1. In a survey of 120 people, 45 like to play football, 40 like to play volleyball, 60 like to play cricket, 10 like football as well as volleyball, 25 like football as well as cricket, 20 like volleyball as well as cricket, and 5 like to play all three games.
a) If F, V, and C denote the set of people who like to play football, volleyball, and cricket respectively, write set notation to represent who do not like all three games.
b) Represent the above information in a Venn diagram.
c) Find the number of people who like at least one game.
d) What percentage of people did not like all three games?

See the below works. ;

Answered by karmenchong | 2025-07-05

The people who do not like all three games is represented as (F' ∩ V' ∩ C'). A Venn diagram can be created using given data to find that 95 people like at least one game, and approximately 20.83% do not like all three games.
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Answered by karmenchong | 2025-07-05

To solve the problem, let's break it down into parts and use set theory to represent the information given.
Let's define:

F as the set of people who like to play football.
V as the set of people who like to play volleyball.
C as the set of people who like to play cricket.

From the given information, we have:

∣ F ∣ = 45 ,
∣ V ∣ = 40 ,
∣ C ∣ = 60 ,
∣ F ∩ V ∣ = 10 (people who like both football and volleyball),
∣ F ∩ C ∣ = 25 (people who like both football and cricket),
∣ V ∩ C ∣ = 20 (people who like both volleyball and cricket),
∣ F ∩ V ∩ C ∣ = 5 (people who like all three games).

We can find the number of people who like at least one game using the formula:
∣ F ∪ V ∪ C ∣ = ∣ F ∣ + ∣ V ∣ + ∣ C ∣ − ∣ F ∩ V ∣ − ∣ F ∩ C ∣ − ∣ V ∩ C ∣ + ∣ F ∩ V ∩ C ∣.
Plugging in the values: ∣ F ∪ V ∪ C ∣ = 45 + 40 + 60 − 10 − 25 − 20 + 5 = 95.
Thus, 95 people like at least one of the games.
a) The set of people who do not like all three games is F ∪ V ∪ C − ( F ∩ V ∩ C ) . This can be expressed as:
∣ F ∪ V ∪ C ∣ − ∣ F ∩ V ∩ C ∣ = 95 − 5 = 90.
So, 90 people do not like all three games.
b) When representing this information in a Venn diagram, use three overlapping circles, where each circle represents one game:

The intersection of all three circles represents the 5 people who like all three games.
The intersections of two circles, excluding the overlap with the third circle, represent those who like exactly two of the games.
The non-overlapping parts of each circle represent those who like only that specific game.

c) As found previously, 95 people like at least one game.
d) To find the percentage of people who did not like all three games:

There are a total of 120 people surveyed.
The number who do not like all three games is 90.
So, the percentage is ( 90/120 ) × 100 = 75% .

Therefore, 75% of the people did not like all three games.

Answered by BenjaminOwenLewis | 2025-07-08