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This pdf covers:-

Sets

Methods of Representing Sets

Representing a Set Using a Description

Representing a Set Using the Roster Method

Set-Builder Notation

Converting from Set-Builder to Roster Notation

The Empty Set

Recognizing the Empty Set

Sets

Methods of Representing Sets

Representing a Set Using a Description

Representing a Set Using the Roster Method

Set-Builder Notation

Converting from Set-Builder to Roster Notation

The Empty Set

Recognizing the Empty Set

1.
Section 2.1

Basic Set Concepts

Objectives

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.

02/02/22 Section 2.1 1

Basic Set Concepts

Objectives

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.

02/02/22 Section 2.1 1

2.
Sets

• A collection of objects whose contents can be clearly

determined.

• Elements or members are the objects in a set.

• A set must be well defined, meaning that its contents can

be clearly determined.

• The order in which the elements of the set are listed is

not important.

02/02/22 Section 2.1 2

• A collection of objects whose contents can be clearly

determined.

• Elements or members are the objects in a set.

• A set must be well defined, meaning that its contents can

be clearly determined.

• The order in which the elements of the set are listed is

not important.

02/02/22 Section 2.1 2

3.
Methods for Representing Sets

Capital letters are generally used to name sets.

• Word description: Describing the members:

Set W is the set of the days of the week.

• Roster method: Listing the members:

W = {Monday, Tuesday, Wednesday, Thursday, Friday, Saturday,

Sunday}

Commas are used to separate the elements of the set.

Braces are used to designate that the enclosed elements form

a set.

02/02/22 Section 2.1 3

Capital letters are generally used to name sets.

• Word description: Describing the members:

Set W is the set of the days of the week.

• Roster method: Listing the members:

W = {Monday, Tuesday, Wednesday, Thursday, Friday, Saturday,

Sunday}

Commas are used to separate the elements of the set.

Braces are used to designate that the enclosed elements form

a set.

02/02/22 Section 2.1 3

4.
Example 1

Representing a Set Using a Description

• Write a word description of the set:

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

• Solution:

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

02/02/22 Section 2.1 4

Representing a Set Using a Description

• Write a word description of the set:

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

• Solution:

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

02/02/22 Section 2.1 4

5.
Example 2

Representing a Set Using the Roster Method

• Write using the roster method:

Set C is the set of U.S. coins with a value of less than a

dollar.

• Solution:

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

02/02/22 Section 2.1 5

Representing a Set Using the Roster Method

• Write using the roster method:

Set C is the set of U.S. coins with a value of less than a

dollar.

• Solution:

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

02/02/22 Section 2.1 5

6.
Set-Builder Notation

– Before the vertical line is the variable x, which

represents an element in general

– After the vertical line is the condition x must meet

in order to be an element of the set.

02/02/22 Section 2.1 6

– Before the vertical line is the variable x, which

represents an element in general

– After the vertical line is the condition x must meet

in order to be an element of the set.

02/02/22 Section 2.1 6

7.
Example 3

Converting from Set-Builder to Roster Notation

• Express set

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

Using the roster method.

• Solution:

There are two months, namely March and May.

Thus,

A = { March, May}

02/02/22 Section 2.1 7

Converting from Set-Builder to Roster Notation

• Express set

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

Using the roster method.

• Solution:

There are two months, namely March and May.

Thus,

A = { March, May}

02/02/22 Section 2.1 7

8.
The Empty Set

• Also called the null set

• Set that contains no elements

• Represented by { } or Ø

• The empty set is NOT represented by { Ø }. This notation

represents a set containing the element Ø.

• These are examples of empty sets:

– Set of all numbers less than 4 and greater than 10

– {x | x is a fawn that speaks}

02/02/22 Section 2.1 8

• Also called the null set

• Set that contains no elements

• Represented by { } or Ø

• The empty set is NOT represented by { Ø }. This notation

represents a set containing the element Ø.

• These are examples of empty sets:

– Set of all numbers less than 4 and greater than 10

– {x | x is a fawn that speaks}

02/02/22 Section 2.1 8

9.
Example 4

Recognizing the Empty Set

• Which of the following is the empty set?

a. {0}

No. This is a set containing one element.

b. 0

No. This is a number, not a set

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

No. This set contains all numbers that are either less

than 4, such as 3, or greater than 10, such as 11.

d. { x | x is a square with three sides}

Yes. There are no squares with three sides.

