students who scored 90% or above on at least

two tests89. The set of students who scored 90% or above on exam 2 and

not on exam 1 and exam 390. The set of students who scored 90% or above on exam 1 and

not on exam 2 and exam 391. Use the Venn diagram shown at the bottom of the previous

page to describe a set of students that is the empty set.92. Use the Venn diagram shown at the bottom of the previous

page to describe the set

The chart shows the most popular shows on television in 2006,2007, and 2008.

Sheila, Ellen, Gary6.5Fred, Ben,

In Exercises 93–98, use the Venn diagram to indicate in whichregion, I through VIII, each television show should be placed.

I

U

II III

IV

V

VI

VII

VIII

2006 2007

2008

MOST POPULAR TELEVISION SHOWS

2006 2007 2008

1. American Idol 1. American Idol 1. American Idol

2. CSI 2. Dancing withthe Stars

2. Dancing withthe Stars

3. DesperateHousewives

3. CSI 3. NBC SundayNight Football

4. Grey’sAnatomy

4. Grey’sAnatomy

4. CSI

5. Without a Trace 5. House 5. Grey’s Anatomy

6. Dancing withthe Stars

6. DesperateHousewives

6. SamanthaWho?

Source: Nielsen Media Research

93. Desperate Housewives

94. Samantha Who?

95. CSI

96. Grey’s Anatomy

97. House

98. 60 Minutes

The chart shows the top single recordings of all time.

TOP SINGLE RECORDINGS

TitleArtist orGroup Sales

YearReleased

“Candle in theWind” Elton John 37 million 1997

“WhiteChristmas” Bing Crosby 30 million 1942

“Rock Aroundthe Clock”

Bill Haley andHis Comets 17 million 1954

“I Want to HoldYour Hand” The Beatles 12 million 1963

“It’s Now orNever” Elvis Presley 10 million 1960

“Hey Jude” The Beatles 10 million 1968

“I Will AlwaysLove You”

WhitneyHouston 10 million 1992

“Hound Dog” Elvis Presley 9 million 1956

“Diana” Paul Anka 9 million 1957

“I’m aBeliever” The Monkees 8 million 1966

Source: RIAA

In Exercises 99–104, use the Venn diagram to indicate in whichregion, I through VIII, each recording should be placed.

99. “Candle in the Wind”

100. “White Christmas”

101. “I Want to Hold Your Hand”

102. “Hey Jude”

103. “Diana”

104. “I’m a Believer”

I

Recordedby a group

USales exceeded

11 million

Recordedbefore 1990

IIIII

IV

V

VI

VII

VIII

BLTZMC02_043-102-hr1 2-11-2009 9:30 Page 86


105. The chart shows three health indicators for seven countriesor regions.


109. I constructed a Venn diagram for three sets by placingelements into the outermost region and working inward.

110. This Venn diagram showing color combinations from red,green, and blue illustrates that white is a combination of allthree colors and black uses none of the colors.

111. I used a Venn diagram to prove that and are not equal.

112. I found 50 examples of two sets, and for whichand resulted in the same set, so this

proves that

The eight blood typesdiscussed in the Blitzer Bonuson page 83 are shown onceagain in the Venn diagram. Inblood transfusions, the set ofantigens in a donor’s bloodmust be a subset of the set ofantigens in a recipient’s blood.Thus, the recipient must haveall or more of the antigenspresent in the donor’s blood.Use this information to solveExercises 113–116.

113. What is the blood type of a universal recipient?

114. What is the blood type of a universal donor?

115. Can an person donate blood to an person?

116. Can an person donate blood to an person?

Group Exercises117. Each group member should find out his or her blood type.

(If you cannot obtain this information, select a blood typethat you find appealing!) Read the introduction toExercises 113–116. Referring to the Venn diagram forthese exercises, each group member should determine allother group members to whom blood can be donated andfrom whom it can be received.

118. The group should define three sets, each of whichcategorizes the set of students in the group, in differentways. Examples include the set of students with blonde hair,the set of students no more than 23 years old, and the set ofstudents whose major is undecided. Once you have definedthe sets, construct a Venn diagram with three intersectingsets and eight regions. Each student should determine whichregion he or she belongs to. Illustrate the sets by writing eachfirst name in the appropriate region.

U,

A+A-

A-A+

1A ´ B2¿ = A¿ ¨ B¿.A¿ ¨ B¿1A ´ B2¿

B,A

A¿ ´ B¿1A ´ B2¿

FILMS WITH THE MOST OSCAR NOMINATIONS

Film Nominations Awards Year

All About Eve 14 6 1950

Titanic 14 11 1997

Gone with the Wind 13 8 1939

From Here to Eternity 13 8 1953

Shakespeare in Love 13 7 1998

Mary Poppins 13 5 1964

Who’s Afraid of VirginiaWoolf? 13 5 1966

Forrest Gump 13 6 1994

The Lord of the Rings:The Fellowship of the Ring 13 4 2001

Chicago 13 6 2004

Source: Academy of Motion Picture Arts and Sciences

WORLDWIDE HEALTH INDICATORS

Country/Region

Male LifeExpectancy

Female LifeExpectancy

Persons perDoctor

United States 74.8 80.1 360

Italy 77.6 83.2 180

Russia 59.9 73.3 240

East Africa 46.9 48.2 13,620

Japan 78.6 85.6 530

England 75.9 81.0 720

Iran 68.6 71.4 1200

Source: Time Almanac 2009

Let set of countries/regions shown in the chart,set of countries/regions with male life expectancy

that exceeds 75 years, set of countries/regions withfemale life expectancy that exceeds 80 years, and set of countries/regions with fewer than 400 persons perdoctor. Use the information in the chart to construct aVenn diagram that illustrates these sets.

106. The chart shows the ten films nominated for the most Oscars.

C = theB = the

A = theU = the

Using abbreviated film titles, let Eve, Titanic, Wind,Eternity, Love, Poppins, Woolf, Gump, Ring, Chicago

set of films nominated for 14 Oscars, setof films that won at least 7 Oscars, and set of filmsthat won Oscars after 1965. Use the information in the chartto construct a Venn diagram that illustrates these sets.

Writing in Mathematics107. If you are given four sets, and describe what is

involved in determining Be as specific aspossible in your description.

108. Describe how a Venn diagram can be used to prove thatand are equal sets.A¿ ¨ B¿1A ´ B2¿

1A ´ B2¿ ¨ C.U,C,B,A,

C = theB = theA = the

6,U = 5

A−

U

Rh

AB− B−

A+AB+

B+

O−O+

A B

Human blood types

BLTZMC02_043-102-hr1 2-11-2009 9:30 Page 87

2.5 Survey Problems

Mexico’s 2006 election, its closest-ever race for president, exposed an emergingtrend in Latin America: a sharpening divide between the rich and the poor.Although Mexico’s economy expanded by 4.4% in 2004 and 3% in 2005, at the endof 2005, almost half of the country’s 107 million people still lived in poverty. (Source:Newsweek, July 17, 2006) Figure 2.25 shows that Mexicans tend to see societalinjustice, rather than personal laziness, as the primary cause of poverty.


