The document provides examples and explanations of sets, Venn diagrams, and solving word problems using sets and Venn diagrams. It defines what a set is, the elements and members of a set. Examples are given of drawing and interpreting Venn diagrams to represent relationships among sets and solve problems involving sets, unions, intersections, complements and differences of sets. Word problems involving 2 or 3 sets are presented along with their Venn diagram representations and solutions.

2. MODULE 1

3. What is a set?
A group or collection of objects
a set is a well-defined collection
of distinct objects, considered as
an object in its own right.

4. Each objects in a set is called a
MEMBER or an ELEMENT of a set.
𝜀

5. (ellipsis)

6. Given: U = {1, 2, 3, 4, 5, 6, 7, 8 , 9, 10, 11, 12, 13, 14}
A={1, 2, 3, 4, 5, 6}
B = {2, 4, 6, 8,10, 12, 14}
C = {2, 3, 5, 7, 9, 11, 13}
Draw a Venn diagram showing the
relationships among the given sets.

7. A B
C
2
6
41
5
3
12
10
8
14
13
11
9
7

8. Given: U = {1, 2, 3, 4, 5, 6, 7, 8 , 9, 10, 11, 12, 13, 14}
A={1, 2, 3, 4, 5, 6}
B = {2, 4, 6, 8,10, 12, 14}
C = {2, 3, 5, 7, 9, 11, 13}
Find: A’ ∪ B
A’ = {7, 8 , 9, 10, 11, 12, 13, 14}
A’ ∪ B = {2, 4, 6, 7, 8 , 9, 10, 11, 12, 13, 14}
B = {2, 4, 6, 8,10, 12, 14}

9. Given: U = {1, 2, 3, 4, 5, 6, 7, 8 , 9, 10, 11, 12, 13, 14}
A={1, 2, 3, 4, 5, 6}
B = {2, 4, 6, 8,10, 12, 14}
C = {2, 3, 5, 7, 9, 11, 13}
Find: A ∩ U
A ∩ U = {1, 2, 3, 4, 5, 6}

10. Given: U = {1, 2, 3, 4, 5, 6, 7, 8 , 9, 10, 11, 12, 13, 14}
A={1, 2, 3, 4, 5, 6}
B = {2, 4, 6, 8,10, 12, 14}
C = {2, 3, 5, 7, 9, 11, 13}
Find: A – B
A – B = {1, 3, 5}

11. Given: U = {1, 2, 3, 4, 5, 6, 7, 8 , 9, 10, 11, 12, 13, 14}
A={1, 2, 3, 4, 5, 6}
B = {2, 4, 6, 8,10, 12, 14}
C = {2, 3, 5, 7, 9, 11, 13}
Find: B’
B’ = {1, 3, 5, 7, 9, 11, 13}

12. Given: U = {1, 2, 3, 4, 5, 6, 7, 8 , 9, 10, 11, 12, 13, 14}
A={1, 2, 3, 4, 5, 6}
B = {2, 4, 6, 8,10, 12, 14}
C = {2, 3, 5, 7, 9, 11, 13}
Find: B’ – C
B’ = {1, 3, 5, 7, 9, 11, 13}
B’ – C = {1}
C = {2, 3, 5, 7, 9, 11, 13}

13. MODULE 2

14. Venn Diagrams

15. • use Venn diagrams to illustrate sets, subsets and set operations.
• solve word problems involving sets with the use of Venn diagrams
• apply set operations to solve a variety of word problems.

16. Venn Diagram is a diagram representing
mathematical or logical sets pictorially as
circles or closed curves within an enclosing
rectangle (the universal set), common
elements of the sets being represented by the
areas of overlap among the circles.

17. John Venn

21. More Examples on Venn Diagram

22. 𝐴 = 𝑝, 𝑛, 𝑜, 𝑦
Elements which belong to set A

23. Elements which belong to set B
𝐵 = 𝑝, 𝑜, 𝑡

24. 𝐴 ∩ 𝐵
𝐴 𝑎𝑛𝑑 𝐵
𝐴 intersection 𝐵
Elements are common to set A and set B 𝐴 ∩ 𝐵

25. 𝐴 ∪ 𝐵
𝐴 𝑜𝑟 𝐵
𝐴 union 𝐵
Elements which belong to set A, or
set B or to both sets
𝐴 ∪ 𝐵

26. A′
Complement of A
Elements of U that do
not belong to A

27. 𝐵′
Complement of B
Elements of U that do
not belong to B

28. 𝐴 − 𝐵
Difference of A and B
Elements which belong to set A
but which do not belong to set B

31. Another example…

32. A class of 25 students were surveyed and asked
if they have a brother or a sister. Eight students said
they have only a brother, 6 students said they have
only a sister, another six said they have both a
brother and a sister and 5 said they don’t have a
brother or a sister.

33. Set-up of the Venn diagram based on the given problem

34. There are problems which involve 3 sets. The shaded parts of
the sets are named and identified.

40. Let’s apply….

41. A group of students were interviewed on what colors
they like. Below are their responses.

42. How many students like blue color?
How many students like yellow and red?
How many students like yellow or blue?
How many students like yellow only?
How many students like yellow, blue and red?
How many students like blue or red but not yellow?
How many students like neither yellow, blue nor red?
25
8
39
5
30
2
11

43. More example…

44. The Venn diagram
displays the results of a
survey to 100 students in
Muntinlupa Science High
School of whom owns a
pet at home.

