INTRODUCTION TO SETS

1. INTRODUCTION TO
SETS

2. Activity: Sort the countries listed below according to
their location on the map.
Northern Hemisphere Southern Hemisphere
Philippines
Italy
Nepal
Australia
Bangladesh
New Zealand
Ecuador
Indonesia
Guatemala
Kenya

3. COUNTRIES IN THE
NORTHERN HEMISPHERE
Nepal
Philippines
Guatemala
Bangladesh
Italy

4. COUNTRIES IN THE
SOUTHERN HEMISPHERE
Australia
Ecuador
New Zealand
Indonesia
Kenya
Name of a
Set
Elements

5. How do we write sets?
Roster Method
Set Builder Notation

6. Roster Method – the
members or elements are
separated by a comma
and enclosed between a
pair of braces.

7. Example 1
Let P be the set of terrestrial
planets Mercury, Venus, Earth and
Mars.
P = {Mercury, Venus, Earth,
Mars}

8. Example 2
Let A be the set of numbers from
1-10.
A = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}

9. Example 3
Let X be the set of numbers from
1-100
Are you going to write all the numbers
from one to 100? - NO
We can use the ELLIPSIS ( ...)

10. Example 3
Let X be the set of numbers from
1-100.
X = {1, 2 . . . 100}
DO NOT
FORGET TO
INCLUDE THE
LAST NUMBER,
WITHOUT IT, IT
WOULD MEAN
INFINITY.

11. Your turn!
Get your whiteboard and
marker and write the following
sets in Roster Method

12. Let P be the set of cluster
leaders in College of Saint
Anthony.

13. P = {Ms. Casiño,
Mr. De Vera}

14. Let A be the set of prime
numbers from 1-50

15. A = {2, 3, 5, 7, 11,
13, 17, 19, 23, 29,
31, 37, 41, 43, 47}

16. Let C be the set of multiples
of 2 from 2-100.

17. C = {2, 4, 6 . . . 100}

18. Let Y be the set of odd
numbers.
Are you having a hard time?

19. We can use the second
method with that. It’s
called the Set Builder
Notation.

20. Set Builder Notation– the
variable x may be use
arbitrarily to represent the
elements of a set along with
a vertical line and description
enclosed by braces.

21. Let Y be the set of odd
numbers.
Y = {x | x is odd}
“Y is the set of numbers x such that x is
odd”

22. Write Set A as the names of the
student of CSA in set builder
notation.
A = {x | x is a name of
a student in CSA}
“Y is the set of names x such that x is a
name of student in CSA”

23. Your turn!
Get your whiteboard and
marker and write the
following sets in Set
Builder Notation

24. Let Y be the set of all the stars
in the universe.
Y = {x | x is a star in
the universe}

25. Let U be the set of all the
multiples of 10.
U = {x | x is a multiple of
10}

26. Let B be the set of all the title
holders in the Miss Universe
Pageant.
U = {x | x is a title holder in
Miss Universe Pageant}

27. GOOD JOB! ☺

28. Practice Reading
of Sets

29. A= {2,4,6,8,10}
Let A be the set of the
numbers, 2, 4, 6, 8 and 10.

30. B= {x|x is even}
B is the set of x such
that x is even.

31. C= {Saturn, Neptune, Jupiter, Uranus}
Let C be the set of the
planets Saturn, Neptune,
Jupiter and Uranus.

32. D= {x|x is a country and x is in Asia}
D is the set of x such
that x is a country
and x is in Asia.

33. A={x|0 ≤ x ≤ 10 and x is a counting
number}
Let A be the set of x such that x
is greater than or equal to 0 and
less than or equal to 10 and is a
counting number.

34. A={x|10 ≤ x ≤ 100 and x is an odd
number}
Let A be the set of x such that x
is greater than or equal to 10
and less than or equal to 100
and is an odd number.

35. E= {Tony, Jose, Albert, Diego}
Let E be the set of Tony,
Jose, Albert and Diego.

36. E= {Tony, Jose, Albert, Diego}
Can you give an element in
this set?
Tony is an element
of E

37. Tony is an element of E can also b
written as Tony Є E

38. The symbol Є denotes
that an object is an
element of a set.
Example:
G= {Tony, Jose, Albert, Diego
Tony Є G
}

39. ACTIVITY #1
(10 pts)
Answer pages 9-10
Letters A and B

40. Finite or Infinite
Sets

41. A set is said to be finite
if and only if, it consists of
a specific number of
distinct elements.
Otherwise, it’s infinite.