1Use Venn diagrams to visualize

a survey’s results.

O B J E C T I V E S

1 Use Venn diagrams to visualize

a survey’s results.

2 Use survey results to complete

Venn diagrams and answer

questions about the survey.

Per

cent

Assessing the Causes of Poverty

40

30

10

0

20

70

50

60

Primary cause is personal laziness.Primary cause is societal injustice.

U.S.

40%

60%

Japan

41%

59%

Australia

51%

49%

Mexico

66%

25%

Sweden

61%

17%

F I G U R E 2.25 Percentages for eachcountry may not total 100% because lessfrequently identified primary causes ofpoverty were omitted from the graph.Source: Ronald Inglehart et al., World ValuesSurveys and European Values Surveys

Suppose a survey is taken that asks randomly selected adults in the UnitedStates and Mexico the following question:

Do you agree or disagree that the primary cause of poverty is societal injustice?In this section, you will see how sets and Venn diagrams are used to tabulateinformation collected in such a survey. In survey problems, it is helpful to rememberthat and means intersection, or means union, and not means complement.Furthermore, but means the same thing as and. Thus, but means intersection.

Visualizing the Results of a Survey

In Section 2.1, we defined the cardinal number of set denoted by as thenumber of elements in set Venn diagrams are helpful in determining a set’scardinality.

Using a Venn Diagram to Visualize the Results of a Survey

We return to the campus survey in which students were asked two questions:Would you be willing to donate blood?Would you be willing to help serve a free breakfast to blood donors?

EXAMPLE 1

A.n1A2,A,

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S E C T I O N 2 . 5 Survey Problems 89

A B

A: Set of students willing to donate bloodB: Set of students willing to serve breakfast to donors

I II III

120 220

290

370

IV

F I G U R E 2.26 Results of a survey

Set represents the set of students willing to donate blood. Set represents the setof students willing to help serve breakfast to donors. The survey results aresummarized in Figure 2.26. Use the diagram to answer the following questions:

a. How many students are willing to donate blood?b. How many students are willing to help serve a free breakfast to blood donors?c. How many students are willing to donate blood and serve breakfast?d. How many students are willing to donate blood or serve breakfast?e. How many students are willing to donate blood but not serve breakfast?f. How many students are willing to serve breakfast but not donate blood?g. How many students are neither willing to donate blood nor serve breakfast?h. How many students were surveyed?

Solutiona. The number of students willing to donate blood can be determined by adding

the numbers in regions I and II. Thus, There are490 students willing to donate blood.

b. The number of students willing to help serve a free breakfast to blooddonors can be determined by adding the numbers in regions II and III.Thus, There are 340 students willing to helpserve breakfast.

c. The number of students willing to donate blood and serve breakfastappears in region II, the region representing the intersection of the two sets.Thus, There are 120 students willing to donate blood andserve breakfast.

d. The number of students willing to donate blood or serve breakfast is foundby adding the numbers in regions I, II, and III, representing the union of thetwo sets. We see that Therefore,710 students in the survey are willing to donate blood or serve breakfast.

e. The region representing students who are willing to donate blood but notserve breakfast, is region I. We see that 370 of the students surveyedare willing to donate blood but not serve breakfast.

f. Region III represents students willing to serve breakfast but not donateblood: We see that 220 students surveyed are willing to help servebreakfast but not donate blood.

g. Students who are neither willing to donate blood nor serve breakfast,fall within the universal set, but outside circles and These

students fall in region IV, where the Venn diagram indicates that there are290 elements. There are 290 students in the survey who are neither willing todonate blood nor serve breakfast.

h. We can find the number of students surveyed by adding the numbers inregions I, II, III, and IV. Thus,There were 1000 students surveyed.

In a survey on musical tastes, respondents were asked: Do you listen toclassical music? Do you listen to jazz? The survey results are summarizedin Figure 2.27. Use the diagram to answer the following questions.

a. How many respondents listened to classical music?b. How many respondents listened to jazz?c. How many respondents listened to both classical music and jazz?d. How many respondents listened to classical music or jazz?e. How many respondents listened to classical music but not jazz?f. How many respondents listened to jazz but not classical music?g. How many respondents listened to neither classical music nor

jazz?h. How many people were surveyed?

1CHEC

KPOIN

T

n1U2 = 370 + 120 + 220 + 290 = 1000.

B.AA¿ ¨ B¿,

B ¨ A¿.

A ¨ B¿,

n1A ´ B2 = 370 + 120 + 220 = 710.

n1A ¨ B2 = 120.

n1B2 = 120 + 220 = 340.

n1A2 = 370 + 120 = 490.

BA

30

ClassicalMusic

Jazz

55 20 70

F I G U R E 2.27

BLTZMC02_043-102-hr1 2-11-2009 9:30 Page 89


Solving Survey Problems

Venn diagrams are used to solve problems involving surveys. Here are the stepsneeded to solve survey problems:

2Use survey results to complete

Venn diagrams and answer

questions about the survey.

SOLVING SURVEY PROBLEMS

1. Use the survey’s description to define sets and draw a Venn diagram.

2. Use the survey’s results to determine the cardinality for each region in theVenn diagram. Start with the intersection of the sets, the innermost region,and work outward.

3. Use the completed Venn diagram to answer the problem’s questions.

Surveying People’s Attitudes

A survey is taken that asks 2000 randomly selected U.S. and Mexican adults thefollowing question:

Do you agree or disagree that the primary cause of poverty is societal injustice?The results of the survey showed that

1060 people agreed with the statement.400 Americans agreed with the statement.Source: World Values Surveys

If half the adults surveyed were Americans,a. How many Mexicans agreed with the statement?b. How many Mexicans disagreed with the statement?

SolutionStep 1 Define sets and draw a Venn diagram. The Venn diagram in Figure 2.28shows two sets. Set U.S. is the set of Americans surveyed. Set (labeled “Agree”) isthe set of people surveyed who agreed with the statement. By representing theAmericans surveyed with circle U.S., we do not need a separate circle for theMexicans. The group of people outside circle U.S. must be the set of Mexicans.Similarly, by visualizing the set of people who agreed with the statement as circle we do not need a separate circle for those who disagreed.The group of people outsidecircle (Agree) must be the set of people disagreeing with the statement.

