# Sets: Number Sets and their Members

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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
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
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
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
4. Example 1
Representing a Set Using a Description
• Write a word description of the set:
• 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
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
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
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
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
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
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
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
13. Inequality Notation and Sets
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
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
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
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
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
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
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
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
• 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