## Sets - NCERT Exercise 1.6 Answers

Class 11 Maths

Q1: If X and Y are two sets such that n( X ) = 17, n( Y ) = 23 and n( X ∪ Y ) = 38, find n ( X ∩ Y ).

n( X ) = 17,
n( Y ) = 23 and
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) = 17 + 23 - 38 = 40 - 38 = 2
n (X ∩ Y) = 2.

Q2: 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?

n( X ) = 8,
n( Y ) = 15 and
n( X ∪ Y ) = 18
∵ n(X ∪ Y) = n(X)  + n(Y) - n (X ∩ Y)
∴  18 = 8 + 15  - n (X ∩ Y)
n (X ∩ Y) = 8 + 15 - 18 = 23 - 18 = 5
n (X ∩ Y) = 5.

Q3: In a group of 400 people, 250 can speak Hindi and 200 can speak English. How many people can speak both Hindi and English?

Answer: Let H be the set of people who speak Hindi, and E be the set of people who speak English.
∴ n(H) = 250
n(E)  = 200
n( H ∪ E ) =400
n (H ∩ E) = ?

∵  n(H ∪ E) = n(H)  + n(E) - n (H ∩ E)
∴  400 = 250 + 400 - n(H ∩ E)
∴   n(H ∩ E) = 450 - 400 = 50.
Thus 50 people can speak Hindi and English both.

Q4: 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?

n(S) = 21,
n(T) = 32
n(S ∩ T) = 11
∵  n(S ∪ T) = n(S)  + n(T) - n (S ∩ T)
∴  n(S ∪ T) = 21 + 32 - 11 = 42            (answer)

Q5: 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?

n(X) = 40,
n(Y) = ?
n(X ∪ Y) = 60
X ∩ Y = 10
∵ n(X ∪ Y) = n(X)  + n(Y) - n (X ∩ Y)
∴ 60 = 40 + n(Y) - 10
∴   n(Y) = 60 -30 = 30

Thus, the set Y has 30 elements.

Q6: 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?

Answer: Let C denote the set of people who like coffee, and T denote the set of people who like tea.
∴ n(C) = 37, n(T) = 52 and n(C ∪ T ) = 70.

∵ n(C ∪ T) = n(C)  + n(T) - n (C ∩ T)

∴     70 = 37 + 52 - n (C ∩ T)

⇒     n (C ∩ T) = 89 - 70 = 19

Thus, 19 people like both coffee and tea.

Q7: 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?

Answer: Let C denote the set of people who like cricket, and T denote the set of people who like tennis
∴ n(C ∪ T) = 65, n(C) = 40, n(T) = ? and n(C ∩ T ) = 10.

∵ n(C ∪ T) = n(C)  + n(T) - n (C ∩ T)

∴  65 = 40 + n(T) - 10

⇒ n(T) = 65 - 30 = 35
Therefore, 35 people like tennis.

∵ n(T) = n(T - C)  + n(T ∩ C)
∴ n(T - C) = n(T) - n(T ∩ C)
= 35 - 10 = 25
Thus, 25 people like only tennis.

Q8: 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 F be the set of people in the committee who speak French, and S be the set of people in the committee who speak Spanish.

∴  n(F) = 50, n(S) = 20 and n(S ∩ F ) = 10.

∵ n(S ∪ F) = n(S)  + n(F) - n (S ∩ F)

∴  n(S ∪ F) = 20 + 50 - 10 = 70 - 10 = 60

Thus, 60 people in the committee speak at least one of the two languages.

Special Mathematical Constants
SETS (Unit Test Paper)
SETS (VENN DIAGRAMS)
SETS (Operations of Sets)
SETS (NCERT Ex 1.4 Q1-Q5)
SETS (NCERT Ex 1.4 Q6 - Q8)
SETS (NCERT Ex 1.4 Q9 - Q12)
SETS (NCERT Ex 1.5)
Laws of Set Operations  