42. Example 1:
Let H be the set of months in
the Gregorian calendar.
H = {January, February... December}
FINITE

43. Example 2:
Let M = {x|x is even}
INFINITE

44. Example 3:
Let N be the set of counting
numbers.
N = {x|x is a counting number}
INFINITE

45. Example 4:
Let O be the set of the stars
in the Solar System.
O = {y|y is a star in the Solar System}
FINITE

46. Example 5:
Let V = {a,e,i,o,u}
V is FINITE

47. EQUAL SETS
Two sets are said to
be equal if and only if,
the two sets contain
the same elements. (=)

48. Are A and B equal sets ?
A={1,2,3,4,5}
B={1,2,3,4,5}
YES!

49. Are C and D equal sets ?
C={2,4,6,7,5}
D={4,2,5,7,6}
YES!

50. Are X and Y equal sets ?
X={m,y,e,g,a,b,f}
Y={b,a,m,g,y}
NO!

51. Are X and Y equal sets ?
X={x|x is even number
from 1-10}
Y={2,4,6,8,10}
YES!

52. Are A and B equal sets ?
A={x|x is a country in Asia}
B={Japan, Korea, Taiwan,
Philippines, Germany}
NO!

53. Example 1:
Suppose that
A={x|0 ≤ x ≤ 10 and x is a
counting number} and
B={0,1,2, ... 10}
A = B

54. Example 2:
A={x|x is a multiple of 2 and x is
a multiple of 3} and
D={x|x is a multiple of 6}
A = D

55. EMPTY/NULL SETS
A set that contains no
element is called an
empty set. It is denoted
by the symbol Ø or { }.

56. Is set A an empty
set?
A= { }
YES!

57. Is set X an empty
set?
X= {1,2,3 }
NO!

58. Is set X an empty
set?
X= Ø
YES!

59. Let Y be the set of
months with 32 days.
EMPTY SET

60. Let T be the set of
triangles with four
sides.
EMPTY SET

61. Let Z be the set of
integers that are
both odd and even.
EMPTY SET

62. CARDINALITY OF A SET
-the number of distinct
elements in a set
-denoted by the symbol n(A)

63. Example:
What is the cardinality of set X?
X= {1,2,3 }
n(X) = 3

64. How many elements???
1.P = {r,s,t}
2.C = {months in a calendar}
3.D = {Filipino Alphabet}
n(P) = 3
n(C) = 12
n(D) = 28

65. Determine the cardinal number of sets in A and B.
Set A consists all the planets in the solar system.
Set B are multiples of 10 from 11 - 100
n(A) = 8 n(B) = 9

66. SUBSET
-a set may contain another set in it
and it is called a subset. Set B may
be called a subset of A if, and only
if, all elements of B are in A.
-denoted by ⊆

67. Example:
A = {1,2,3,4,5 }
B = {1,2,3}
C = {2,3,5 }
D = {5}
E = {1,2,3,6 }
Is set B a subset of Set A?
Yes, we can write it as B ⊆ A
How about, is set E a subset of A?
No, we can write that as E ⊄ A

68. How many subsets are there in Set A?
A = {1,2,3}
Ø {1} {2} {3} {1,2}
{1,3} {2,3} {1,2,3}
There are 8 subsets in Set A

69. How many subsets are there In Set B?
B= {2,4,6,8,10}

70. 𝑛 = 7
Step 1:Determine
the cardinality of
the set.
Step 2:Plug it
in the formula
and simplify.
2𝑛
27=2 × 2 × 2 × 2 × 2 × 2 × 2
128 𝑠𝑢𝑏𝑠𝑒𝑡𝑠

71. How many subsets are there in Set X?
Set X is equal to the number of months in
the calendar.

72. 𝑛 = 12
2𝑛
212 = 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2
× 2 × 2 × 2 × 2
4,096 𝑠𝑢𝑏𝑠𝑒𝑡𝑠

73. WORKBOOK 1
(10 pts)
Firm Up Letter C
#s 1-5