Step 2 Determine the cardinality for each region in the Venn diagram, starting withthe innermost region and working outward. We are given the following cardinalities:

There were 2000 people surveyed:Half the people surveyed were Americans:The number of people who agreed with the statement was 1060:There were 400 Americans who agreed with the statement:

Now let’s use these numbers to determine the cardinality of each region, startingwith region II, moving outward to regions I and III, and ending with region IV.

II400

I III

U.S.U

IV

A

Start with region II.

II400600

I III

U.S.U

IV

A

Move out to region I.

II represents the set of Americanswho agreed with the statement.

We are given that n(U.S. � A) = 400.

We are given that set U.S.,regions I and II, contains 1000 people:

n(U.S.) = 1000. With 400 elements in II,this leaves 1000 − 400 = 600 people in I.

n1U.S. ¨ A2 = 400.n1A2 = 1060.

n1U.S.2 = 1000.n1U2 = 2000.

A

A,

A

EXAMPLE 2

II

U.S.

UA

(Agree)

III

IV

I

F I G U R E 2.28

BLTZMC02_043-102-hr1 2-11-2009 9:30 Page 90


Step 3 Use the completed Venn diagram to answer the problem’s questions. Thecompleted Venn diagram that illustrates the survey’s results is shown in Figure 2.29.

a. The Mexicans who agreed with the statement are those members of the setof people who agreed who are not Americans, shown in region III. Thismeans that 660 Mexicans agreed that societal injustice is the primary causeof poverty.

b. The Mexicans who disagreed with the statement can be found outside thecircles of people who agreed and people who are Americans. Thiscorresponds to region IV, whose cardinality is 340. Thus, 340 Mexicansdisagreed that societal injustice is the primary cause of poverty.

In a Gallup poll, 2000 U.S. adults were selected at random and asked toagree or disagree with the following statement:

Job opportunities for women are not equal to those for men.The results of the survey showed that

1190 people agreed with the statement.700 women agreed with the statement.Source: The People’s Almanac

If half the people surveyed werewomen,

a. How many men agreed withthe statement?

b. How many men disagreedwith the statement?

2CHEC

KPOIN

T

Move out to region III.

II400600 660

I III

U.S.U

IV

A

II400600 660

I III

U.S.U

IV340

A

End with IV, the outer region.

We are given that set U, regions I, II,III, and IV, contains 2000 people:

n(U) = 2000. With 600 + 400 + 660= 1660 elements in I, II, and III, this

leaves 2000 − 1660 = 340 people in IV.

We are given that set A, regions II and III,contains 1060 people: n(A) = 1060.With 400 elements in II, this leaves1060 − 400 = 660 people in III.

II400600 660

I III

U.S.U

IV340

A

Is the Primary Cause of PovertySocietal Injustice?

F I G U R E 2.29

II

W(Women)

UA

(Agree)

III

IV

I

When tabulating survey results, more than two circles within a Venn diagramare often needed. For example, consider a Time/CNN poll that sought to determinehow Americans felt about reserving a certain number of college scholarshipsexclusively for minorities and women. Respondents were asked the following question:

Do you agree or disagree with the following statement: Colleges shouldreserve a certain number of scholarships exclusively for minorities and women?Source: Time Almanac

Suppose that we want the respondents to the poll to be identified by gender (man orwoman), ethnicity (African American or other), and whether or not they agreedwith the statement. A Venn diagram into which the results of the survey can betabulated is shown in Figure 2.30.

Based on our work in Example 2, we only used one circle in the Venn diagramto indicate the gender of the respondent. We used for men, so the set of womenrespondents, consists of the regions outside circle Similarly, we used forthe set of African-American respondents, so the regions outside circle account forall other ethnicities. Finally, we used for the set of respondents who agreed withthe statement. Those who disagreed lie outside circle A.

AB

BM.M¿,M

I

M (Men)U

B (African American)

A (Agree)

II III

IVV

VI

VIII

VII

F I G U R E 2.30

BLTZMC02_043-102-hr1 2-11-2009 9:30 Page 91


In the next example, we create a Venn diagram with three intersecting sets toillustrate a survey’s results. In our final example, we use this Venn diagram to answerquestions about the survey.

Constructing a Venn Diagram for a Survey

Sixty people were contacted and responded to a movie survey. The followinginformation was obtained:

a. 6 people liked comedies, dramas, and science fiction.b. 13 people liked comedies and dramas.c. 10 people liked comedies and science fiction.d. 11 people liked dramas and science fiction.e. 26 people liked comedies.f. 21 people liked dramas.g. 25 people liked science fiction.

Use a Venn diagram to illustrate the survey’s results.

Solution The set of people surveyed is a universal set with 60 elementscontaining three subsets:

We draw these sets in Figure 2.31. Now let’s use the numbers in (a) through (g), aswell as the fact that 60 people were surveyed, which we call condition (h), todetermine the cardinality of each region in the Venn diagram.

S = the set of those who like science fiction. D = the set of those who like dramas C = the set of those who like comedies

EXAMPLE 3

CU

D

S

7

6

10

9 3

16

4 5

(e) 26 people liked comedies:n(C) = 26. With 7 + 6 + 4 = 17counted, there are 26 − 17 = 9people in this region.

(f) 21 people liked dramas: n(D) = 21.With 7 + 6 + 5 = 18 counted,there are 21 − 18 = 3 people in thisregion.

(g) 25 people liked science fiction:n(S) = 25. With 4 + 6 + 5 = 15counted, there are 25 − 15 = 10people in this region.

(h) 60 people were surveyed: n(U) = 60.With 9 + 7 + 3 + 4 + 6 + 5 + 10 = 44counted, there are 60 − 44 = 16 peoplein this region.

S

7

64 5

UC D

(c) 10 people liked comediesand science fiction:n(C � S) = 10.With 6 counted, there are10 − 6 = 4 people in thisregion.

(d) 11 people liked drama andscience fiction:n(D � S) = 11.With 6 counted, there are11 − 6 = 5 people in thisregion.

S

7

6

CU

D

(a) 6 people liked comedies,drama, and science fiction:n(C � D � S) = 6.

(b) 13 people liked comediesand drama: n(C � D) = 13. With 6 counted, there are13 − 6 = 7 people inthis region.

CU

D

S

F I G U R E 2.31

STUDY TIP

After entering the cardinality ofthe innermost region, workoutward and use subtraction toobtain subsequent cardinalities.The phrase

SURVEY SUBTRACT

is a helpful reminder of theserepeated subtractions.

With a cardinality in each region, we have completed the Venn diagram thatillustrates the survey’s results.