45. How many students owned only a dog? 51
How many students owned only a cat? 17
How many students owned only a bird? 7
How many students owned a dog but not a cat? 56
How many students owned a cat? 35
How many students owned a cat and a bird? 9
How many students owned a dog and a bird but not a cat? 5
How many students owned a dog or a cat but not a bird? 77
How many students owned none of the three pets? 2
How many students own all of the three pets? 6

46. More example…

47. In a survey, seventy-five students were interviewed on which
subject they like better, English or Mathematics? Here is the
result.
45 students like English
32 students like Mathematics
12 students like both Mathematics and English
How many students like Mathematics only?
How many students like English only?
How many students dislike the two subjects?

48. English Math
201233
10
U 45 students like English 32 students like Mathematics
12 students like both Mathematics and English
32 - 1245 - 1275 - 65
75 students

49. In a survey, seventy-five students were interviewed on which
subject they like better, English or Mathematics? Here is the
result.
45 students like English
32 students like Mathematics
12 students like both Mathematics and English
How many students like Mathematics only?
How many students like English only?
How many students dislike the two subjects?
20
33
10

50. Another example…

51. Among the 50 pupils of Muntinlupa Elementary School, 32
likes gumamela flower while 26 likes rose flower.
How many pupils like gumamela flower and rose flower?
How many pupils like gumamela flower only?

52. GumamelaRose
U
818 24
32 likes gumamela flower
26 likes rose flower
50 pupils of Muntinlupa Elementary School

53. Among the 50 pupils of Muntinlupa Elementary School, 32
likes gumamela flower while 26 likes rose flower.
How many pupils like gumamela flower and rose flower?
How many pupils like gumamela flower only?
8
24

54. Another example…

55. Out of fifty students, 23 joined Mathematics club and 32 joined
English club. If 8 joined in both Mathematics and English club,
How many have joined the English club only?
How about in Mathematics club only?
How many are neither in Mathematics nor in English club?

56. 8 joined in both Mathematics and English club
32 joined English club
23 joined Mathematics club
Out of 50 students

57. 8
8 joined in both Mathematics and English club
32 joined English club 23 joined Mathematics club
English Math
23 – 8
1524
32 – 8
U
Out of 50 students
50 – 47
3

58. Out of fifty students, 23 joined Mathematics club and 32 joined
English club. If 8 joined in both Mathematics and English club,
How many have joined the English club only?
How about in Mathematics club only?
How many are neither in Mathematics nor in English club?
24
15
3

59. Another example…

60. “start inside out”

61. A group of 50 students went to a tour in Palawan
province. Out of the 50 students, 24 joined the trip to
Coron, 18 went to Tubbataha Reef, 20 visited El Nido,
12 made a trip to Coron and Tubbataha Reef, 15 saw
Tubbataha Reef and El Nido, 11 made a trip to Coron
and El Nido and 10 saw the three tourist spots.
Questions:
a. How many students went to Coron only?
b. How many students went to Tubbataha Reef only?
c. How many joined the El Nido trip only?
d. How many did not go to any of the tourist spots?

62. 50 students went in a tour in Palawan province.
24 joined the trip to Coron,(C)
18 went to Tubbataha Reef, (T)
20 visited El Nido, (E)
12 made a trip to Coron and Tubbataha Reef,
15 saw Tubbataha Reef and El Nido,
11 made a trip to Coron and El Nido
10 saw the three tourist spots.

63. 10
CoronU
Tubbataha
El Nido
1 5
2
4
1
11
16
10 saw the three tourist spots
50 students went in a tour in Palawan province.
24 joined the trip to Coron
18 went to Tubbataha Reef
20 visited El Nido
12 made a trip to Coron and Tubbataha Reef
15 saw Tubbataha Reef and El Nido
11 made a trip to Coron and El Nido

64. “side to side”

65. Among the 50 pupils of Muntinlupa Elementary School, 32
likes gumamela flower while 26 likes rose flower.
How many pupils like gumamela flower and rose flower?
How many pupils like gumamela flower only?

66. GumamelaRose
U
x26 – x 32 – x
32 likes gumamela flower
26 likes rose flower
50 pupils of Muntinlupa Elementary School

67. 26 – x + x 32 – x+ = 50

69. Another example…

70. Among the 40 students of section Arrhenius, 23 loves singing
and 25 loves dancing.
How many students love singing and dancing?
How many students love singing only?

71. Dance Sing
25 loves dancing 23 loves singing
x25 – x 23 – x
U
40 students of section Arrhenius

72. x25 – x 23 – x+ + = 40

73. Dance Sing
8
25 loves dancing 23 loves singing
17 15
U

74. Among the 40 students of section Arrhenius, 23 loves singing
and 25 loves dancing.
How many students love singing and dancing?
How many students love singing only?
8
15

76. 113 Munscian students were surveyed and it was found out that all of them have visited at least
one of the following three provinces: Abra, Bataan, and Cavite.
66 of the students visited Abra.
52 of the students visited Bataan.
76 of the students visited Cavite
19 of the students visited Abra and Bataan.
28 of the students visited Bataan and Cavite.
42 of the students visited Abra and Cavite.
40 of the students visited exactly one of the following three cities: Abra, Bataan, and Cavite.
a. How many students visited only Abra?
b. How many students visited only Bataan?
c. How many students visited only Cavite?
d. How many students visited both Abra and Bataan, but not Cavite?
e. How many students visited both Bataan and Cavite, but not Abra?
f. How many students visited both Abra and Cavite, but not Bataan?
g. How many students visited all of the following three cities: Abra, Bataan, and Cavite?

77. In a group of 60 people, 27 like cold drinks and
42 like hot drinks and each person likes at
least one of the two drinks. How many like
both hot and cold drinks?
3 of the students in Sir G's class have been to
Archimedes. 4 students have been to Ampere, and 1
student has been to both Archimedes and Ampere.
How many students have been to Archimedes but
not Ampere?