Question

In a survey of 60 people, it was found that 25 people read newspaper H, 

26 read newspaper T, 26 read newspaper I, 9 read both H and I,

11 read both H and T, 8 read both T and I, 

3 read all three newspapers. Find: 

(i) the number of people who read at least one of the newspapers. 

(ii) the number of people who read exactly one newspaper. 

Answer 1

Let A be the set of people who read newspaper H. 

Let  B be the set of people who read newspaper T. 

Let C be the set of people who read newspaper I. 

Accordingly,

 n(A) = 25, n(B) = 26, n(C) = 26 , 

n(A ∩ C) = 9, n(A ∩ B) = 11, 

n(B ∩ C) = 8 , 

n(A ∩ B ∩ C) 

= 3 Let U be the set of people who took part in the survey.

 (i) Accordingly,

 n(A B C) = n(A) + n(B) + n(C) – n(A ∩ B) – n(B ∩ C) – n(C ∩ A) + n(A ∩ B ∩ C)

= 25 + 26 + 26 – 11 – 8 – 9 + 3 = 52

Hence,

 52 people read at least one of the newspapers. 

(ii) Let a be the number of people who read newspapers H and T only. 

                                           

Let b denote the number of people who read newspapers I and H only. 

Let c denote the number of people who read newspapers T and I only.

Let d denote the number of people who read all three newspapers. 

Accordingly,

d = n(A ∩ B ∩ C) = 3 

Now, 

n(A ∩ B) = a + d 

n(B ∩ C) = c + d

 n(C ∩ A) = b + d 

∴ a + d + c + d + b + d 

= 11 + 8 + 9 = 28 

 ⇒ a + b + c + d 

= 28 – 2d = 28 – 6 = 22 

Hence, 

(52 – 22) = 30 people read exactly one newspaper.