02/02/22 Section 2.1 9

Recognizing the Empty Set

• Which of the following is the empty set?

a. {0}

No. This is a set containing one element.

b. 0

No. This is a number, not a set

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

No. This set contains all numbers that are either less

than 4, such as 3, or greater than 10, such as 11.

d. { x | x is a square with three sides}

Yes. There are no squares with three sides.

02/02/22 Section 2.1 9

10.
Notations for Set Membership

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.”

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.”

02/02/22 Section 2.1 10

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.”

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.”

02/02/22 Section 2.1 10

11.
Example 5

Using the symbols and

Determine whether each statement is true or false:

a. r {a,b,c,…,z}

True

b. 7 {1,2,3,4,5}

True

c. {a} {a,b}

False. {a} is a set and the set {a} is not an element of

the set {a,b}.

02/02/22 Section 2.1 11

Using the symbols and

Determine whether each statement is true or false:

a. r {a,b,c,…,z}

True

b. 7 {1,2,3,4,5}

True

c. {a} {a,b}

False. {a} is a set and the set {a} is not an element of

the set {a,b}.

02/02/22 Section 2.1 11

12.
Example 6

Sets of Natural Numbers

= {1,2,3,4,5,…}

• Ellipsis, the three dots after the 5 indicate that there is

no final element and that the listing goes on forever.

Express each of the following sets using the roster method

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

A = {1,2,3,4}

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

equal to 25.

B = {25, 26, 27, 28,…}

c. E = { x| x and x is even}.

E = {2, 4, 6, 8,…}

02/02/22 Section 2.1 12

Sets of Natural Numbers

= {1,2,3,4,5,…}

• Ellipsis, the three dots after the 5 indicate that there is

no final element and that the listing goes on forever.

Express each of the following sets using the roster method

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

A = {1,2,3,4}

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

equal to 25.

B = {25, 26, 27, 28,…}

c. E = { x| x and x is even}.

E = {2, 4, 6, 8,…}

02/02/22 Section 2.1 12

13.
Inequality Notation and Sets

Inequality Symbol Set Builder Roster

and Meaning Notation Method

02/02/22 Section 2.1 13

Inequality Symbol Set Builder Roster

and Meaning Notation Method

02/02/22 Section 2.1 13

14.
Example 7

Representing Sets of Natural Numbers

Express each of the following sets using the roster method:

a. { x | x and x ≤ 100}

Solution: {1, 2, 3, 4,…,100}

b. { x | x and 70 ≤ x <100 }

Solution: {70, 71, 72, 73, …, 99}

02/02/22 Section 2.1 14

Representing Sets of Natural Numbers

Express each of the following sets using the roster method:

a. { x | x and x ≤ 100}

Solution: {1, 2, 3, 4,…,100}

b. { x | x and 70 ≤ x <100 }

Solution: {70, 71, 72, 73, …, 99}

02/02/22 Section 2.1 14

15.
Example 8

Cardinality of Sets

• The cardinal number of set A, represented by n(A), is the

number of distinct elements in set A.

– The symbol n(A) is read “n of A.”

– Repeating elements in a set neither adds new elements

to the set nor changes its cardinality.

Find the cardinal number of each set:

a. A = { 7, 9, 11, 13 }

n(A) = 4

b. B = { 0 }

n(B) = 1

c. C = { 13, 14, 15,…,22, 23}

n(C)=11

02/02/22 Section 2.1 15

Cardinality of Sets

• The cardinal number of set A, represented by n(A), is the

number of distinct elements in set A.

– The symbol n(A) is read “n of A.”

– Repeating elements in a set neither adds new elements

to the set nor changes its cardinality.

Find the cardinal number of each set:

a. A = { 7, 9, 11, 13 }

n(A) = 4

b. B = { 0 }

n(B) = 1

c. C = { 13, 14, 15,…,22, 23}

n(C)=11

02/02/22 Section 2.1 15

16.
Equivalent Sets

• Set A is equivalent to set B if set A and set B contain the

same number of elements. For equivalent sets, n(A) =

n(B).

These are equivalent sets:

The line with arrowheads, , indicate that each element of

set A can be paired with exactly one element of set B and

each element of set B can be paired with exactly one

element of set A.

02/02/22 Section 2.1 16

• Set A is equivalent to set B if set A and set B contain the

same number of elements. For equivalent sets, n(A) =

n(B).

These are equivalent sets:

The line with arrowheads, , indicate that each element of

set A can be paired with exactly one element of set B and

each element of set B can be paired with exactly one

element of set A.

