Practical problems on set | Part-1

Set Theory Problem: Tea and Coffee Preferences

Question: In a group of 60 people, 37 like tea, 35 like coffee, and each person likes at least one of the two drinks. Find the number of persons who like: i) both tea and coffee ii) only tea iii) only coffee iv) neither tea nor coffee v) exactly one of the drinks.

Solve:

Let:

• Total number of people = 60
• Number of people who like tea, n(T) = 37
• Number of people who like coffee, n(C) = 35
• Since everyone likes at least one drink, n(T ∪ C) = 60

We use the formula:

n(T ∪ C) = n(T) + n(C) − n(T ∩ C)

Substitute the known values:

60 = 37 + 35 − n(T ∩ C)

n(T ∩ C) = 72 − 60 = 12

i) Number of people who like both tea and coffee:

Answer: 12

ii) Number of people who like only tea:

n(Only Tea) = n(T) − n(T ∩ C) = 37 − 12 = 25

iii) Number of people who like only coffee:

n(Only Coffee) = n(C) − n(T ∩ C) = 35 − 12 = 23

iv) Number of people who like neither tea nor coffee:

Since everyone likes at least one, the answer is 0

v) Number of people who like exactly one of the drinks:

n(Exactly one) = n(Only Tea) + n(Only Coffee) = 25 + 23 = 48

Summary:

• i) Both tea and coffee: 12
• ii) Only tea: 25
• iii) Only coffee: 23
• iv) Neither: 0
• v) Exactly one: 48