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2 Set Theory > 2.1 Basic Set Concepts

2.1 Basic Set Concepts

What am I Supposed to Learn?After you have read this section, you should be able to:

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.

WE TEND TO PLACE THINGS IN categories, which allows us to order and structure the world. For example, to which populations do you belong? Do you categorizeyourself as a college student? What about your gender? What about your academic major or your ethnic background? Our minds cannot find order and meaning withoutcreating collections. Mathematicians call such collections sets. A set is a collection of objects whose contents can be clearly determined. The objects in a set are calledthe elements, or members, of the set.

A set must be well defined, meaning that its contents can be clearly determined. Using this criterion, the collection of actors who have won Academy Awards is a set.We can always determine whether or not a particular actor is an element of this collection. By contrast, consider the collection of great actors. Whether or not a personbelongs to this collection is a matter of how we interpret the word great. In this text, we will only consider collections that form well-defined sets.

Methods for Representing Sets

1 Use three methods to represent 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 W to represent the set of the days of the week.

Three methods are commonly used to designate a set. One method is a word description. We can describe set W as the set of the days of the week. A second methodis the roster method. This involves listing the elements of a set inside a pair of braces, The braces at the beginning and end indicate that we are representing aset. The roster form uses commas to separate the elements of the set. Thus, we can designate the set W by listing its elements:

Grouping symbols such as parentheses, and square brackets, are not used 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 the elements are listed in a set is not important. Thus, another way of expressing theset of the days of the week is

Example 1 Representing a Set Using a DescriptionWrite a word description of the set

SOLUTION

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∈ ∉ .

{ } .

W = {Monday, Tuesday, Wednesday, Thursday, Friday, Saturday, Sunday} .

( ) , [ ] ,

W = {Saturday, Sunday, Monday, Tuesday, Wednesday, Thursday, Friday} .

P = {Washington, Adams, Jefferson, Madison, Monroe} .

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Page 48

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

Check Point 1Write a word description of the set

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L = {a, b, c, d, e, f} .



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2 Set Theory > 2.1 Basic Set Concepts

Example 2 Representing a Set Using the Roster MethodSet C is the set of U.S. coins with a value of less than a dollar. Express this set using the roster method.

SOLUTION

Check Point 2Set M is the set of months beginning with the letter A. Express this set using the roster method.

Great Question!Do I have to use x to represent the variable in set-builder notation?

No. Any letter can be used to represent the variable. Thus, and allrepresent the same set.

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

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

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

TABLE 2.1 Sets Using Three DesignationsWord Description Roster Method Set-Builder Notation

B is the set of members of the Beatles in 1963.

S is the set of states whose names begin with the letter A.

Example 3 Converting from Set-Builder to Roster NotationExpress set



C = {penny, nickel, dime, quarter, half-dollar}

{x|x is a day of the week} , {y|y is a day of the week} , {z|z is a day of the week}

B ={George Harrison,

John Lennon, Paul McCartney, Ringo Starr}

B = {x|x was a member

of the Beatles in 1963}

S = {Alabama, Alaska,

Arizona, Arkansas}

S = {x|x is a U.S.

state whose name begins with the letter A}

A = {x|x is a month that begins with the letter M}







9

using the roster method.

SOLUTION

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

Check Point 3Express the set

using the roster method.


A = {x|x is a month that begins with the letter M}

A = {March, May} .

O = {x|x is a positive odd number less than 10}








0

2 Set Theory > 2.1 Basic Set Concepts > The Empty Set

The 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 of alllowercase letters of the English alphabet. If L is chosen as a name for this set, we can use set-builder notation to represent L as follows:

A complete listing using the roster method is rather tedious:

We can shorten the listing in set L by writing

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

Blitzer BonusThe Loss of Sets

Have you ever considered what would happen if we suddenly lost our ability to recall categories and the names that identify them? This is precisely what happened toAlice, 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 arms clasped lovingly round the soft neck of the Fawn, till they came out into another open field, andhere the Fawn gave a sudden bound into the air, and shook itself free from Alice's arm. “I'm a Fawn!” it cried out in a voice of delight. “And, dear me! you're a humanchild!” A sudden look of alarm came into its beautiful brown eyes, and in another moment it had darted away at full speed.

