# Solving problems using Venn diagram

The following examples will illustrate how a Venn diagram can be used
to solve certain mathematical problem.
Example
Use Venn diagram to show that:
a) (A n B)’ = A’ U B’
b) (A u B)’ = A’ n B’

Example
There are 40 men working in a company. 38 of them own either a car
or a house or both. 33 men own a car, 24 of whom also own a house.
Represent this information in a Venn diagram and use it to state:
a) the number of men who do not own a house.
b) the number of men who own a house but do not own a car.
c) the number of men who own neither a car nor a house.
Solution
Let, C represent those who own a car.
H represents those who own a house.
And let: x be the number of men who own a car only.
y be the number of men who own a house only.
Z be the number of men who own neither a car nor a house.

Given:

n(ℇ) = 40
n(C U H) = 38 ,n(C) = 33
n(C n H) = 24, n(H) = ?

From the diagram:
X + 24 = 33
X = 33 – 24 = 9
n(C U H) = X + 24 +y = 38
9 + 24 +y = 38
y = 5
a) Number of men who do not own a house
n (H) =x + Z = 9 + 5 = 14
b) Number of men who own a house but not a car = y = 5
c) n(C’ U H’) = Z =2