02/02/22 Section 2.1 16

17.
One-To-One Correspondences and

Equivalent Sets

• If set A and set B can be placed in a one-to-one

correspondence, then A is equivalent to B: n(A) = n(B).

• If set A and set B cannot be placed in a one-to-one

correspondence, then A is not equivalent to B:

n(A) ≠n(B).

02/02/22 Section 2.1 17

Equivalent Sets

• If set A and set B can be placed in a one-to-one

correspondence, then A is equivalent to B: n(A) = n(B).

• If set A and set B cannot be placed in a one-to-one

correspondence, then A is not equivalent to B:

n(A) ≠n(B).

02/02/22 Section 2.1 17

18.
Example 9

Determining if Sets are Equivalent

• This Table shows the Most Frequent Host of

celebrities who hosted NBC’s Saturday Night Live

Saturday Night Live most Celebrity Number of

frequently and the number of Shows

times each starred on the Hosted

show. Steve Martin 14

A = the set of the five most Alec Baldwin 12

frequent hosts. John Goodman 12

B = the set of the number of Buck Henry 10

times each host starred on Chevy Chase 9

the show.

• Are the sets equivalent?

02/02/22 Section 2.1 18

Determining if Sets are Equivalent

• This Table shows the Most Frequent Host of

celebrities who hosted NBC’s Saturday Night Live

Saturday Night Live most Celebrity Number of

frequently and the number of Shows

times each starred on the Hosted

show. Steve Martin 14

A = the set of the five most Alec Baldwin 12

frequent hosts. John Goodman 12

B = the set of the number of Buck Henry 10

times each host starred on Chevy Chase 9

the show.

• Are the sets equivalent?

02/02/22 Section 2.1 18

19.
Example 9 continued

• Method 1: Trying to set up a One-to-One Correspondence.

• Solution:

The lines with the arrowheads indicate that the

correspondence between the sets in not one-to-one. The

elements Baldwin and Goodman from set A are both paired

with the element 12 from set B. These sets are not

02/02/22 Section 2.1 19

• Method 1: Trying to set up a One-to-One Correspondence.

• Solution:

The lines with the arrowheads indicate that the

correspondence between the sets in not one-to-one. The

elements Baldwin and Goodman from set A are both paired

with the element 12 from set B. These sets are not

02/02/22 Section 2.1 19

20.
Example 9 continued

• Method 2: Counting Elements

• Solution:

Set A contains five distinct elements: n(A) = 5. Set B

contains four distinct elements: n(B) = 4. Because the

sets do not contain the same number of elements, they

are not equivalent.

02/02/22 Section 2.1 20

• Method 2: Counting Elements

• Solution:

Set A contains five distinct elements: n(A) = 5. Set B

contains four distinct elements: n(B) = 4. Because the

sets do not contain the same number of elements, they

are not equivalent.

02/02/22 Section 2.1 20

21.
Finite and Infinite Sets,

Equal Sets

• Finite set: Set A is a finite set if n(A) = 0 ( that is, A is the

empty set) or n(A) is a natural number.

• Infinite set: A set whose cardinality is not 0 or a natural

number. The set of natural numbers is assigned the infinite

cardinal number 0 אread “aleph-null”.

• Equal sets: Set A is equal to set B if set A and set B

contain exactly the same elements, regardless of order or

possible repetition of elements. We symbolize the equality

of sets A and B using the statement A = B.

If two sets are equal, then they must be equivalent!

02/02/22 Section 2.1 21

Equal Sets

• Finite set: Set A is a finite set if n(A) = 0 ( that is, A is the

empty set) or n(A) is a natural number.

• Infinite set: A set whose cardinality is not 0 or a natural

number. The set of natural numbers is assigned the infinite

cardinal number 0 אread “aleph-null”.

• Equal sets: Set A is equal to set B if set A and set B

contain exactly the same elements, regardless of order or

possible repetition of elements. We symbolize the equality

of sets A and B using the statement A = B.

If two sets are equal, then they must be equivalent!

02/02/22 Section 2.1 21

22.
Example 10

Determining Whether Sets are Equal

Determine whether each statement is true or false:

a. { 4, 8, 9 } = { 8, 9, 4 }

True

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

False

02/02/22 Section 2.1 22

Determining Whether Sets are Equal

Determine whether each statement is true or false:

a. { 4, 8, 9 } = { 8, 9, 4 }

True

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

False

02/02/22 Section 2.1 22