By realizing that Alice is a member of the set of human beings, which in turn is part of the set of dangerous things, the fawn is overcome by fear. Thus, the fawn'sexperience is determined by the way it structures the world into sets with various characteristics.

The Empty Set

2 Define and recognize the empty set.

Consider the following sets:

Can you see what these sets have in common? They both contain no elements. There are 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.

The Empty SetThe empty set, also called the null set, is the set that contains no elements. The empty set is represented by or

Notice that and have the same meaning. However, the empty set is not represented by This notation represents a set containing the element



L = {x|x is a lowercase letter of the English alphabet} .

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

L = {a, b, c, d, … , z} .

{x|x is a fawn that speaks}

{x|x is a number greater than 10 and less than 4} .

{ } Ø.

{ } Ø {Ø} . Ø.








2 Set Theory > 2.1 Basic Set Concepts > Notations for Set Membership

Example 4 Recognizing the Empty SetWhich one of the following is the empty set?

a.

b. 0

c.

d.

SOLUTION

a. 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 that are either less than 4, such as 3, or greater than 10, such as 11. Becausesome elements belong to this set, it cannot be the empty set.

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

Check Point 4Which one of the following is the empty set?

a.

b.

c. nothing

d.

Blitzer BonusThe Musical Sounds of the Empty Set

John Cage (1912–1992), the American avant-garde composer, translated the empty set into the quietest piece of music ever written. His piano composition requires the musician to sit frozen in silence at a piano stool for 4 minutes, 33 seconds, or 273 seconds. (The significance of 273 is that at approximately all molecular motion stops.) The set

is the empty set. There are no musical sounds in the composition. Mathematician Martin Gardner wrote, “I have not heard performed, but friends who have tellme it is Cage's finest composition.”

Notations for Set Membership

3 Use the symbols and

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

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

Example 5 Using the Symbols and Determine whether each statement is true or false:

a.

b.

c.

SOLUTION



{0}

{x|x is a number less than 4 or greater than 10}

{x|x is a square with exactly three sides}

{0}


{x|x is a square with exactly three sides}


{x|x is a number less than 3 and greater than 5}

{Ø}

4'33''

−273°C,

{x|x is a musical sound from 4'33'' }

4'33''

∈ ∉ .

∈ ∉∈ ∈

∉ ∉

∈ ∉

r ∈ {a, b, c, … , z}

7 ∉ {1, 2, 3, 4, 5}

{a} ∈ {a, b} .

{a, b, c, … , z} ,








Page 51

a. 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 three dots appear even though the element is not listed.


{a, b, c, … , z} ,

r ∈ {a, b, c, … , z}








2 Set Theory > 2.1 Basic Set Concepts > Sets of Natural Numbers

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 the statement

is false.

Check Point 5Determine whether each statement is true or false:

a.

b.

c.

Great Question!Can a set ever belong to another set—sort of a set within a set?

Yes. A set can be an element of another set. For example, is a set with two elements. One element is the set and the other element is the letter c.Thus, and

Sets of Natural Numbers

4 Apply set notation to sets of natural numbers.

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

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

The Set of Natural Numbers

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

Example 6 Representing Sets of Natural NumbersExpress each of the following sets using the roster method:

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

b. Set B 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 A can be expressed using the roster method as

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

The three dots show that the listing goes on forever.

c. The set-builder notation



{1, 2, 3, 4, 5} ,

7 ∉ {1, 2, 3, 4, 5}

{a} {a} {a, b} ,

{a} ∈ {a, b}

8 ∈ {1, 2, 3, … , 10}

r ∉ {a, b, c, z}

{Monday} ∈ {x|x is a day of the week} .

{{a, b} , c} {a, b}{a, b} ∈ {{a, b} , c} c ∈ {{a, b} , c} .

{1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, …} .

N = {1, 2, 3, 4, 5, …}

E = {x|x ∈ N and x is even} .

A = {1, 2, 3, 4} .

B = {25, 26, 27, 28, …} .

E = {x|x ∈ N and x is even}








Page 52

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


E = {2, 4, 6, 8, …} .








Page 53

2 Set Theory > 2.1 Basic Set Concepts > Sets of Natural Numbers

Check Point 6Express each of the following sets using the roster method:

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

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

c.

