In a survey of students about favorite sports, the results include 18 who like tennis, 25 who like football, 10 who like tennis and football, 13 who like tennis and baseball, 20 who like football and baseball, 8 who like all three sports, and 9 who like none of the sports. How many students like only tennis and football? How many students like only tennis and baseball? How many students like only baseball and football?

Number of students who like only tennis and football = 2

Number of students who like only tennis and baseball = 5

Number of students who like only baseball and football = 12

Solution:

Let students who likes tennis be represented by set T.

Let students who likes football be represented by set F.

Let students who likes baseball be represented by set B.

Given that

Number of students who likes tennis that is n (T) = 18

Number of students who likes football that is n (F) = 25

Number of students who likes tennis and football = n (T ∩ F) = 10

Number of students who likes tennis and baseball = n (T ∩ B) = 13

Number of students who likes football and baseball = n (F ∩ B) = 20

Number of students who likes all three sports = n (T ∩ F ∩ B) = 8

Number of students who likes none of the sports = 9.

Determining number of students who like only tennis and football:

number of students who like only tennis and football = Number of students who likes tennis and football - Number of students who likes all three sports

so number of students who like only tennis and football = n (T ∩ F) - n (T ∩ F ∩ B)

= 10 - 8 = 2

Determining number of students who like only tennis and baseball:

number of students who like only tennis and baseball = Number of students who likes tennis and baseball - Number of students who likes all three sports

so number of students who like only tennis and baseball = n (T ∩ B) - n (T ∩ F ∩ B)

= 13 - 8 = 5

Determining number of students who like only baseball and football:

number of students who like only baseball and football = Number of students who likes baseball and football - Number of students who likes all three sports

so number of students who like only baseball and football = n (B∩ F) - n (T ∩ F ∩ B)

= 20 - 8 = 12

Summarizing the result:

number of students who like only tennis and football = 2

number of students who like only tennis and baseball = 5

number of students who like only baseball and football = 12.