Three sets problem

So far, we have seen how to represent two sets in a Venn diagram.
We shall also use Venn diagram to represent or solve problems that
may involve three or more sets.
Example
Given that A = {1, 2, 3, 8, 9, 10}, B = {3, 4, 5, 6, 7, 8, 9}, and C = {7, 8, 9,
10, 11}
Represent the above information on a Venn diagram, hence find:

a) n (A n B n C)
b) n (A u B u C)
c) n (A n B n C’)

bb

n(A n B n C) = 2
n(A u B u c) = 11
n(A n B n C’) = 1

Example
On the Venn diagram, shade the following:
a) (A u B n C)
b) (A n B u C)
Solution
a) For (A u B n C)

A u B u C

A u B is the shaded vertically and C is shaded horizontally.
A u B n C is the area shaded both ways.

b) For A n C u B

A n B u C 1

A n C is shaded vertically and B is shaded horizontally.
A n C u B is the area shaded both ways.
Example
The Venn diagram below represents the members of students taking
History (H), Geography (G), and French (F) in a certain class.

HG

a) How many take history?
b) How many take French and Geography?
c) How many students take all the three subjects?
d) How many students are there altogether?
e) Write down:
i) n(H n F)
ii) n(F u G)
iii) n (H n F n G’)
iv) n(H u F u G)’

Solution
a) Number taking history = 2 + 4 + 8 + 10 = 24
b) Number taking French and geography = 8 + 7 = 15
c) n(H n F n G) = 8
d) n(E) = (2 + 4 + 6 + 8 + 7 + 10 + 11 + 12) = 60
There are 60 students altogether.
e) i) n (H n F) = 10 + 8 = 18
ii) n (F u G) = 4 + 6 + 8 + 7 + 10 + 11 = 46
iii) n(H n F n G’) = 10
iv) n(H u F u G)’ = 12

Example
50 students were asked whether they liked football (F), volleyball (V) or
basketball (B). 8 like football and volleyball, 11 liked basketball and
volleyball, 19 liked football and basketball, 6 liked basketball only, 7
liked football only and 8 liked volleyball only.
How many liked:
a) all the three games if three of the students liked none of the games.
b) basketball and football only.
Solution
n(E) = 50
n (B n F’ n V’) = 6 n(F n V) = 8
n (v n B’ n F’) = 8 n(B n v) = 11
n (F n B’ n V’) = 7 n(F n B) = 19
n (F u V u B)’ = 3
Let n(F n V n B) = x

e 50

a) For all the three games:
7 + 8 – x +8 +x + 19 – x + 11 – x + 6 + 3 = 50
62 – 2x = 50
-2x = -12 divide both sides by -2
x = 6
Therefore, 6 students liked all the three games.
b) Basketball and football only 19 – x = 19 – 6 = 13

Example
The Venn diagram below shows representation of members of
community council to three different committees of Finance (F),
Production (P), and Security (S)

setss

a) Determine the value of a, b and c.
b) Find:
i) The total number of members who make up the community council.
ii) Number of members who belong to more than one committee.
Solution
a) For Production:
a + c + 2 + 3 = 10
a+ c = 10 – 5
a + c = 5 …………………………..(1)
For Finance:
a + b + 3 +3 = 10
a + b = 4…………………….(2)

For Security:
b + c + 3 + 1 = 7
b +c =7 – 4
b + c = 3 ………………………(3)

From equation (1): a = 5 – c and substituting for a from (1) in
equation (2):
5 – c + b = 4
b – c = -1 ………………………..(1)

Solving (3) and (4) simultaneously
b + c = 3
b – c = -1
2b = 2
b = 1
c = 3 – b = 3 – 1 = 2
a = 5 – c = 5 – 2 = 3 there for a = 3, b = 1, c = 2
b) i) The total number of members making up the community council
2 + a + 3 + 3 + c + b + 1 + 3
2 + 3 + 3 + 3 + 2 + 1 + 1 + 3
= 18
ii) Number of members who belong to more than one committee
a + c + b + 3
3 + 2 + 1 + 3 = 9