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

Table 2.2 Inequality Notation and Sets

Inequality Symbol and MeaningExample

Set-Builder Notation Roster Method

{1, 2, 3}

{1, 2, 3, 4}

{5, 6, 7, 8, …}

{4, 5, 6, 7, …}

{5, 6, 7}

{4, 5, 6, 7, 8}

{4, 5, 6, 7}

{5, 6, 7, 8}



O = {x|x ∈ N and x is odd} .

x < a ◀ x is less than a.

{x|x ∈ N and x < 4}

▲

x is a natural number

less than 4.

x ≤ a ◀x is less than

or equal to a.

{x|x ∈ N and x ≤ 4}

▲


less than or equal to 4.

x > a ◀ x is greater than a.

{x|x ∈ N and x > 4}

▲


greater than 4.

x ≥ a ◀x is greater than

or equal to a.

{x|x ∈ N and x ≥ 4}

▲


greater than or equal to 4.

a < x < b ◀x is greater than a

and less than b.

{x|x ∈ N and 4 < x < 8}

▲

x is a natural number greater

than 4 and less than 8.

a ≤ x ≤ b ◀

x is greater than or

equal to a and less

than or equal to b.

{x|x ∈ N and 4 ≤ x ≤ 8}

▲

x is a natural number greater than or

equal to 4 and less than or equal to 8.

a ≤ x < b ◀

x is greater than or

equal to a and

less than b.

{x|x ∈ N and 4 ≤ x < 8}

▲

x is a natural number greater

than or equal to 4 and less than 8.

a < x ≤ b ◀

x is greater than a

and less than

or equal to b.

{x|x ∈ N and 4 < x ≤ 8}

▲

x is a natural number greater than

4 and less than or equal to 8.








Page 54

2 Set Theory > 2.1 Basic Set Concepts > Cardinality and Equivalent Sets

Example 7 Representing Sets of Natural NumbersExpress each of the following sets using the roster method:

a.

b.

SOLUTION

a. 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 numbers greater than or equal to 70 and less than 100. This set in roster form is

Check Point 7Express each of the following sets using the roster method:

a.

b.

Cardinality and Equivalent Sets

5 Determine a set's cardinal number.

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

Definition of a Set's Cardinal NumberThe cardinal number of set A, represented by is the number of distinct elements in set A. The symbol is read “n of

Notice that the cardinal number of a set refers to the number of distinct, or different, elements in the set. Repeating elements in a set neither adds new elements tothe set nor changes its cardinality. For example, and represent the same set with three distinct elements, 3, 5, and 7. Thus,

and

Example 8 Determining a Set's Cardinal NumberFind 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 number of elements in the set.

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

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

c. Set lists only five elements. However, the three dots 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 numbers in set C. The cardinality of set C is 11, and

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



{x|x ∈ N and x ≤ 100}

{x|x ∈ N and 70 ≤ x < 100} .

{x|x ∈ N and x ≤ 100}

{1, 2, 3, 4, … , 100} .

{x|x ∈ N and 70 ≤ x < 100}{70, 71, 72, 73, … , 99} .

{x|x ∈ N and x < 200}

{x|x ∈ N and 50 < x ≤ 200} .

{a, e, i, o, u}

n (A) , n (A) A.”

A = {3, 5, 7} B = {3, 5, 5, 7, 7, 7}n (A) = 3 n (B) = 3.

A = {7, 9, 11, 13}

B = {0}

C = {13, 14, 15, … , 22, 23}

Ø.

A = {7, 9, 11, 13} n (A) = 4.

B = {0} n (B) = 1.

C = {13, 14, 15, … , 22, 23}n (C) = 11.

Ø, n (Ø) = 0.








2 Set Theory > 2.1 Basic Set Concepts > Cardinality and Equivalent Sets

Check Point 8Find 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.

6 Recognize equivalent sets.

Definition of Equivalent SetsSet A is equivalent to set B means that set A and set B contain the same number of elements. For equivalent sets,

Here is an example of two equivalent sets:

It is not necessary to count elements and arrive at 5 to determine that these sets are equivalent. The lines with arrowheads, indicate that each element of set A canbe paired with exactly one element of set B and each element of set B can be paired with exactly one element of set A. We say that the sets can be placed in a one-to-one correspondence.

