Sets - NCERT Exercise 1.6 Answers
Class 11 Maths
Answer: Given,
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?
Answer: Given,
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?
Answer: Given
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?
Answer: Given,
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?
Answer:
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.
☞See also:
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
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