A' is the set of all triangles whose angle is in other words A' is set of all equilateral triangles.

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

(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′

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

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.

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

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, . . . }
...

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 +...

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.

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...

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...

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}

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.

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.

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}

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.

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

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
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}

Q.4 If A = {1, 2, 3, 4}, B = {3, 4, 5, 6}, C = {5, 6, 7, 8 }and D = { 7, 8, 9, 10 }; find

(VI) A B D

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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