BLTZMC02_043-102-hr1 2-11-2009 9:30 Page 92


A survey of 250 memorabilia collectors showed the following results:108 collected baseball cards. 92 collected comic books. 62 collectedstamps. 29 collected baseball cards and comic books. 5 collectedbaseball cards and stamps. 2 collected comic books and stamps.2 collected all three types of memorabilia. Use a Venn diagram toillustrate the survey’s results.

Using a Survey’s Venn Diagram

The Venn diagram in Figure 2.32 shows the results of the movie survey inExample 3. How many of those surveyed liked

a. comedies, but neither dramas nor science fiction?b. dramas and science fiction, but not comedies?c. dramas or science fiction, but not comedies?d. exactly one movie style?e. at least two movie styles?f. none of the movie styles?

Solutiona. Those surveyed who liked comedies, but neither dramas nor science fiction,

are represented in region I. There are 9 people in this category.b. Those surveyed who liked dramas and science fiction, but not comedies, are

represented in region VI. There are 5 people in this category.c. We are interested in those surveyed who liked dramas or science fiction, but

not comedies:

EXAMPLE 4

3CHEC

KPOIN

T

(D � S) � C'

C � (D' � S')

The intersection of the regions in the voice balloons consists of the commonregions shown in red, III, VI, and VII. There are elementsin these regions. There are 18 people who liked dramas or science fiction, butnot comedies.

d. Those surveyed who liked exactly one movie style are represented in regionsI, III, and VII. There are elements in these regions. Thus,22 people liked exactly one movie style.

e. Those surveyed who liked at least two movie styles are people who liked twoor more types of movies. People who liked two movie styles are representedin regions II, IV, and VI. Those who liked three movie styles are representedin region V. Thus, we add the number of elements in regions II, IV, V, and VI:

Thus, 22 people liked at least two movie styles.f. Those surveyed who liked none of the movie styles are represented in region

VIII. There are 16 people in this category.

Use the Venn diagram you constructed in Check Point 3 to determinehow many of those surveyed collected

a. comic books, but neither baseball cards nor stamps.b. baseball cards and stamps, but not comic books.c. baseball cards or stamps, but not comic books.d. exactly two types of memorabilia.e. at least one type of memorabilia.f. none of the types of memorabilia.

4CHEC

KPOIN

T

7 + 4 + 6 + 5 = 22.

9 + 3 + 10 = 22

3 + 5 + 10 = 18

Dramas or science fiction, but not comedies

(D � S) � C '.

Regions II, III, IV, V, VI, VII Regions III, VI, VII, VIII

CU

D

S

II7

V6

VII10

I9

III3

VIII16

IV4

VI5

F I G U R E 2.32

BLTZMC02_043-102-hr1 2-11-2009 9:30 Page 93


Practice ExercisesUse the accompanying Venn diagram, which shows the numberof elements in regions I through IV, to answer the questions inExercises 1–8.

1. How many elements belong to set

2. How many elements belong to set

3. How many elements belong to set but not set

4. How many elements belong to set but not set

5. How many elements belong to set or set

6. How many elements belong to set and set

7. How many elements belong to neither set nor set

8. How many elements are there in the universal set?

Use the accompanying Venn diagram, which shows the number ofelements in region II, to answer Exercises 9–10.

9. If and find the numberof elements in each of regions I, III, and IV.

10. If and find the numberof elements in each of regions I, III, and IV.

Use the accompanying Venn diagram, which shows the cardinalityof each region, to answer Exercises 11–26.

11. How many elements belong to set 12. How many elements belong to set 13. How many elements belong to set but not set 14. How many elements belong to set but not set 15. How many elements belong to set or set 16. How many elements belong to set or set B?A

C?A

A?B

C?A

A?B?

5

U

1 6

27

3

8

10

BA

C

n1U2 = 53,n1A2 = 23, n1B2 = 27,

n1U2 = 48,n1A2 = 21, n1B2 = 29,

UA B

I II7

IV

III

B?A

B?A

B?A

A?B

B?A

B?

A?

II

AU

B

III

IV7

I

17 9 11

17. How many elements belong to set and set 18. How many elements belong to set and set 19. How many elements belong to set and set but not to

set20. How many elements belong to set and set but not to

set21. How many elements belong to set or set but not to

set22. How many elements belong to set or set but not to

set23. Considering sets and how many elements belong to

exactly one of these sets?24. Considering sets and how many elements belong to

exactly two of these sets?25. Considering sets and how many elements belong to

at least one of these sets?26. Considering sets and how many elements belong to

at least two of these sets?

The accompanying Venn diagram shows the number of elements inregion V. In Exercises 27–28, use the given cardinalities to determinethe number of elements in each of the other seven regions.

27.

28.

Practice PlusIn Exercises 29–32, use the Venn diagram and the given conditionsto determine the number of elements in each region, or explainwhy the conditions are impossible to meet.

29.

n1A ¨ B ¨ C2 = 7

n1A ¨ B2 = 17, n1A ¨ C2 = 11, n1B ¨ C2 = 8,

n1U2 = 38, n1A2 = 26, n1B2 = 21, n1C2 = 18,

I

U

II III

IVV

VI

VIII

VII

BA

C

n1A ¨ B2 = 6, n1A ¨ C2 = 7, n1B ¨ C2 = 8

n1U2 = 32, n1A2 = 21, n1B2 = 15, n1C2 = 14,

n1A ¨ B2 = 3, n1A ¨ C2 = 5, n1B ¨ C2 = 3

n1U2 = 30, n1A2 = 11, n1B2 = 8, n1C2 = 14,

I

U

II III

IV

V2 VI

VIII

VII

BA

C

C,A, B,

C,A, B,

C,A, B,

C,A, B,B?

C,A

A?C,B

B?C,A

A?C,B

B?A

C?A

Exercise Set 2.5

BLTZMC02_043-102-hr1 2-11-2009 9:30 Page 94


(In Exercises 30–32, continue to refer to the Venn diagram at thebottom of the previous page.)

30.

31.

32.

Application ExercisesAs discussed in the text on page 91, a poll asked respondents if theyagreed with the statement

Colleges should reserve a certain number of scholarshipsexclusively for minorities and women.

Hypothetical results of the poll are tabulated in the Venn diagram.Use these cardinalities to solve Exercises 33–38.

33. How many respondents agreed with the statement?

34. How many respondents disagreed with the statement?

35. How many women agreed with the statement?

36. How many people who are not African American agreedwith the statement?

37. How many women who are not African American disagreedwith the statement?

38. How many men who are not African American disagreedwith the statement?

39. A pollster conducting a telephone poll of a city’s residentsasked two questions:

1. Do you currently smoke cigarettes?

2. Regardless of your answer to question 1, would yousupport a ban on smoking in all city parks?

a. Construct a Venn diagram that allows the respondents tothe poll to be identified by whether or not they smokecigarettes and whether or not they support the ban.

b. Write the letter b in every region of the diagram thatrepresents smokers polled who support the ban.

c. Write the letter c in every region of the diagram thatrepresents nonsmokers polled who support the ban.

d. Write the letter d in every region of the diagram thatrepresents nonsmokers polled who do not support theban.

