11th NCERT Sets Exercise 1.6 Questions 8

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Question (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).

Solution

n( X ∪ Y ) = n(X) + n(Y) - n( X ∩ Y)
38 = 17 + 23 - - n( X ∩ Y)
n( X ∩ Y) = 40 - 38
n( X ∩ Y) = 2

Question (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?

Solution

n( X ∪ Y ) = n(X) + n(Y) - n( X ∩ Y)
18 = 8 + 15 - - n( X ∩ Y)
n( X ∩ Y) = 23 - 18
n( X ∩ Y) =5

Question (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?

Solution

H : person speak Hindi. n(H) = 250,
E : person speak English . n(E) = 200,
n( H ∪ E ) = 400, Find n( H ∩ E).
n( H ∪ E ) = n(H) + n(E) - n( H ∩ E)
400 = 250 + 200 - n( H ∩ E)
n(H ∩E) = 450 - 400
n( H ∩ E) = 50
So 50 people can speak both Hindi and English.

Question (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?

Solution

n(S) = 21, n(T) = 32, n(s ∩ T) = 11.
n(S ∪ T ) = n(S) + n(T) - n( S ∩ T)
= 21 + 32 - 11
= 42.

Question (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?

Solution

n(X) = 40, n(X ∪ Y) = 60, n(X ∩ Y) = 10. find n(Y).
n( X ∪ Y ) = n(X) + n(Y) - n( X ∩ Y)
60 = 40 +n(Y) - 10
n( Y) = 60 - 30
n( Y) = 30
Y has 30 elements.

Question (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?

Solution

C ; person like coffee, n(C) = 37
T : person likes tea. n(T) = 52
Since each person likes at least one of the two drinks.and there are 70 people in a group. n(C ∪ T ) = 70.
Find n(C ∩ T).
n(C ∪ T ) = n(C) + n(T) - n( C ∩ T)
70 = 37 + 52 - n( C ∩ T)
70 = 89 n( C ∩ T)
n( C ∩ T) = 89 - 70
= 19
19 persons like both coffee and tea.

Question (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?

Solution

C : people likes Cricket. , n(C) = 40.
T : people likes tennis., n(T) = ?
n(C ∪ T)= 65, n( C ∩ T ) = 10
n(C ∪ T ) = n(C) + n(T) - n( C ∩ T)
65 = 40 + n(T) - 10
65 = 30+ n( T)
n(T) = 65 - 30
= 35
So 35 likes tennis. number of person like tennis not cricket = n(T ∩ C')
= n(T) - n( T ∩ C)
= 35 - 10
= 25.
So 25 persons like only tennis not cricket. and 35 likes tennis.

Question (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?

Solution

Let F : person speaks French , n(F) = 50
S : person speaks Spanish, n(S) = 20
n( F ∩ S ) = 10, Find n( F ∪ S ).
n( F ∪ S ) = n(F) + n(S) - n( F ∩ S)
= 50 + 20 - 10
= 60
60 people speaks atleast one of these two languages.