No part of this E-module/LMS Content can be reproduced, or transported or
shared to others without permission from the University. Unauthorized use of the
materials, other than personal learning use, will be penalized.
LEARNING OUTCOMES
At the end of the lesson, the students should be able to:
1. Define a Venn Diagram;
2. Explain components and purpose of a Venn Diagram;
3. Correctly create a Venn Diagram; and
4. Recognize relationships between the concepts.
VENN DIAGRAM
A Venn diagram uses
overlapping circles or
other shapes to illustrate
the logical relationships
between two or more sets
of items.
REPRESENTING SUBSETS ON A VENN DIAGRAM
U= { 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10 }
S = { 0, 1, 2 }
T = { 0, 1, 2, 3, 4 }
From the sets above, it can be noticed that all the elements of S and T,
respectively, are elements of the Universal Set U.
Hence, S and T are subsets of U, (S ⊆ U and T ⊆ U).
Also, since all the elements of S are elements of T, then S ⊆ T.
REPRESENTING SUBSETS ON A VENN DIAGRAM
U= { 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10 }
S = { 0, 1, 2 }
T = { 0, 1, 2, 3, 4 }
Notice that the smaller circle is
enclosed in the bigger one.
This implies that all the elements
enclosed on the smaller circle, in
this case Set S, are also elements of
the bigger circle, in this case Set T.
Hence, S ⊆ T.
In the same manner, both circles are
enclosed in a rectangle, in this case
Universal Set U.
Thus, both sets S and T are subsets
of U.
It must be noted that the elements
of U include all the elements in the
two circles.
INTERSECTION OF TWO SETS ON A VENN DIAGRAM
The intersection of two sets A and B, A ∩ B, consists of all elements
belonging to both A and B. This can be represented in Venn diagram by:
A only
B only
Region outside the
overlapping circles
EXAMPLE
If M = {vowels in the alphabet} and L = {letters in ‘algebra’ },
determine M ∩ L.
M = {a, e, i, o, u}
L = {a, l, g, e, b, r}
The elements in the overlapping areas of the two circles are ‘e’ and ‘a’.
Therefore, M ∩ L = {a, e}
UNION OF TWO SETS ON A VENN DIAGRAM
Let A and B be the two sets. The union of A and B is the set of all those
elements which belong either to A or to B or both A and B. It is denoted
as A U B.
A U B = {all the elements found on the shaded portion of the two circles}
EXAMPLE
If M = {vowels in the alphabet} and L = {letters in ‘algebra’},
determine M U L.
M = {a, e, i, o, u}
L = {a, l, g, e, b, r}
To get the union of M and L, just list ALL the elements found in the two
circles. Hence, M U L = {a, e, i, o, u, l, g, b, r}
COMPLEMENT OF A SET ON A VENN DIAGRAM
The complement of a set, denoted A', is the set of all elements in
the given universal set U that are not in A.
A' = {set of all elements located in the shaded portion of the rectangle }
EXAMPLE
If U = {days of the week } and B = { days not included on weekdays},
determine B’.
The complement of B is the set of elements not located in the circle
(located in the shaded portion of the rectangle), hence
B’ = {Monday, Tuesday, Wednesday, Thursday, Friday}.
DISJOINT SETS ON A VENN DIAGRAM
Two sets are called disjoint if they have no elements in common. Thus,
the intersection of these two sets is an empty set, { }.
From the Venn diagram, the
two circles have no overlapping
area. This suggests that there
are no elements common to
both sets. Hence, it can be said
that A and B are disjoint.
EXAMPLE
Determine whether sets A and B are disjoint.
A = { x │ x is a positive even number < 10}
B = { x │ x is a positive odd number < 10}
A = {2, 4, 6, 8}
B = {1, 3, 5, 7, 9}
From the Venn diagram above,
there are no elements common
to both A and B.
Thus, A and B are disjoint.
SAMPLE PROBLEM 1
Use the Venn Diagram below to identify the elements of the
indicated sets
1. A ∩ (B U C)
2. (A’ ∩ B) U C
U = {K, P, N, V, R, T, Q, J, H, X, E}
A = {K, P, N, V, R}
B = {N, V, J, H, T, Q}
C = {R, V, H, J, X}
SAMPLE PROBLEM 1
1. A ∩ (B U C)
A = {K, P, N, V, R}
B = {N, V, J, H, T, Q}
C = {R, V, H, J, X}
Step 1:
Perform first the operation within a parenthesis.