8

M (Men)U

B (African American)

A (Agree)

2 3

4

5

7

9

2

n1A ¨ B ¨ C2 = 5

n1A ¨ B2 = 6, n1A ¨ C2 = 9, n1B ¨ C2 = 8,

n1U2 = 25, n1A2 = 8, n1B2 = 9, n1C2 = 10,

n1A ¨ B ¨ C2 = 2

n1A ¨ B2 = 6, n1A ¨ C2 = 9, n1B ¨ C2 = 7,

n1U2 = 40, n1A2 = 10, n1B2 = 11, n1C2 = 12,

n1A ¨ B ¨ C2 = 5

n1A ¨ B2 = 17, n1A ¨ C2 = 11, n1B ¨ C2 = 9,

n1U2 = 42, n1A2 = 26, n1B2 = 22, n1C2 = 25,

40. A pollster conducting a telephone poll at a college campusasked students two questions:

1. Do you binge drink three or more times per month?2. Regardless of your answer to question 1, are you

frequently behind in your school work?a. Construct a Venn diagram that allows the respondents to

the poll to be identified by whether or not they bingedrink and whether or not they frequently fall behind inschool work.

b. Write the letter b in every region of the diagram thatrepresents binge drinkers who are frequently behind inschool work.

c. Write the letter c in every region of the diagram thatrepresents students polled who do not binge drink butwho are frequently behind in school work.

d. Write the letter d in every region of the diagram thatrepresents students polled who do not binge drink andwho do not frequently fall behind in their school work.

41. A pollster conducting a telephone poll asked three questions:1. Are you religious?

2. Have you spent time with a person during his or herlast days of a terminal illness?

3. Should assisted suicide be an option for terminallyill people?

a. Construct a Venn diagram with three circles that canassist the pollster in tabulating the responses to the threequestions.

b. Write the letter b in every region of the diagram thatrepresents all religious persons polled who are not infavor of assisted suicide for the terminally ill.

c. Write the letter c in every region of the diagram thatrepresents the people polled who do not considerthemselves religious, who have not spent time with aterminally ill person during his or her last days, and whoare in favor of assisted suicide for the terminally ill.

d. Write the letter d in every region of the diagram thatrepresents the people polled who consider themselvesreligious, who have not spent time with a terminally illperson during his or her last days, and who are not infavor of assisted suicide for the terminally ill.

e. Write the letter e in a region of the Venn diagram otherthan those in parts (b)–(d) and then describe who in thepoll is represented by this region.

42. A poll asks respondents the following question:

Do you agree or disagree with this statement: In order toaddress the trend in diminishing male enrollment,colleges should begin special efforts to recruit men?

a. Construct a Venn diagram with three circles that allowsthe respondents to be identified by gender (man orwoman), education level (college or no college), andwhether or not they agreed with the statement.

b. Write the letter b in every region of the diagram thatrepresents men with a college education who agreedwith the statement.

c. Write the letter c in every region of the diagram thatrepresents women who disagreed with the statement.

d. Write the letter d in every region of the diagram thatrepresents women without a college education whoagreed with the statement.

BLTZMC02_043-102-hr1 2-11-2009 9:30 Page 95


e. Write the letter e in a region of the Venn diagram otherthan those in parts (b)–(d) and then describe who in thepoll is represented by this region.

In Exercises 43–48, construct a Venn diagram and determine thecardinality for each region. Use the completed Venn diagram toanswer the questions.

43. A survey of 75 college students was taken to determinewhere they got the news about what’s going on in the world.Of those surveyed, 29 students got the news fromnewspapers, 43 from television, and 7 from both newspapersand television.

Of those surveyed,a. How many got the news from only newspapers?b. How many got the news from only television?c. How many got the news from newspapers or

television?d. How many did not get the news from either newspapers

or television?

44. A survey of 120 college students was taken at registration. Ofthose surveyed, 75 students registered for a math course,65 for an English course, and 40 for both math and English.

Of those surveyed,a. How many registered only for a math course?b. How many registered only for an English course?c. How many registered for a math course or an English

course?d. How many did not register for either a math course or

an English course?

45. A survey of 80 college students was taken to determine themusical styles they listened to. Forty-two students listenedto rock, 34 to classical, and 27 to jazz. Twelve studentslistened to rock and jazz, 14 to rock and classical, and 10 toclassical and jazz. Seven students listened to all threemusical styles.

Of those surveyed,a. How many listened to only rock music?b. How many listened to classical and jazz, but not rock?c. How many listened to classical or jazz, but not rock?d. How many listened to music in exactly one of the

musical styles?e. How many listened to music in at least two of the

musical styles?f. How many did not listen to any of the musical styles?

46. A survey of 180 college men was taken to determineparticipation in various campus activities. Forty-three studentswere in fraternities, 52 participated in campus sports, and35 participated in various campus tutorial programs. Thirteenstudents participated in fraternities and sports, 14 in sports andtutorial programs, and 12 in fraternities and tutorial programs.Five students participated in all three activities.

Of those surveyed,

a. How many participated in only campus sports?

b. How many participated in fraternities and sports, but nottutorial programs?

c. How many participated in fraternities or sports, but nottutorial programs?

d. How many participated in exactly one of theseactivities?

e. How many participated in at least two of theseactivities?

f. How many did not participate in any of the threeactivities?

47. An anonymous survey of college students was taken todetermine behaviors regarding alcohol, cigarettes, andillegal drugs. The results were as follows: 894 drank alcoholregularly, 665 smoked cigarettes, 192 used illegal drugs,424 drank alcohol regularly and smoked cigarettes, 114drank alcohol regularly and used illegal drugs, 119 smokedcigarettes and used illegal drugs, 97 engaged in all threebehaviors, and 309 engaged in none of these behaviors.

Source: Jamie Langille, University of Nevada Las Vegas

a. How many students were surveyed?Of those surveyed,b. How many drank alcohol regularly or smoked

cigarettes?c. How many used illegal drugs only?d. How many drank alcohol regularly and smoked

cigarettes, but did not use illegal drugs?e. How many drank alcohol regularly or used illegal drugs,

but did not smoke cigarettes?f. How many engaged in exactly two of these behaviors?g. How many engaged in at least one of these

behaviors?

