1. If X and Y are two sets such that n(X) = 17, n( Y ) = 23 and n (X ∪ Y) = 38, find n ( X ∩ Y ).
It is given that,
n(X) = 17, n(Y) = 23, n(X∪Y) = 38
We know that,
n(X∪Y) = n(X) + n(Y) - n(X∩Y)
38 = 17 + 23 - n(X∩Y)
n(X∩Y) = 40-38 = 2

2. If X and Y are two sets such that X ∪ Y has 18 elements, X has 8 elements and Y has 15 elements. how many elements does X ∩ Y have?
It is given than,
n(X∪Y) = 18, n(X) = 8, n(Y) = 15
We know that,
n(X∪Y) = n(X) + n(Y) - n(X∩Y)
18 = 8 + 15 - n(X∩Y)
n(X∩Y) = 23-18 = 5

3. In a group of 400 people, 250 can speak Hindi and 200 can speak English. How many people can speak both Hindi and English?
Let H and E be set of people who speak Hindi and English respectively.
It is given that,
n(H∪E) = 400, n(H) = 250, n(E) = 200
We know that,
n(H∪E) = n(H) + n(E) - n(H∩E)
400 = 250 + 200 - n(H∩E)
n(H∩E) = 450 - 400 = 50
Thus, 50 people can speak both Hindi and English.

4. If S and T are two sets such that S has 21 elements, T has 32 elements, and S∩T has 11 elements, how many elements does S∪T have?
It is given that,
n(S∩T) = 11, n(S) = 21, n(T) = 32
We know that,
n(S∪T) = n(S) + n(T) - n(S∩T)
n(S∪T) = 21 + 32 - 11
n(S∪T) = 42

5. If X and Y are two sets such that X has 40 elements, X ∪ Y has 60 elements and X ∩ Y has 10 elements, how many elements does Y have?
It is given that,
n(X) = 40,  n(X∪Y) = 60, N(X∩Y)=10
We know that,
n(X∪Y) = n(X) + n(Y) - n(X∩Y)
60 = 40 + n(Y) -10
n(Y) = 60-30 =30

6. In a group of 70 people, 37 like coffee, 52 like tea and each person likes at least one of the two drinks. How many people like both coffee and tea? 
Let C and T be set of people who likes coffee and tea respectively.
It is given that,
n(C∪T) = 70, n(C) = 37, n(T) = 52
We know that,
n(C∪T) = n(C) + n(T) - n(C∩T)
70 = 37 + 52 - n(C∩T)
n(C∩T) = 89 - 70 = 19

7. In a group of 65 people, 40 like cricket, 10 like both cricket and tennis. How many like tennis only and not cricket? How many like tennis?
Let C and T be set of people who likes cricket and tennis respectively.
It is given that,
n(C∪T) = 65, n(C) = 40, n(C∩T) =10
We know that,
n(C∪T) = n(C) + n(T) - n(C∩T)
65 = 40 + n(T) - 10
n(T) = 65 - 30 = 35
∴  35 people like tennis.
Number of people who like tennis only and not cricket = n(T - C)
n(T) = n(T - C) + n (T∩C)
35 = n(T - C) + 10
n(T - C) = 25
∴ 25 people like only tennis and not cricket.

8. In a committee, 50 people speak French, 20 speak Spanish and 10 speak both Spanish and French. How many speak at least one of these two languages?
Let S and F be set of people who speak Spanish and French respectively.
It is given that,
n(S∩F) = 10, n(S) = 20, n(F) = 50
We know that,
n(S∪F) = n(S) + n(F) - n(S∩F)
n(S∪F) = 20 + 50 - 10
n(S∪F) = 60