A ∩ (B U C)
B U C = {R, V, J, H, X, N T, Q}
Step 2:
Get the intersection of the union of B and C and A.
A ∩ (B U C)
{K, P, N, V, R} ∩ {R, V, J, H, X, N T, Q} = {N, R, V}
Therefore, A ∩ (B U C) = {N, R, V}.
SAMPLE PROBLEM 1
2. (A’ ∩ B) U C
Step 1:
Step 2:
Step 3:
U = {K, P, N, V, R, T, Q, J, H, X, E}
A = {K, P, N, V, R}
B = {N, V, J, H, T, Q}
C = {R, V, H, J, X}
Determine first the complement of A, A’.
A’ = {X, J, H, Q,T, E }
Perform the operation within the parenthesis.
(A’ ∩ B) U C
(A’ ∩ B) = {X, J, H, Q, T, E} ∩ {N, V, J, H, T, Q}
(A’ ∩ B) = {J, H, Q, T}
Get the union of {J, H, Q, T} and C.
{J, H, Q, T } U C = {R, X, V, J, H, Q, T }
Therefore, (A’ ∩ B) U C ={R, X, V, J, H, Q, T }.
SAMPLE PROBLEM 2
The areas of a Venn Diagram are often represented as regions
using Roman Numerals. For each item below, identify the
region in which it would be placed.
a. 6
b. 20
c. 30
d. 9
e. 15
f. 42
g. 210
SAMPLE PROBLEM 2
a. 6
Notice that 6 is an even number and
divisible by three. Therefore, it is
present on the intersection of even
numbers and multiples of three.
6 is in REGION II
b. 20
Since 20 is an even number and it is
divisible by 5, therefore 20 is an
intersection of even numbers and
multiples of 5.
20 is in REGION IV
SAMPLE PROBLEM 2
c. 30
Thirty (30) is an intersection of all the
sets present in the problem because it is
divisible by 5 and 3 and it also an even
number.
30 is in REGION V
d. 9
Nine (9) is only divisible by three and it
is an odd number.
9 is in REGION III
SAMPLE PROBLEM 2
e. 15
Fifteen (15) is a multiple of both 3 and 5.
Thus, it represents an intersection
between them.
15 is in REGION VI
SAMPLE PROBLEM 2
f. 42
Since 42 is an even number and is
divisible by three, therefore it is
present on the intersection of even
numbers and multiples of three.
42 is in REGION II
g. 210
210 is an intersection of all the sets
present in the problem because it is
divisible by 5 and 3 and it is also an
even number.
210 is in REGION V
SAMPLE PROBLEM 3
A travel agent surveyed 100 people to find out how many of them
had visited the cities of Tagaytay and Baguio. Thirty-one people
had visited Tagaytay City, 26 people had been to Baguio City, and
12 people had visited both cities. Draw a Venn diagram to find the
number of people who had visited:
a. Either Tagaytay City or Baguio City
b. Baguio City but not Tagaytay City
c. Only one of the two cities
d. Neither of the cities
SAMPLE PROBLEM 3
Let
M be the set of people who had visited Tagaytay City
B be the set of people who had visited Baguio City
U be the set of people surveyed
| U | = 100
| M | = 31
| B | = 26
| M ∩ B | = 12
Hence number in
M only = 31 – 12 = 19
and number in
B only = 26 – 12 = 14.
12
19
14
SAMPLE PROBLEM 3
Find the number of people who had visited either Tagaytay City or
Baguio City
Let n be the number of people who had visited either Tagaytay or
Baguio City
n = 19 + 14 +12
n = 45
SAMPLE PROBLEM 3
Find the number of people who had visited Baguio City but not Tagaytay
City
Let n be the number of people who had visited Baguio City but not
Tagaytay
n = 14
SAMPLE PROBLEM 3
Find the number of people who had visited only one of the two cities
Let n be the number of people who had visited only one city
n = 19 + 14
n = 33
SAMPLE PROBLEM 3
Find the number of people who had visited neither of the cities
Let n be the number of people who had visited neither of the cities
n = 100 – 45
n = 55
SAMPLE PROBLEM 4
In a survey of 500 investors, it was reported that 270 invested in real
estate, 300 invested in forex trading, and 100 invested in both real
estate and forex trading. Use Venn diagram to answer the following.
a. How many invested in real estate only?
b. How many invested in forex trading only?
c. How many invested in both?
d. How many invested in neither real estate nor forex trading?
e. How many invested in either real estate or forex trading?