48. In the August 2005 issue of Consumer Reports, readerssuffering from depression reported that alternativetreatments were less effective than prescription drugs.Suppose that 550 readers felt better taking prescriptiondrugs, 220 felt better through meditation, and 45 felt bettertaking St. John’s wort. Furthermore, 95 felt better usingprescription drugs and meditation, 17 felt better usingprescription drugs and St. John’s wort, 35 felt better usingmeditation and St. John’s wort, 15 improved using all threetreatments, and 150 improved using none of thesetreatments. (Hypothetical results are partly based onpercentages given in Consumer Reports.)a. How many readers suffering from depression were

included in the report?Of those included in the report,b. How many felt better using prescription drugs or

meditation?c. How many felt better using St. John’s wort only?d. How many improved using prescription drugs and

meditation, but not St. John’s wort?e. How many improved using prescription drugs or

St. John’s wort, but not meditation?f. How many improved using exactly two of these

treatments?g. How many improved using at least one of these

treatments?

Writing in Mathematics49. Suppose that you are drawing a Venn diagram to sort and

tabulate the results of a survey. If results are beingtabulated along gender lines, explain why only a circlerepresenting women is needed, rather than two separatecircles representing the women surveyed and the mensurveyed.

BLTZMC02_043-102-hr1 2-11-2009 9:30 Page 96


Percentage of United States Adults Suffering from Various Ailments

Smokers Nonsmokers

0Percentage of Adults

5% 10% 20%15% 25% 30%

Depression

FrequentHangovers

Anxiety/PanicDisorder

SeverePain

28%12%

20%10%

19%8%

14%7%

Source: MARS OTC/DTC

50. Suppose that you decide to use two sets, and to sortand tabulate the responses for men and women in asurvey. Describe the set of people represented by regionsII and IV in the Venn diagram shown. What conclusion canyou draw?

Critical Thinking ExercisesMake Sense? In Exercises 51–54,determine whether each statementmakes sense or does not make sense, and explain your reasoning.

51. A survey problem must present the information in exactlythe same order in which I determine cardinalities frominnermost to outermost region.

Exercises 52–54 are based on the graph that shows the percentageof smokers and nonsmokers suffering from various ailments. Usethe graph to determine whether each statement makes sense.

II

MU

W

III

IV

I

W,M 53. I improved the Venn diagram in Exercise 52 by adding athird circle for nonsmokers.

54. I used a single Venn diagram to represent all the datadisplayed by the bar graph.


55. In a survey, 110 students were taking mathematics, 90 weretaking psychology, and 20 were taking neither. Thus,220 students were surveyed.

56. If then

57. When filling in cardinalities for regions in a two-set Venndiagram, the innermost region, the intersection of the twosets, should be the last region to be filled in.

58. can be obtained by subtracting from

59. In a survey of 150 students, 90 were taking mathematics and30 were taking psychology.

a. What is the least number of students who could havebeen taking both courses?

b. What is the greatest number of students who could havebeen taking both courses?

c. What is the greatest number of students who could havebeen taking neither course?

60. A person applying for the position of college registrarsubmitted the following report to the college president on90 students: 31 take math; 28 take chemistry; 42 takepsychology; 9 take math and chemistry; 10 take chemistry andpsychology; 6 take math and psychology; 4 take all threesubjects; and 20 take none of these courses. The applicantwas not hired. Explain why.

Group Exercise61. This group activity is intended to provide practice in the

use of Venn diagrams to sort responses to a survey. Thegroup will determine the topic of the survey. Although youwill not actually conduct the survey, it might be helpful toimagine carrying out the survey using the students on yourcampus.

a. In your group, decide on a topic for the survey.

b. Devise three questions that the pollster will ask to thepeople who are interviewed.

c. Construct a Venn diagram that will assist the pollster insorting the answers to the three questions. The Venndiagram should contain three intersecting circles withina universal set and eight regions.

d. Describe what each of the regions in the Venn diagramrepresents in terms of the questions in your poll.

n1U2.n1A2n1A¿2

n1A ´ B2 = n1A2 + n1B2.A ¨ B = �,

52. I represented the data for depression using the followingVenn diagram:

28%

SmokersU Suffer

Depression

12%

88%

72%

BLTZMC02_043-102-hr1 2-11-2009 9:30 Page 97


Chapter Summary, Review, and Test

Summary – Definitions and Concepts Examples2.1 Basic Set Concepts

a. A set is a collection of objects whose contents can be clearly determined. The objects in a set are called theelements, or members, of the set.

b. Sets can be designated by word descriptions, the roster method (a listing within braces, separating elementswith commas), or set-builder notation:

The setof

allelements x

suchthat

x meetsthese conditions

{ x | condition (s) }.

Ex. 1, p. 44;Ex. 2, p. 45;Ex. 3, p. 45

c. The empty set, or the null set, represented by or is a set that contains no elements.�,5 6 Ex. 4, p. 47

d. The symbol means that an object is an element of a set. The symbol means that an object is not anelement of a set.

xH Ex. 5, p. 47

e. The set of natural numbers is Inequality symbols, summarized in Table 2.2 on page 49,are frequently used to describe sets of natural numbers.

N = 51, 2, 3, 4, 5, Á 6. Ex. 6, p. 48;Ex. 7, p. 49

f. The cardinal number of a set is the number of distinct elements in set Repeating elements in a setneither adds new elements to the set nor changes its cardinality.

A.n1A2,A, Ex. 8, p. 50

g. Equivalent sets have the same number of elements, or the same cardinality. A one-to-one correspondencebetween sets and means that each element in can be paired with exactly one element in and viceversa. If two sets can be placed in a one-to-one correspondence, then they are equivalent.

B,ABAEx. 9, p. 51

h. Set is a finite set if or if is a natural number. A set that is not finite is an infinite set.n1A2n1A2 = 0A

i. Equal sets have exactly the same elements, regardless of order or possible repetition of elements. If two setsare equal, then they must be equivalent.

Ex. 10, p. 53

2.2 Subsets

a. Set is a subset of set expressed as if every element in set is also in set The notation means that set is not a subset of set so there is at least one element of set that is not an element of set B.AB,A

A h BB.AA 8 B,B,A Ex. 1, p. 57

b. Set is a proper subset of set expressed as if is a subset of and A Z B.BAA ( B,B,A Ex. 2, p. 59

c. The empty set is a subset of every set. Ex. 3, p. 60

d. A set with elements has subsets and proper subsets.2n- 12nn Ex. 4, p. 61

2.3 Venn Diagrams and Set Operations

a. A universal set, symbolized by is a set that contains all the elements being considered in a given discussionor problem.

U, Ex. 1, p. 66

b. Venn Diagrams: Representing Two Subsets of a Universal Set Ex. 2, p. 67

UBA

No A are B.A and B are disjoint.

U BA

All A are B.

UA = B

A and B areequal sets.

