Open for improvements. Thanks God bless us all.

2. Lesson 1.1

3. Sets

4. SETS
- are that
falls under a certain
These objects are called
or of the set.

5. Try this examples:
A set of silverware

6. Try this examples:
A set of tires for a car

7. Try this examples:
A set of encyclopedias

8. A = {0, 1, 2, 3, 4, 5, …}
In Mathematics..
A set of whole numbers

9. B = {… -3, -2, -1, 0, 1, 2, 3, …}
In Mathematics..
A set of integers

10. Well –
Defined
Sets

11. Mathematics on Sets
The main characteristics of a set in
Mathematics is that it is
.

12. Example of a well-defined set
A set of whole numbers up to 9
A = { 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 }

13. Example of a well-defined set
A set of positive even integers starting
from 2 to 10
A = {2, 4, 6, 8, 10}

14. Not Well-
Defined
Sets

15. Example of a not well-defined set
1. A set of good books to read.
2. A set of fragrant perfume.
3. A set of enjoyable social networking
sites.
4. The set of nice people in your school
5. The set of female teachers in SSM

17. TRY THIS!

18. State whether each of the following
sets is or .
1. The set of multiples of 8.
2. The set of pretty ladies.
3. The set of all large numbers.
4. The set of integers between 0
and 10.
5. The set of intelligent students.

19. ANSWERS!

20. State whether each of the following
sets is or .
1. The set of multiples of 8.

21. State whether each of the following
sets is or .
2. The set of pretty ladies.

22. State whether each of the following
sets is or .
3. The set of all large numbers.

23. State whether each of the following
sets is or .
4. The set of integers between 0 and
10.

24. State whether each of the following
sets is or .
5. The set of intelligent students.

26. Reading and
Symbols Used
in Sets

27. A = { 1, 2, 3 }
What are these?
Set name
Braces
Elements
-members of
the set Commas
- it denotes a
set using a
CAPITAL
LETTER
Used for
enclosing
elements
of a set
Separator for
elements

28. Take note:
The only correct fence or
enclosure for elements are
, not parenthesis (
), or brackets [ ].

29. A = { 1, 2, 3 }
The Set A contains the elements 1, 2,
and 3.
We read this as..

30. Read what elements this
set has..
1) M = { Math, Science,
English, Music, Health }

31. Read what elements this
set has..
2) A = { 4, 3, 2, 1, 0 }

32. Read what elements this
set has..
3) Z = { Tito Sotto, Joey
de Leon, Vic Sotto}

33. A = { 1, 2, 3 }
What are these again?
Set name
Braces
Elements
Commas
- it denotes a
set using a
CAPITAL
LETTER
Used for
enclosing
elements
of a set
Separator for
elements

35. Next.. 

36. Symbol for “is an element of”
We use this
symbol to
indicate that an
element is part of
that set.
A = { 1, 2, 3, }

37. Symbol for “is an element of”
The symbol is
read as:
or
A = { 1, 2, 3, }

38. Symbol for “is an element of”
Thus, we write
and is read
as:1 ∈ A
A = { 1, 2, 3, }

39. Symbol for “is an element of”
Same goes for
2 ∈ A and 3 ∈ A.
A = { 1, 2, 3, }
2 ∈ A
3 ∈ A

40. Symbol for “is an element of”
∴ 1 is an element of A
2 is an element of A
3 is an element of A
A = { 1, 2, 3, }
2 ∈ A
3 ∈ A
1 ∈ A

41. But how about
those who are
not an element of
a particular set?

42. For those who are not an element..
In that case, we use
this symbol to
indicate that it is
of the
set.
A = { 1, 2, 3, }

43. For those who are not an element..
For example, 4 is not
an element. We write
Read as
A = { 1, 2, 3, }
4 ∈ A

44. For those who are not an element..
Same goes for
those who are not
part of the set.
A = { 1, 2, 3, }
5 ∈ A
11 ∈ A

45. What do you call
a with one
and only one
element?

46. Unit Set
We call a set with
a .
B = { apple }

47. How about those
set with no
elements?

48. Empty Sets or Null Sets
We call sets with no
elements as
or . We
use this symbol to
indicate it.
∅
or { }
A = { } or A = ∅

49. Wait.. One more
thing! 

50. Finite and Infinite Sets
is a set that
has a
of elements.
A = { a, b, c, d, ... y, z }
is a set
that elements never
comes to an end
B = { 2, 4, 6, 8 ... }

51. Ellipsis
We use three
dots or ellipsis to
indicate that
there are
elements in the
set that have not
been written
down.
…

52. Ellipsis
It is read as

53. C = { 1, 2, 3, 4, …, 82 }
The Set C contains
the elements 1, 2, 3, 4
Let’s Try To Read This Set
With An Ellipsis..
and so on and so forth
up to 82.