SAMPLE PROBLEM 4
Let:
Real Estate = S = |270|
100
Forex Trading = B = |300|
S ∩ B = 100
Hence,
S only = (270 – 100) = 170
B only = (300 - 100) = 200
170
200
SAMPLE PROBLEM 4
How many invested in real estate only?
Let n be the number of investors who invested in real estate only
n = 170
SAMPLE PROBLEM 4
How many invested in forex trading only?
Let n be the number of investors who invested in forex trading
only
n = 200
SAMPLE PROBLEM 4
How many invested in both?
Let n be the number of investors who invested in both
investments
n = 100
SAMPLE PROBLEM 4
How many invested in neither real estate nor forex trading?
Let n be the number of investors who invested in neither real
estate nor forex trading
n = 500 – (170 + 100 + 200)
n = 30
SAMPLE PROBLEM 4
How many invested in either real estate or forex trading?
Let n be the number of investors who invested in either real
estate nor forex trading
n = 170 + 200 + 100
n = 470
SAMPLE PROBLEM 5
A group of 62 students were surveyed, and it was found that each of
the students surveyed liked at least one of the following three fruits:
apples, bananas, and oranges.
34 liked apples
30 liked bananas
33 liked oranges
11 liked apples and bananas
15 liked bananas and oranges
17 liked apples and oranges
19 liked exactly two of the following fruits: apples, bananas, and
oranges
SAMPLE PROBLEM 5
a. How many students liked apples, but not bananas or oranges?
b. How many students liked oranges, but not bananas or apples?
c. How many students liked all of the following three fruits:
apples, bananas, and oranges?
d. How many students liked apples and oranges, but not bananas?
SAMPLE PROBLEM 5
Let x be the number of students who liked
all the fruits
11 liked apple and bananas
15 liked bananas and oranges
17 liked apple and oranges
11 - x
17 - x
STEP 1
11 liked apples and bananas
: 11 – x
15 liked bananas and oranges
: 15 – x
17 liked apples and oranges
: 17 – x
15 - x
SAMPLE PROBLEM 5
STEP 2
From the given statement that
“19 liked exactly two of the
following fruits: apples, bananas,
and oranges”, then
(11 – x) + (15 – x) + (17 – x) = 19
43 – 3x = 19
24 = 3x
x=8
8
SAMPLE PROBLEM 5
STEP 3
Substitute the obtained value of x
to the following regions,
11 − 𝑥 → 11 − 8 = 𝟑
15 − 𝑥 → 15 − 8 = 𝟕
17 − 𝑥 → 17 − 8 = 𝟗
3
9
7
SAMPLE PROBLEM 5
STEP 4
Solve for the number of the students who like only 1 fruit
Apple only = 34 – (9 + 8 + 3)
Apple only = 14 students
Banana only = 30 – (7 + 8 + 3)
Banana only = 12 students
Orange only = 33 – (9 + 8 + 7)
Orange only = 9 students
14
15
12
12
9
Oranges
SAMPLE PROBLEM 5
How many students liked
apples, but not bananas or
oranges?
Let n be the number of
students who liked apples,
but not bananas or oranges
n = 14
SAMPLE PROBLEM 5
How many students liked
oranges, but not bananas or
apples?
Let n be the number of
students who liked oranges,
but not bananas or apples
n=9
SAMPLE PROBLEM 5
How many students liked
all of the three fruits:
apples,
bananas,
and
oranges?
Let n be the number of
students who liked all of
the three fruits
n=8
SAMPLE PROBLEM 5
How many students liked
apples and oranges, but not
bananas?
Let n be the number of
students who liked apples
and oranges, but not
bananas
n=9
POLL QUESTION
WHAT
IS
YOUR
LEVEL
UNDERSTANDING ABOUT THIS TOPIC?
A. NONE
B. A LITLE
C.SOME
D. A LOT
OF