UBA

Some (at leastone) A are B.A B

BLTZMC02_043-102-hr1 2-11-2009 9:30 Page 98

Chapter Summary, Review, and Test 99

c. (the complement of set ), which can be read prime or not is the set of all elements in the universalset that are not in A.

A,AAA¿ Ex. 3, p. 68

d. ( intersection ), which can be read set and set is the set of elements common to both set andset B.

AB,ABAA ¨ B Ex. 4, p. 69

e. ( union ), which can be read set or set is the set of elements that are members of set or of set or of both sets.B

AB,ABAA ´ B Ex. 5, p. 69

f. Some problems involve more than one set operation. Begin by performing any operations inside parentheses. Ex. 6, p. 71

g. Elements of sets involving a variety of set operations can be determined from Venn diagrams. Ex. 7, p. 71

h. Cardinal Number of the Union of Two Finite Sets

n1A ´ B2 = n1A2 + n1B2 - n1A ¨ B2

Ex. 8, p. 73

2.4 Set Operations and Venn Diagrams with Three Sets

a. When using set operations involving three sets, begin by performing operations within parentheses. Ex. 1, p. 77

b. The figure below shows a Venn diagram with three intersecting sets that separate the universal set, into eightregions. Elements of sets involving a variety of set operations can be determined from this Venn diagram.

I

U

II III

IVV

VI

VIII

VII

BA

C

U, Ex. 2, p. 79

c. To construct a Venn diagram illustrating the elements in and first find andThen place elements into the eight regions shown above, starting with the innermost region, region

V, and working outward to region VIII.A ¨ B ¨ C.

A ¨ B, A ¨ C, B ¨ C,U,C,B,A, Ex. 3, p. 80

d. If two specific sets represent the same regions of a general Venn diagram, then this deductively proves that thetwo sets are equal.

Ex. 4, p. 81;Ex. 5, p. 82

2.5 Survey Problems

a. Venn diagrams can be used to organize information collected in surveys. When interpreting cardinalities insuch diagrams, and and but mean intersection, or means union, and not means complement.

Ex. 1, p. 88

b. To solve a survey problem,1. Define sets and draw a Venn diagram.2. Fill in the cardinality of each region, starting with the innermost region and working outward.3. Use the completed diagram to answer the problem’s questions.

Ex. 2, p. 90;Ex. 3, p. 92;Ex. 4, p. 93

Review Exercises

2.1

In Exercises 1–2, write a word description of each set. (More thanone correct description may be possible.)

1.

2.

In Exercises 3–5, express each set using the roster method.

3.

4.

5. 5x ƒ x H N and x … 306

5x ƒ x H N and 8 … x 6 136

5x ƒ x is a letter in the word miss6

51, 2, 3, Á , 106

5Tuesday, Thursday6

In Exercises 6–7, determine which sets are the empty set.

6. 7.

In Exercises 8–9, fill in the blank with either or to make eachstatement true.8. 93 _____

9. _____

In Exercises 10–11, find the cardinal number for each set.

10.

11. B = 518, 19, 20, Á , 31, 326

A = 5x ƒ x is a month of the year6

5a, b, c, d, e65d6

51, 2, 3, 4, Á , 99, 1006

xH

5x ƒ x 6 4 and x Ú 665�6

BLTZMC02_043-102-hr1 2-11-2009 9:30 Page 99


In Exercises 12–13, fill in the blank with either or to makeeach statement true.

12. _____

13. _____

In Exercises 14–15, determine if the pairs of sets are equivalent,equal, both, or neither.

14. is a lowercase letter that comes before f in the

15.

In Exercises 16–17, determine whether each set is finite or infinite.

16.

17.

2.2

In Exercises 18–20, write or in each blank so that theresulting statement is true.

18.

_____

19. _____

20. _____

In Exercises 21–22, determine whether both, or neither canbe placed in each blank to form a true statement.

21. _____

22.

_____

In Exercises 23–29, determine whether each statement is true orfalse. If the statement is false, explain why.

23.

24.

25.

26.

27.

28. has subsets.

29.

30. List all subsets for the set Which one of these subsetsis not a proper subset?

In Exercises 31–32, find the number of subsets and the number ofproper subsets for each set.

31.

32.

2.3

In Exercises 33–37, let and Find each of the following sets.

33. 34.

35. 36.

37. A¿ ¨ B¿

1A ´ B2¿A¿ ¨ B

A ´ B¿A ¨ B

B = 51, 2, 4, 56.A = 51, 2, 3, 46,U = 51, 2, 3, 4, 5, 6, 7, 86,

5x ƒ x is a month that begins with the letter J6

52, 4, 6, 8, 106

51, 56.

� 8 5 6265six6

53, 7, 96 8 59, 7, 3, 16

5� , �6 ( 5�, � 6

5e, f, g6 ( 5d, e, f, g, h, i6

4 8 52, 4, 6, 8, 10, 126

Texas H 5Oklahoma, Louisiana, Georgia, South Carolina6

5y ƒy is a person living on planet Earth6

5x ƒ x is a person living in the United States6

51, 1, 2, 2651, 26

8 , ( ,

5x ƒ x is an odd natural number6�

5-3, -2, -1, 1, 2, 365-1, 0, 16

5half-dollar, quarter, dime, nickel, penny6

5penny, nickel, dime6

h8

5x ƒ x H N and x is even6

5x ƒ x H N and x 6 50,0006

B = 54, 5, 66

A = 5x ƒ x H N and 3 6 x 6 76

B = 52, 4, 6, 8, 106

English alphabet6

A = 5x ƒ x

58, 9, 10, Á , 10065x ƒ x H N and x 7 76

58, 2, 6, 4650, 2, 4, 6, 86

Z= In Exercises 38–45, use the Venn diagram to represent each set inroster form.

38. 39.

40. 41.

42. 43.

44. 45.


2.4



47. 48.


49. 50.

51. 52.

53. 54.

55. Construct a Venn diagram illustrating the following sets:and

56. Use a Venn diagram with two intersecting circles to prove that

57. Use a Venn diagram with three intersecting circles todetermine whether the following statement is a theorem:

A ¨ 1B ´ C2 = A ´ 1B ¨ C2.

1A ´ B2¿ = A¿ ¨ B¿.

U = 5p, q, r, s, t, u, v, w, x, y6.A = 5q, r, s, t, u6, B = 5p, q, r6, C = 5r, u, w, y6,

A ¨ B ¨ C1A ¨ C2¿

A ¨ C¿1A ¨ B2 ´ C

B ¨ CA ´ C

AU

B

C

dc

ab

f

r

pk

eg

h

1A ¨ C2¿ ´ BA ´ 1B ¨ C2

C = 51, 56.