54. D = { 15, 30, 45, …, 180 }
The Set D contains
the elements 15, 30, 45
Let’s Try To Read This Set
With An Ellipsis..
and so on and so forth
up to 180.

55. E = { a, b, c, …, z }
The Set E contains
the elements a, b, c
Let’s Try To Read This Set
With An Ellipsis..
and so on and so forth
up to z.

58. Determine whether each of the
following statement is true or false.
Explain your answer.
If K = { 3, 4, 7, 8}
L = { 4, 7, 9 }
M = { }
1) 7 ∈ K

59. Determine whether each of the
following statement is true or false.
Explain your answer.
If K = { 3, 4, 7, 8}
L = { 4, 7, 9 }
M = { }
2) 8 ∈ L

60. Determine whether each of the
following statement is true or false.
Explain your answer.
If K = { 3, 4, 7, 8}
L = { 4, 7, 9 }
M = { }
2) M = ∅

61. Determine whether each of the
following statement is true or false.
Explain your answer.
If K = { 3, 4, 7, 8}
L = { 4, 7, 9 }
M = { }
4) 7 ∈ L
9 ∈ L
4 ∈ L

62. Determine whether each of the
following statement is true or false.
Explain your answer.
If K = { 3, 4, 7, 8}
L = { 4, 7, 9 }
M = { }
5) 4 ∈ K
7 ∈ K 9 ∈ K
8 ∈ K

64. Three
Ways of
Describing
Sets

65. There are three ways in which
we can describe a set. These are the
following:
1. The Roster Notation or Listing
Method
2. The Verbal Description Method.
3. The Set Builder Notation

66. Three Ways of Describing Sets
The Roster Notation or Listing Method
𝐴 = 𝑎, 𝑒, 𝑖, 𝑜, 𝑢
The Verbal Description Method
𝑇ℎ𝑒 𝑠𝑒𝑡 𝐴 𝑖𝑠 𝑡ℎ𝑒 𝑠𝑒𝑡 𝑜𝑓
𝑣𝑜𝑤𝑒𝑙𝑠 𝑖𝑛 𝑡ℎ𝑒 𝐸𝑛𝑔𝑙𝑖𝑠ℎ 𝑎𝑙𝑝ℎ𝑎𝑏𝑒𝑡
The Set Builder Notation [Rule Method]
𝐴 = 𝑥 𝑥 𝑖𝑠 𝑎 𝑣𝑜𝑤𝑒𝑙 𝑖𝑛 𝑡ℎ𝑒 𝐸𝑛𝑔𝑙𝑖𝑠ℎ 𝑎𝑙𝑝ℎ𝑎𝑏𝑒𝑡}

67. The Roster
Notation or
Listing
Method

68. Roster Notation or Listing Method
This is a method describing a set
by each element of the set inside
the symbol { }.
In listing the elements of the set,
each and
the order of the elements does not
matter.

69. Roster Notation or Listing Method
E = { 3, 2, 1, 4 }
Example: The set integers between
0 and 5.
Set Name
Equal sign
Opening Brace
Elements separated by comma
Closing Brace

70. Roster Notation or Listing Method
The Set E contains the elements
3, 2, 1 and 4.
Read as:

71. Roster Notation or Listing Method
J = { a, e, i, o, u }
Example: The set of vowel letters
in the English alphabet.
Set Name
Equal sign
Opening Brace
Elements separated by comma
Closing Brace

72. Roster Notation or Listing Method
The Set J contains the elements
a, e, i, o and u.
Read as:

73. Roster Notation or Listing Method
L = { 2, 4, 6, 8 }
Example: The set of positive even
integers less than 10.
Set Name
Equal sign
Opening Brace
Elements separated by comma
Closing Brace

74. Roster Notation or Listing Method
The Set Lcontains the elements
2, 4, 6, and 8.
Read as:

75. Roster Notation or Listing Method
L = { a, l, g, e, b, r}
Example: The set of letters in the
word ALGEBRA
Set Name
Equal sign
Opening Brace
Elements separated by comma
Closing Brace
Once again, in listing the elements of the set,
each and the order
of the elements does not matter.

77. The Verbal
Description
Method

78. The Verbal Description Method
It is a method of
. In here, you’re going to think or
what would be the best
description that suits the case.
Let’s try to describe the sets in the
previous slides.