One-to-One Correspondences and Equivalent Sets1. If set A and set B can be placed in a one-to-one correspondence, then A is equivalent to

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

Example 9 Determining If Sets Are EquivalentIn most societies, women say they prefer to marry men who are older than themselves, whereas men say they prefer women who are younger. Figure 2.1 shows thepreferred age difference in a mate in five selected countries.

dFIGURE 2.1 Source: Carole Wade and Carol Tavris, Psychology, Sixth Edition, Prentice Hall, 2000.

Let



A = {6, 10, 14, 15, 16}

B = {872}

C = {9, 10, 11, … , 15, 16}

D = { } .

n (A) = n (B) .

n(A) = n(B) = 5

▶

▶

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

B = {x|x ∈ N and 3 ≤ x ≤ 7} = { , , , , }.3↕

4↕

5↕

6↕

7↕

↕ ,

B : n (A) = n (B) .

B : n (A) ≠ n (B) .








Page 55

Are these sets equivalent? Explain.


A

B

=

=

the set of the five countries shown in Figure 2.1

the set of the average number of years women in each of these

countries prefer men who are older than themselves.








2 Set Theory > 2.1 Basic Set Concepts > Finite and Infinite Sets

SOLUTION

Let's begin by expressing each set in roster form.

dFIGURE 2.1 (repeated)

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 the correspondence between the sets is not one-to-one. The elements Poland and Italy fromset A are both paired with the element 3.3 from set B. These sets are not equivalent.

Method 2. Counting Elements

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

Check Point 9Let

Are these sets equivalent? Explain.

Finite and Infinite Sets

7 Distinguish between finite and infinite sets.

Example 9 illustrated that to compare the cardinalities of two sets, pair off their elements. 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 of finite sets, some unusual conclusions emerge when dealing with infinite sets.

Finite Sets and Infinite Sets



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

B = {

↕

4.2,

↕

4.5,

↕ ⤢

3.3,▲

Do not write 3.3 twice.

We are interested in each

set's elements.distinct− −−−−−

↕

2.5}

n (A) = 5. n (B) = 4.

A

B

=

=

the set of the five countries shown in Figure 2.1.

the set of the average number of years men in each of these

countries prefer women who are younger than themselves.

n (A) = 0 n (A)








Page 56

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

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 sethave a cardinality? The answer is yes, albeit one of the strangest numbers you've ever seen. The set of natural numbers is assigned the infinite cardinal number (read: “aleph-null,” aleph being the first letter of the Hebrew alphabet). What follows is a succession of mind-boggling results, including a hierarchy of different infinitenumbers in which is the smallest infinity:

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


n (A) = 0 n (A)

N = {1, 2, 3, 4, 5, 6, …} ,ℵ0

ℵ0

< < < < < … .ℵ0 ℵ1 ℵ2 ℵ3 ℵ4 ℵ5








Page 57

2 Set Theory > 2.1 Basic Set Concepts > Equal Sets

Equal Sets

8 Recognize equal sets.

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

Definition of Equality of SetsSet A is equal to set B means that set A and set B contain exactly the same elements, regardless of order or possible repetition of elements. We symbolize theequality of sets A and B using the statement

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

Because equal sets contain the same elements, they also have the same cardinal number. For example, the equal sets and havefour elements each. Thus, both sets have the same cardinal number: 4. Notice that a possible one-to-one correspondence between the equal sets A and B can beobtained by pairing each element with itself:

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

Great Question!Can you clarify the difference between equal sets and equivalent sets?

In English, the words equal and equivalent often mean the same thing. This is not the case in set theory. Equal sets contain the same elements. Equivalent setscontain the same number of elements. If two sets are equal, then they must be equivalent. However, if two sets are equivalent, they are not necessarily equal.

Example 10 Determining Whether Sets Are EqualDetermine whether each statement is true or false:

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 an element 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.

Check Point 10Determine whether each statement is true or false:

a.

b.




A = B.

A = {w, x, y, z} B = {z, y, w, x} , A = B

A = {w, x, y, z} B = {z, y, w, x}

{4, 8, 9} = {8, 9, 4}

{1, 3, 5} = {0, 1, 3, 5} .