B = 51, 2, 4, 56

A = 51, 2, 3, 46

U = 51, 2, 3, 4, 5, 6, 7, 86

A ´ B?B.A

BA

UA ¨ B¿

1A ´ B2¿1A ¨ B2¿

A ¨ BA ´ B

B¿A

AU B

6 189

237

45

BLTZMC02_043-102-hr1 2-11-2009 9:30 Page 100

Chapter Summary, Review, and Test 101

58. The Penguin Atlas of Women in the World uses maps andgraphics to present data on how women live acrosscontinents and cultures. The table is based on data fromthe atlas.

2.5

59. A pollster conducting a telephone survey of college studentsasked two questions:

1. Are you a registered Republican?2. Are you in favor of the death penalty?

a. Construct a Venn diagram with two circles that can assist thepollster in tabulating the responses to the two questions.

b. Write the letter b in every region of the diagram thatrepresents students polled who are registered Republicanswho are not in favor of the death penalty.

c. Write the letter c in every region of the diagram thatrepresents students polled who are not registeredRepublicans and who are in favor of the death penalty.

In Exercises 60–61, construct a Venn diagram and determine thecardinality for each region. Use the completed Venn diagram toanswer the questions.

60. A survey of 1000 American adults was taken to analyzetheir investments. Of those surveyed, 650 had invested instocks, 550 in bonds, and 400 in both stocks and bonds. Ofthose surveyed,a. How many invested in only stocks?b. How many invested in stocks or bonds?c. How many did not invest in either stocks or bonds?

61. A survey of 200 students at a nonresidential college wastaken to determine how they got to campus during the fallterm. Of those surveyed, 118 used cars, 102 used publictransportation, and 70 used bikes. Forty-eight students usedcars and public transportation, 38 used cars and bikes, and26 used public transportation and bikes. Twenty-twostudents used all three modes of transportation.Of those surveyed,a. How many used only public transportation?b. How many used cars and public transportation, but

not bikes?c. How many used cars or public transportation, but not

bikes?d. How many used exactly two of these modes of

transportation?e. How many did not use any of the three modes of

transportation to get to campus?

WOMEN IN THE WORLD: SCHOOL, WORK, AND LITERACY

Percentageof CollegeStudentsWho AreWomen

Percentageof Women

Working forPay

Percentageof WomenWho AreIlliterate

United States 57% 59% 1%

Italy 57% 37% 2%

Turkey 43% 28% 20%

Norway 60% 63% 1%

Pakistan 46% 33% 65%

Iceland 65% 71% 1%

Mexico 51% 40% 10%

Source: The Penguin Atlas of Women in the World, 2009

The data can be organized in the following Venn diagram:

Use the data to determine the region in the Venn diagraminto which each of the seven countries should be placed.

I

II

III

IV

V

VI

Percentage ofcollege students

exceeds 50%

UPercentageworking forpay exceeds 50%

Percentage ofilliteracy less

than 5%

Women in the World

VII

VIII

Chapter 2 Test

1. Express the following set using the roster method:

In Exercises 2–9, determine whether each statement is true orfalse. If the statement is false, explain why.

2.

3. If and then sets and are equivalent.

4. 52, 4, 6, 86 = 58, 8, 6, 6, 4, 4, 26

BAB = 52, 4, 6, Á , 146,A = 5x ƒ x is a day of the week6

566 H 51, 2, 3, 4, 5, 6, 76

5x ƒ x H N and 17 6 x … 246.

5.

6.

7.

8. has 25 subsets.

9. The empty set is a proper subset of any set, includingitself.

10. List all subsets for the set Which of these subsets isnot a proper subset?

56, 96.

5a, b, c, d, e6

14 x 51, 2, 3, 4, Á , 39, 406

53, 4, 56 ( 5x ƒ x H N and x 6 66

5d, e, f, g6 8 5a, b, c, d, e, f6

Step-by-step test solutions are found on the Chapter Test Prep Videos available via the VideoResources on DVD, in , or on (search "Blitzer, Thinking Mathematically"and click on "Channels").

BLTZMC02_043-102-hr1 2-11-2009 9:30 Page 101



Find each of the following sets or cardinalities.

11.

12.

13.

14.

15.


16.

17.

18.

19. Construct a Venn diagram illustrating the followingsets: and

20. Use the Venn diagram shown to determine whether thefollowing statement is a theorem:

Show work clearly as you develop the regions representingeach side of the statement.

I

AU

B

C

II III

IV

V

VI

VIII

VII

A¿ ¨ 1B ´ C2 = 1A¿ ¨ B2 ´ 1A¿ ¨ C2.

U = 51, 2, 3, 4, 5, 6, 76.A = 51, 4, 56, B = 51, 5, 6, 76,

1A ¨ B2 ´ 1A ¨ C2

A ¨ B ¨ C

A¿

eg

AU

B

C

f c

h

ajk

bi

d

n1A ´ B¿2

1A ´ B2 ¨ C

A ¨ C¿

1B ¨ C2¿

A ´ B

C = 5a, e, g6.

B = 5c, d, e, f6

A = 5a, b, c, d6

U = 5a, b, c, d, e, f, g6

21. Here is a list of some famous people on whom the FBIkept files:

Famous Person Length of FBI File

Bud Abbott (entertainer) 14 pages

Charlie Chaplin (entertainer) 2063 pages

Albert Einstein (scientist) 1800 pages

Martin Luther King, Jr. (civil rightsleader)

17,000 pages

Elvis Presley (entertainer) 663 pages

Jackie Robinson (athlete) 131 pages

Eleanor Roosevelt (first lady; U.N.representative)

3000 pages

Frank Sinatra (entertainer) 1275 pages

Source: Paul Grobman, Vital Statistics, Plume, 2005.

The data can be organized in the following Venn diagram:

Use the data to determine the region in the Venn diagraminto which each of the following people should be placed.a. Chaplin b. Einstein c. Kingd. Roosevelt e. Sinatra

22. A winter resort took a poll of its 350 visitors to see whichwinter activities people enjoyed. The results were as follows:178 liked to ski, 154 liked to snowboard, 57 liked to ice skate, 49liked to ski and snowboard, 15 liked to ski and ice skate, 2 likedto snowboard and ice skate, and 2 liked all three activities.a. Use a Venn diagram to illustrate the survey’s results.Use the Venn diagram to determine how many of thosesurveyed enjoyedb. exactly one of these activities.c. none of these activities.d. at least two of these activities.e. snowboarding and ice skating, but not skiing.

f. snowboarding or ice skating, but not skiing.

g. only skiing.

I

File exceeded2000 pages

U = set of people on whom the FBI kept files

Entertainers

Men

IIIII

IV

V

VI

VII

VIII

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