Q30 of 58 Page 12

Two finite sets have m and n elements. The number of subsets of the first set is 112 more than that of the second set. The values of m and n are, respectively,

Given: Two finite sets have m and n elements.


To find: value of m and n


Formula used:


The number of subsets of a set containing x elements is given by 2x


According to question:


2m – 2n = 112


⇒ 2n (2m-n – 1) = 16 × 7


⇒ 2n (2m-n – 1) = 24 × 7


On comparing:


2n = 24 and 2m-n – 1 = 7


⇒ n = 4 and 2m-n = 8


⇒ 2m-n = 23


⇒ m – n = 3


⇒ m – 4 = 3


⇒ m = 7


Hence, value of m and n is 7 and 4 respectively

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28

In a group of 50 students, the number of students studying French, English, Sanskrit were found to be as follows:

French = 17, English = 13, Sanskrit = 15


French and English = 09, English and Sanskrit = 4


French and Sanskrit = 5, English, French and Sanskrit = 3. Find the number of students who study


(i) French only


(ii) English only


(iii) Sanskrit only


(iv) English and Sanskrit but not French


(v) French and Sanskrit but not English


(vi) French and English but not Sanskrit


(vii) at least one of the three languages


(viii) none of the three languages

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Suppose A1, A2, ..., A30 are thirty sets each having 5 elements and B1, B2, ..., Bn are n sets each with 3 elements, let union _ i = 1 ^30a_i = union _ j = 1 ^nb_j = s and each element of S belongs to exactly 10 of the Ai’s and exactly 9 of the B,’S. then n is equal to

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The set (A ∩ B′)′ ∪ (B ∩ C) is equal to

 32

Let F1 be the set of parallelograms, F2 the set of rectangles, F3 the set of rhombuses, F4 the set of squares and F5 the set of trapeziums in a plane. Then F1 may be equal to