{4, 8, 9} {8, 9, 4}

{4, 8, 9} = {8, 9, 4}

{1, 3, 5} {0, 1, 3, 5} ,

{1, 3, 5} = {0, 1, 3, 5}

{O, L, D} = {D, O, L}

{4, 5} = {5, 4,Ø} .













2 Set Theory > 2.1 Basic Set Concepts > Concept and Vocabulary Check

Concept and Vocabulary CheckFill in each blank so that the resulting statement is true.

1. The set is expressed using the ___________________ method. The set is expressed using ___________________ notation.

2. A set that contains no elements is called the null set or the ___________________ set. This set is represented by or ___________________.

3. The symbol is used to indicate that an object _____________________ of a set.

4. The set is called the set of ______________________.

5. The number of distinct elements in a set is called the ___________________ number of the set. If A represents the set, this number is represented by___________________.

6. Two sets that contain the same number of elements are called ___________________ sets.

7. Two sets that contain the same elements are called ___________________ sets.

Exercise Set 2.1Practice ExercisesIn Exercises 1–6, determine which collections are not well defined and therefore not sets.

1. The collection of U.S. presidents

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

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

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

5. The collection of natural numbers greater than one million

6. The collection of even natural numbers greater than 100

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

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

8.

9.

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.

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.

26.



{California, Colorado, Connecticut}{x|x is a U.S. state whose name begins with the letter C}

{ }

∈

N = {1, 2, 3, 4, 5, …}

{Saturday, Sunday}

{January, June, July}

{April, August}

{6, 7, 8, 9, …}

{9, 10, 11, 12, …}

{6, 7, 8, 9, … , 20}

{9, 10, 11, 12, … , 25}

{x|x is a month that ends with the letters b-e-r}

{x|x is a lowercase letter of the alphabet that follows d and comes before j}

{x|x ∈ N and x ≤ 5}

{x|x ∈ N and x ≤ 4}

{x|x ∈ N and x > 5}

{x|x ∈ N and x > 4}

{x|x ∈ N and 6 < x ≤ 10}








Page 58

27.

28.

29.

30.

31.

32.

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

33.

34.

35.

36.

37.

38.

39.

40.

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.


{x|x ∈ N and 6 < x ≤ 10}

{x|x ∈ N and 7 < x ≤ 11}

{x|x ∈ N and 10 ≤ x < 80}

{x|x ∈ N and 15 ≤ x < 60}

{x|x + 5 = 7}

{x|x + 3 = 9}

{Ø, 0}

{0,Ø}

{x|x is a woman who served as U.S. president before 2016}

{x|x is a living U. S. president born before 1200}

{x|x is the number of women who served as U.S. president before 2016}

{x|x is the number of living U.S. presidents born before 1200}

{x|x is a U. S. state whose name begins with the letter X}

{x|x is a month of the year whose name begins with the letter X}

{x|x < 2 and x > 5}

{x|x < 3 and x > 7}

{x|x ∈ N and 2 < x < 5}

{x|x ∈ N and 3 < x < 7}



3 ∈ {1, 3, 5, 7}

6 ∈ {2, 4, 6, 8, 10}

12 ∈ {1, 2, 3, … , 14}

10 ∈ {1, 2, 3, … , 16}

5 ∈ {2, 4, 6, … , 20}

8 ∈ {1, 3, 5, … 19}

11 ∉ {1, 2, 3, … , 9}

17 ∉ {1, 2, 3, … , 16}

37 ∉ {1, 2, 3, … , 40}

26 ∉ {1, 2, 3, … , 50}

4 ∉ {x|x ∈ N and x is even}

2 ∈ {x|x ∈ N and x is odd}

13 ∉ {x|x ∈ N and x < 13}

20 ∉ {x|x ∈ N and x < 20}

16 ∉ {x|x ∈ N and 15 ≤ x < 20}

19 ∉ {x|x ∈ N and 16 ≤ x < 21}









63.

64.

65.

66.

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

67.

68.

69.

70.

71.

72.

73.

74.

75.

76.

77.

78.

79.

80.

In Exercises 81–90,

a. Are the sets equivalent? Explain.

b. Are the sets equal? Explain.

