Example

600 men are interviewed about which sports they like. It is found that 190 like football, 140 like basketball, 120 like volleyball, 65 like football and basketball, 50 like football and volleyball, 40 like basketball and volleyball, 30 like all three sports.

1. How many of them like at least one of these sports?
2. How many of them like none of these sports?
3. How many of them like only football?
4. How many of them like football and basketball but not volleyball?
5. How many of them like basketball or volleyball but not football?

Solution

Define

F = The set of all men who like football,

B = The set of all men who like basketball,

V = The set of all men who like volleyball.

Then s(F) = 190,    s(B) = 140,   s(V) = 120, s(F ? B) = 65,   s(F ? V) = 50,   s(B ? V) = 40,

s(F ? B ? V) = 30 where, as defined above, the symbol s(.) is used for the number of elements. Below, we give the Venn diagram solution of this problem where the numbers on the regions they are sitting indicate the number of elements of the corresponding set. For  example, the number 30 indicates that he number of elements of the intersection F ? B ? V is 30.

We can easily count the required numbers in the questions a. – e.

1. 105 + 35 + 30 + 20 + 65 +10 + 60 = 325
2. 600 – 325 = 275,
3. 105,
4. 65 - 30 = 35,
5. 60 + 10 + 65 = 135.

As also undersood from the information given, all statements in the options are correct according to the case given, so the correct answer is E.