Solve problems involving sets with the use of Venn Diagram

2. MODULE 2

3. Venn Diagrams

4. • 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.

5. 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.

6. John Venn

7. MODULE 3

8. Solve problems involving sets with the use of
Venn Diagram

9. REVIEW!

10. INTERSECTION
Given: A = {p, u, r, e} and B = {h, e, a, r, t}
the intersection of two sets A and B, denoted by A ∩ B, is
the set containing all elements of A that also belong to B
(or equivalently, all elements of B that also belong to A).
A ∩ B = {r, e}
Given: A = {2, 4, 6, 8} and B = {1, 3, 5, 7}
A ∩ B = {} or ∅

11. Draw a Venn diagram to represent the relationship
between the sets
X = {1, 2, 5, 6, 7, 9, 10} and Y = {1, 3, 4, 5, 6, 8, 10}
X ∩ Y = {1, 5, 6, 10}

12. UNION
The union of two sets is a set containing all elements that
are in A or in B (possibly both)

13. UNION
The union of two sets is a set containing all elements that
are in A or in B (possibly both)
Given: A = {p, u, r, e} and B = {h, e, a, r, t}
A U B = {p, u, r, e, h, a, t}
Given: A = {l, o, v, e} and B = {h, a, t, e}
A U B = {l, o, v, e, h, a, t}

14. Given U = {1, 2, 3, 4, 5, 6, 7, 8, 10}
X = {1, 2, 6, 7} and Y = {1, 3, 4, 5, 8}
Find X ∪ Y and draw a Venn diagram to illustrate X ∪ Y.
X ∪ Y = {1, 2, 3, 4, 5, 6, 7, 8}

15. DIFFERENCE The relative complement or set difference of sets A and
B, denoted A – B, is the set of all elements in A that are
not in B.
A = {d, i, n, e} B = {e, n, d}
A - B = {i}

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

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

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

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

23. “start inside out”

24. 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?

25. 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.

26. 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

27. “side to side”

28. 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?

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

31. Example 1

32. Fifty people are asked about the pets they keep at home.
The Venn diagram shows the results.
Let D = {people who have dogs}
F = {people who have fish}
C = {people who have cats} D F
C
19
12
1
7 6
3
U
How many people have:
a. dogs
b. dogs and fish
c. dogs or cats
d. fish and cats but not dogs
e. dogs or fish but not cats
f. all three
g. neither one of the three
39
8
42
0
32
1
2
n(D)
n(D∩F)
n(DUC)
n(F ∩ C) ∩ D’
n(DUF) ∩ C’
n(D ∩ F ∩ C)
n(D U F U C)’
2

33. Example 2

34. In Munsci, 200 students are randomly selected. 140 like tea, 120
like coffee and 80 like both tea and coffee
• How many students like only tea?
• How many students like only coffee?
• How many students like neither tea nor coffee?
• How many students like only one of tea or coffee?
• How many students like at least one of the beverages?
U
T C
8060 40
20
60
40
20
60 40 = 100+
60 40+ 80+ = 180

35. Example 3

36. In a class, there are 15 students who like chocolate. 13
students like vanilla. 10 students like neither. If there are 35
people in the class, how many students like chocolate and
vanilla?
15 students like chocolate
13 students like vanilla
10 students like neither
35 people in the class
U
10
C V
15 students like chocolate
13 students like vanilla
15 + 13 + 10= 38
38 – 35= 3
35 people in the class
312 10
SOLUTIONS!

37. Example 4

38. In a survey of 200 students of a school it was found that 120 study mathematics, 90 study
physics and 70 study chemistry, 40 study mathematics and physics, 30 study physics and
chemistry, 50 study chemistry and mathematics and 20 study none of these subjects. Find the
number of students who study all three subjects.
M P
C
U
200 students of a school
120 study mathematics
90 study physics
70 study chemistry
40 study mathematics and physics
30 study physics and chemistry
50 study chemistry and mathematics
20 study none of these subjects 20
50 30
40120 90
70
200

39. M P
C
U
20
50 30
40120 90
70
200

40. Try Me

41. A given company has 1500 employees. Of those
employees, 800 are computer science majors. 25%
of those computer science majors are also
mathematics majors. That group of computer
science/math dual majors makes up one third of
the total mathematics majors. How many
employees have majors other than computer
science and mathematics?