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

82. A is the set of states in the United States. B is the set of people who are now governors of the states in the United States.

83.

84.

85.

86.

87.

88.

89.

90.

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



{3} ∈ {3, 4}

{7} ∈ {7, 8}

−1 ∉ N

−2 ∉ N

A = {17, 19, 21, 23, 25}

A = {16, 18, 20, 22, 24, 26}

B = {2, 4, 6, … , 30}

B = {1, 3, 5, … , 21}

C = {x|x is a day of the week that begins with the letter A}

C = {x|x is a month of the year that begins with the letter W}

D = {five}

D = {six}

A = {x|x is a letter in the word five}

A = {x|x is a letter in the word six}

B = {x|x ∈ N and 2 ≤ x < 7}

B = {x|x ∈ N and 3 ≤ x < 10}

C = {x|x < 4 and x ≥ 12}

C = {x|x < 5 and x ≥ 15}

A = {1, 2, 3, 4, 5}

B = {0, 1, 2, 3, 4}

A = {1, 3, 5, 7, 9}

B = {2, 4, 6, 8, 10}

A = {1, 1, 1, 2, 2, 3, 4}

B = {4, 3, 2, 1}

A = {0, 1, 1, 2, 2, 2, 3, 3, 3, 3}

B = {3, 2, 1, 0}

A = {x|x ∈ N and 6 ≤ x < 10}

B = {x|x ∈ N and 9 < x ≤ 13}

A = {x|x ∈ N and 12 < x ≤ 17}

B = {x|x ∈ N and 20 ≤ x < 25}

A = {x|x ∈ N and 100 ≤ x ≤ 105}

B = {x|x ∈ N and 99 < x < 106}

A = {x|x ∈ N and 200 ≤ x ≤ 206}

B = {x|x ∈ N and 199 < x < 207}

{x|x ∈ N and x ≥ 100}








Page 59

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 in order to be a member of the set. (Morethan one correct inequality may be possible.)

97.

98.

99.

100.

In Exercises 101–104, give examples of two sets that meet the given conditions. 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 of their population having used marijuana. In Exercises 105–112, use the information given by the graphto represent each set by the roster method, or use the appropriate notation to indicate that the set is the empty set.

dSource: Organization for Economic Cooperation and Development

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

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

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

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

109.

110.

111.

112.


{x|x ∈ N and x ≥ 100}

{x|x ∈ N and x ≥ 50}

{x|x ∈ N and x ≤ 1, 000, 000}

{x|x ∈ N and x ≤ 2, 000, 000}

{61, 62, 63, 64, …}

{36, 37, 38, 39, …}

{61, 62, 63, 64, … , 89}

{36, 37, 38, 39, … , 59}

{x|x is a country in which 8% ≤ percentage having used marijuana < 12.3%}

{x|x is a country in which 7.6% ≤ percentage having used marijuana < 9%}

{x|x is a country in which the percentage having used marijuana > 22.2%}

{x|x is a country in which the percentage having used marijuana ≥ 22.2%}









A study of 900 working women in Texas showed that their feelings changed throughout the day. The following line graph shows 15 different times in a day and theaverage level of happiness for the women at each time. Based on the information given by the graph, represent each of the sets in Exercises 113–116 using the rostermethod.

dSource: D. Kahneman et al. “A Survey Method for Characterizing Daily Life Experience,” Science

113.

114.

115.

116.

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

Writing in Mathematics118. What is a set?

119. Describe the three methods used to represent a set. Give an 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 finite set and an infinite set.

Critical Thinking ExercisesMake Sense? In Exercises 124–127, determine whether each statement makes sense or does not make sense, and explain your reasoning.

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

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

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

d



{x|x is a time of the day when the average level of happiness was 3}

{x|x is a time of the day when the average level of happiness was 1}

{x|x is a time of the day when 3 < average level of happiness < 4}

{x|x is a time of the day when 3 < average level of happiness ≤ 4}








Page 60

Source: ATUS, Bureau of Labor Statistics

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

In Exercises 128–135, determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce 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 of the following sets:

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

a. is

b. is


n (Ø) = 1

A = {x|x is a man of the town who shaves himself}

B = {x|x is a man of the town who does not shave himself}.

s ∈ A?

s ∈ B?






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