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D Divya Prakash Singh
A' is the set of all triangles whose angle is  in other words A' is set of all equilateral triangles.

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S seema garhwal
  Let      A ⊂ B    and     B ⊂ C There be a element x such that     Let,   x  A          x   B            ( Because A  B )         x  C              ( Because B  C ) Hence,the statement is true that   A  C  

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D Divya Prakash Singh
(i) (A ∪ B)′ (A ∪ B) is in yellow colour (A ∪ B)′ is in green colour ii) A′ ∩ B′ is represented by the green colour in the below figure   iii) (A ∩ B)′ is represented by green colour in the below diagram and white colour represents (A ∩ B) iv)  A′ ∪ B′ The green colour represents   A′ ∪ B′    

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S seema garhwal
The given statement is false, Let ,  A = {1}          B = { 1,2,3}          C = {0,{1,2,3},4} Here, {1}   { 1,2,3} =  A  B and { 1,2,3}  {0,{1,2,3},4} = B  C But,  {1}  {0,{1,2,3},4} = A  C

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D Divya Prakash Singh
U = {1, 2, 3, 4, 5, 6, 7, 8, 9 }, A = {2, 4, 6, 8} and B = { 2, 3, 5, 7} (i) (A  B)′ = A′ B′ L.H.S = (A  B)′ = U - (A  B) = {1,9} R.H.S =  A′ B′ = {1,3,5,7,9}  {1,4,6,8,9} = {1,9} L.H.S = R.H.S Hence,the statement is true. (ii) (A  B)′ = A′  B′   L.H.S = U - (A  B) ={1,3,4,5,6,7,8,9}  R.H.S =  A′  B′ = {1,3,5,7,9}  {1,4,6,8,9} = {1,3,4,5,6,7,8,9}   L.H.S = R.H.S Hence,the statement is true.

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S seema garhwal
The given statment is false, example: Let A = { 2,4 }                      B = { 1,{2,4},5}                      x be 2.           then, 2    { 2,4 } = x  A and { 2,4 }  { 1,{2,4},5} = A  B               But 2  { 1,{2,4},5}  i.e. x  B

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S seema garhwal
Solution of this equation are  – 8x + 12 = 0                                              ( x - 2 ) ( x - 6 ) = 0                                                    X = 2,6                                            A = { 2,6 }                                               B = { 2, 4, 6 }                                               C = { 2, 4, 6, 8, . . . }                                    ...

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D Divya Prakash Singh
Universal set = U = {1,2,3,4,5,6,7,8.............} (vi) { x : x is a perfect square }' = U - { x : x is a perfect square } =  { x : x  N and x is not a perfect square } (vii) { x : x is a perfect cube}' = U - { x : x is a perfect cube} =  { x : x  N and x is not a perfect cube} (viii) { x : x + 5 8 }' = U - { x : x + 5 8 } = U - {3} = { x : x N and x  3 } (ix) { x : 2x + 5 9}' = U - { x : 2x +...

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S seema garhwal
n(french) = 50 n(spanish) = 20 n(speak both french and spanish) = 10 n(speak at least one of these two languages) = n(french) + n(spanish) - n(speak both french and spanish) n(speak at least one of these two languages) = 50  + 20 - 10 n(speak at least one of these two languages) = 70 -10 n(speak at least one of these two languages) = 60 Hence,60 people speak at least one of these two languages.

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S seema garhwal
Total people = 65 n(like  cricket) = 40 n(like both cricket and tennis) = 10 n(like tennis) = ? Total people = n(like  cricket)   +   n(like tennis)  -  n(like both cricket and tennis)  n(like tennis) = 65 - 40 + 10  n(like tennis) = 35 Hence,35 people like tennis. n (people like only tennis) =  n(like tennis) - n(like both cricket and tennis) n (people like only tennis) = 35 - 10 n (people...

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S seema garhwal
n(people like coffee) = 37  n(people like tea ) = 52 Total number of people = 70  Total number of people = n(people like coffee)  +   n(people like tea )  - n(people like both tea and coffee)                              70  =  37  +   52 - n(people like both tea and coffee) n(people like both tea and coffee)  =  89 - 70  n(people like both tea and coffee)  =  19  Hence,19 people like both...

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D Divya Prakash Singh
U = { a, b, c, d, e, f, g, h} (i) A = {a, b, c}      A' = U - A = {d,e,f,g,h} (ii) B = {d, e, f, g}      B' = U - B = {a,b,c,h} (iii) C = {a, c, e, g}       C' = U - C = {b,d,f,h} (iv) D = { f, g, h, a}        D' = U - D = {b,c,d,e}  

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S seema garhwal
n( X  Y) = 60 n( X ∩ Y) = 10 n(X) = 40 n( X  Y) =  n(X)  +  n(Y)   -  n( X ∩ Y)        n(Y) = 60 - 40 + 10        n(Y) = 30 Hence,set Y has 30 elements.

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S seema garhwal
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 n(S T) = 53 - 11 n(S  T)  = 42 Hence, the set  (S  T)  has 42 elements.    

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D Divya Prakash Singh
 U = { 1, 2, 3, 4, 5, 6, 7, 8, 9 }, A = { 1, 2, 3, 4}, B = { 2, 4, 6, 8 } and C = { 3, 4, 5, 6 } (i) A′ = U - A = {5,6,7,8,9} (ii) B′ = U - B = {1,3,5,7,9} (iii) A  C = {1,2,3,4,5,6}      (A  C)′ = U - (A  C) = {7,8,9} (iv) (A  B) = {1,2,3,4,6,8}       (A  B)′ = U - (A  B) = {5,7,9} (v) (A')' = A = { 1, 2, 3, 4} (vi) (B – C) = {2,8}        (B – C)' = U - (B – C) = {1,3,4,5,6,7,9}    

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S seema garhwal
Hindi speaking = 250 = n(A) English speakind = 200 = n(B) Group of people = 400 = n(A  B) People speaking both Hindi and English = n(A  B) n(A  B) = n(A) + n(B) - n(A  B) n(A  B) = 250 + 200 - 400 n(A  B) = 450 - 400 n(A  B)  = 50 Thus,50 people speak hindi and english both.  

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S seema garhwal
n( X  Y ) = 18 n(X) = 8 n(Y) = 15 n( X  Y ) =  n(X) + n(Y) - n(X  Y) n(X  Y) = 8 + 15 - 18  n(X  Y) = 5  

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S seema garhwal
n( X ) = 17,       n( Y ) = 23             and  n( X  Y ) = 38 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.
Here,B = {3, 4, 5, 6},          C = {5, 6, 7, 8 } and           D = { 7, 8, 9, 10 } The union can be written as   B  C  D = {3,4,5,6,7,8,9,10}
Here, A = {1, 2, 3, 4}, B = {3, 4, 5, 6} D = { 7, 8, 9, 10 } The union can be written as A  B  D = {1,2,3,4,5,6,7,8,9,10}  
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