1. Let 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 }. Find
(i) A'
A′ = U-A = {5,6,7,8,9}

(ii) B′
B′ = U-B = {1,3,5,7,9}

(iii) (A ∪ C)′
(A∪C) = {1,2,3,4,5,6}
(A∪C)' = U - (A∪C) = {7,8,9}

(iv) (A ∪ B)′
(A∪B) = {1,2,3,4,6,8}
(A∪B)' = U - (A∪B) = {5,7,9}

(v) (A′)′
A' = {5,6,7,8,9}
(A')' = U - A' = {1,2,3,4}

(vi) (B – C)′
B-C = {2,8}
(B-C)' = U - (B-C) = {1,3,4,5,6,7,9}

2. If  U = { a, b, c, d, e, f, g, h}, find the complements of the following sets :
(i) A = {a, b, c}
A' = {d,e,f,g,h}

(ii) B = {d, e, f, g}
B' = {a,b,c,h}

(iii) C = {a, c, e, g}
C' = {b,d,f,h}

(iv) D = { f, g, h, a}
D' = {b,c,d,e}

3. Taking the set of natural numbers as the universal set, write down the complements of the following sets:
(i) {x : x is an even natural number} = {x: x is an odd natural number}
(ii) { x : x is an odd natural number } = {x: x is an even natural number}
(iii) {x : x is a positive multiple of 3} = {x: x∈N, and x is not a multiple of 3}
(iv) { x : x is a prime number } = {x: x is a positive composite number and x=1}
(v) {x : x is a natural number divisible by 3 and 5} = {x: x∈N and x is not divisible by 3 and 5}
(vi) { x : x is a perfect square } = {x: x∈N and x is not a perfect square}
(vii) { x : x is a perfect cube} = {x: x∈N and x is not a perfect cube}
(viii) { x : x + 5 = 8 } = {x: x∈N and x≠3}
(ix) { x : 2x + 5 = 9} = {x: x∈N and x≠2}
(x) { x : x ≥ 7 } = {x: x∈N and x<7}
(xi) { x : x ∈ N and 2x + 1 > 10 } = {x: x∈N and x ≤ (9/2)}

 4. If U = {1, 2, 3, 4, 5, 6, 7, 8, 9 }, A = {2, 4, 6, 8} and B = { 2, 3, 5, 7}. Verify that
(i) (A ∪ B)′ = A′ ∩ B′ 
(A∪B)' = {2,3,4,5,6,7,8}' = {1,9}
A'∩B' = {1,3,5,7,9}∩{1,4,6,8,9} = {1,9}
∴ (A∪B)' = A'∩B'

(ii) (A ∩ B)′ = A′ ∪ B′
(A∩B)' = {2}' = {1,3,4,5,6,7,8,9}
A'∪B' = {1,3,5,7,9} ∪ {1,4,6,8,9} = {1,3,4,5,6,7,8,9}
∴ (A∩B)' = A'∪B'

5. Draw appropriate Venn diagram for each of the following :
(i) (A ∪ B)′, (ii) A′ ∩ B′, (iii) (A ∩ B)′, (iv) A′ ∪ B′


6. Let U be the set of all triangles in a plane. If A is the set of all triangles with at least one angle different from 60°, what is A′ ?
A is set of all triangles in which, at least one angle is not 60°.
∴ A' is set all equilateral triangles.

7. Fill in the blanks to make each of the following a true statement :
(i) A ∪ A′ = U
(ii) φ′ ∩ A = A
(iii) A ∩ A′ = ϕ
(iv) U′ ∩ A = ϕ