79. Verbal Description Method
Examples:
1. A = { 1, 2, 3, 4 }
2. B = { p, h, i, l, n, e, s }
3. C = { 5, 10, 15… }
4. D = { moon }

80. Verbal Description Method
The Set A counting numbers
less than 5.
is the set of
= { 1, 2, 3, 4 }
You write..

82. Verbal Description Method
The Set A letters in the word
“Philippines”.
is the set of
= { p, h, i, l, n, e, s }
You write..

83. Verbal Description Method
The Set A positive multiples of
5.
is the set of
= { 5, 10, 15… }
You write..

84. Verbal Description Method
The Set A positive multiples of
5
is the set of
= { 5, 10, 15… }
You write..
50
up to 50.

85. Verbal Description Method
The Set A a natural earth
satellite.
is the set of
= { moon }
You write..

86. Verbal Description Method
= {a, b, c, d, e, …, z}
The Set E letters in the English
alphabet.
is the set of
You write..

88. The Set
Builder
Notation
(Rule Method)

89. The Set Builder Notation
It is a method that list the that
determine whether an object is an
element of the set rather than the
actual elements.
Let’s try to build the rules for the
sets in the previous slides.

90. Set Builder Notation
Examples:
1. A = { 1, 2, 3, 4 }
2. B = { p, h, i, l, n, e, s }
3. C = { 5, 10, 15… }
4. D = { moon }

91. Set Builder Notation
A = { x|x is a counting
number less than 5 }
= { 1, 2, 3, 4 }
Example:

92. Set Builder Notation
A = { x|x is a counting
number less than 5 }
= { 1, 2, 3, 4 }

93. Set Builder Notation
A = { x|x is a counting
number less than 5 }
= { 1, 2, 3, 4 }
A = { x|x is a counting
number less than 5 }
The Set A is the set of all x’s such that
x
You read this as..
is a counting number less than 5.

94. Set Builder Notation
A = { x|x is a counting
number less than 5 }
= { 1, 2, 3, 4 }
A = { x|x is a counting
number less than 5 }
TAKENOTE: The vertical bar
after the first x is translated as
“such that”

95. Set Builder Notation
A = { x|x is a counting
number less than 5 }
= { 1, 2, 3, 4 }
A = { x|x is a counting
number less than 5 }
The Set A is the set of all x’s such that
is a counting number less than 5.
Again, let’s read it..
x

96. How Do We Write a
Rule Method?
We do it like this:
E = { x | is a primary color
The Set E is the set of all x’s such that
x is a primary color.
}
Set
Name
Equal Sign
Opening
Brace
first X “ set of all x’s”
vertical bar
“such that”
rule Closing
Brace
x
second X
= { red, yellow, blue }

99. Set Builder Notation
Write:
B = { x | is a letter in the word “Philippines”
The Set B is the set of all x’s such that
x is a letter in the word “Philippines”.
.
}
Set
Name
Equal Sign
Opening
Brace
first X “ set of all x’s”
vertical bar
“such that”
rule Closing
Brace
x
second X
= { p, h, i, l, n, e, s }

100. Set Builder Notation
Write:
C= { x | is a positive multiple of 5
The Set C is the set of all x’s such that
x is a positive multiple of 5.
}
Set
Name
Equal Sign
Opening
Brace
first X “ set of all x’s”
vertical bar
“such that”
rule Closing
Brace
x
second X
= { 5, 10, 15… }

101. Set Builder Notation
Write:
C= { x | is a positive multiple of 5 up to 50
The Set C is the set of all x’s such that
x is a positive multiple of 5
}
Set
Name
Equal Sign
Opening
Brace
first X “ set of all x’s”
vertical bar
“such that”
rule Closing
Brace
x
second X
= { 5, 10, 15… }50
up to 50.

102. Set Builder Notation
Write:
D= { x | is a natural satellite of the earth
The Set D is the set of all x’s such that
x is a natural satellite of the earth.
}
Set
Name
Equal Sign
Opening
Brace
first X “ set of all x’s”
vertical bar
“such that”
rule Closing
Brace
x
second X
= { moon }

103. List the elements of the
following sets.
Answer:
Given:

104. List the elements of the
following sets.
Answer:
Given:

105. List the elements of the
following sets.
Answer:
Given:

106. List the elements of the
following sets.
Answer:
Given:

107. List the elements of the
following sets.
Answer:
Given:

108. List the elements of the
following sets.
Answer:
Given: