Question-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'
(ii) B'
(iii) (A ∪ C)'
(iv) (A ∪ B)'
(v) (A')'
(vi) (B - C)'
Given that 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
A' = {1, 2, 3, 4, 5, 6, 7, 8, 9} - {1, 2, 3, 4}
A' = {5, 6, 7, 8, 9}
(ii) B' = U - B
B' = {1, 2, 3, 4, 5, 6, 7, 8, 9} - {2, 4, 6, 8}
B' = {1, 3, 5, 7, 9}
(iii) (A ∪ C)' = U - (A ∪ C)
A ∪ C = {1, 2, 3, 4} ∪ {3, 4, 5, 6}
A ∪ C = {1, 2, 3, 4, 5, 6}
(A ∪ C)' = {1, 2, 3, 4, 5, 6, 7, 8, 9} - {1, 2, 3, 4, 5, 6}
(A ∪ C)' = {7, 8, 9}
(iv) (A ∪ B)' = U - (A ∪ B)
A ∪ B = {1, 2, 3, 4} ∪ {2, 4, 6, 8}
A ∪ B = {1, 2, 3, 4, 6, 8}
(A ∪ B)' = {1, 2, 3, 4, 5, 6, 7, 8, 9} - {1, 2, 3, 4, 6, 8}
(A ∪ B)' = {5, 7, 9}
(v) (A')' = U - A'
A' = U - A
A' = {1, 2, 3, 4, 5, 6, 7, 8, 9} - {1, 2, 3, 4}
A' = {5, 6, 7, 8, 9}
(A')' = {1, 2, 3, 4, 5, 6, 7, 8, 9} - {5, 6, 7, 8, 9}
(A')' = {1, 2, 3, 4} = A
(vi) (B - C)' = U - (B - C)
B - C = {2, 4, 6, 8} - {3, 4, 5, 6}
B - C = {2, 8}
(B - C)' = {1, 2, 3, 4, 5, 6, 7, 8, 9} - {2, 8}
(B - C)' = {1, 3, 4, 5, 6, 7, 9}
Question-2 :- If U = {a, b, c, d, e, f, g, h}, find the complements of the following sets:
(i) A = {a, b, c}
(ii) B = {d, e, f, g}
(iii) C = {a, c, e, g}
(iv) D = {f, g, h, a}
Given that U = {a, b, c, d, e, f, g, h}.
(i) A = {a, b, c}
A' = U - A
A' = {a, b, c, d, e, f, g, h} - {a, b, c}
A' = {d, e, f, g, h}
(ii) B = {d, e, f, g}
B' = U - B
B' = {a, b, c, d, e, f, g, h} - {d, e, f, g}
B' = {a, b, c, h}
(iii) C = {a, c, e, g}
C' = U - C
C' = {a, b, c, d, e, f, g, h} - {a, c, e, g}
C' = {b, d, f, h}
(iv) D = {f, g, h, a}
D' = U - D
D' = {a, b, c, d, e, f, g, h} - {f, g, h a}
D' = {b, c, d, e}
Question-4 :- 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}
(ii) {x: x is an odd natural number}
(iii) {x: x is a positive multiple of 3}
(iv) {x: x is a prime number}
(v) {x: x is a natural number divisible by 3 and 5}
(vi) {x: x is a perfect square}
(vii) {x: x is perfect cube}
(viii) {x: x + 5 = 8}
(ix) {x: 2x + 5 = 9}
(x) {x: x ≥ 7}
(xi) {x: x ∈ N and 2x + 1 > 10}
Given that U = {x : x is set of natural numbers} = {1, 2, 3,....}
(i) A = {x: x is an even natural number}
A = {2, 4, 6,....}
A' = U - A
A' = {1, 2, 3,....} - {2, 4, 6,....}
A' = {1, 3, 5....}
A' = {x : x is set of odd natural numbers, x ∈ N}
(ii) B = {x: x is an odd natural number}
B = {1, 3, 5,....}
B' = U - B
B' = {1, 2, 3,....} - {1, 3, 5,....}
B' = {2, 4, 6....}
B' = {x : x is set of even natural numbers, x ∈ N}
(iii) C = {x: x is a positive multiple of 3}
C = {3, 6, 9,....}
C' = U - C
C' = {1, 2, 3,....} - {3, 6, 9,....}
C' = {2, 4, 5....}
C' = {x : x is not a positive multiple of 3, x ∈ N}
(iv) D = {x: x is a prime number}
D = {2, 3, 5,....}
D' = U - D
D' = {1, 2, 3,....} - {2, 3, 5,....}
D' = {1, 4, 6....}
D' = {x : x is not a prime number, x ∈ N}
(v) E = {x: x is a natural number divisible by 3 and 5}
E = {15, 30, 45,....}
E' = U - E
E' = {1, 2, 3,....} - {15, 30, 45,....}
E' = {1, 3, 5....}
E' = {x : x is not a natural number divisible by 3 and 5, x ∈ N}
(vi) F = {x: x is a perfect square}
F = {1, 4, 9,....}
F' = U - F
F' = {1, 2, 3,....} - {1, 4, 9,....}
F' = {2, 3, 5....}
F' = {x : x is not a perfect square, x ∈ N}
(vii) G = {x: x is perfect cube}
G = {1, 8, 27,....}
G' = U - G
G' = {1, 2, 3,....} - {1, 8, 27,....}
G' = {2, 3, 4....}
G' = {x : x is not a perfect cube, x ∈ N}
(viii) H = {x: x + 5 = 8}
x + 5 = 8
x = 8 - 5
x = 3
H = {3}
H' = U - H
H' = {1, 2, 3,....} - {3}
H' = {1, 2, 4....}
H' = {x : x ≠ 3, x ∈ N}
(ix) I = {x: 2x + 5 = 9}
2x + 5 = 9
2x = 9 - 5
2x = 4
x = 4/2
x = 2
I = {2}
I' = U - I
I' = {1, 2, 3,....} - {2}
I' = {1, 3, 4,....}
I' = {x : x ≠ 2, x ∈ N}
(x) J = {x: x ≥ 7}
J = {7, 8, 9,....}
J' = U - J
J' = {1, 2, 3,....} - {7, 8, 9,....}
J' = {1, 2, 3, 4, 5, 6}
J' = {x : x < 7, x ∈ N}
(xi) K = {x: x ∈ N and 2x + 1 > 10}
K = {5, 6, 7,....}
K' = U - K
K' = {1, 2, 3,....} - {5, 6, 7,....}
k' = {1, 2, 3, 4}
K' = {x: x ∈ N and x ≤ 9/2}
Question-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'
(ii) (A ∩ B)' = A' ∪ B'
Given that 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)
A ∪ B = {2, 4, 6, 8} ∪ {2, 3, 5, 7}
A ∪ B = {2, 3, 4, 5, 6, 7, 8}
(A ∪ B)' = {1, 2, 3, 4, 5, 6, 7, 8, 9} - {2, 3, 4, 5, 6, 7, 8}
(A ∪ B)' = {1, 9}
R.H.S
A'∩ B' = (U - A) ∩ (U - B)
A' = U - A
A' = {1, 2, 3, 4, 5, 6, 7, 8, 9} - {2, 4, 6, 8}
A' = {1, 3, 5, 7, 9}
B' = U - B
B' = {1, 2, 3, 4, 5, 6, 7, 8, 9} - {2, 3, 5, 7}
B' = {1, 4, 6, 8, 9}
A'∩ B' = {1, 3, 5, 7, 9} ∩ {1, 4, 6, 8, 9}
A'∩ B' = {1, 9}
L.H.S = R.H.S
(ii) (A ∩ B)' = A' ∪ B'
L.H.S
(A ∩ B)' = U - (A ∩ B)
A ∩ B = {2, 4, 6, 8} ∩ {2, 3, 5, 7}
A ∩ B = {2}
(A ∩ B)' = {1, 2, 3, 4, 5, 6, 7, 8, 9} - {2}
(A ∩ B)' = {1, 3, 4, 5, 6, 7, 8, 9}
R.H.S
A'∪ B' = (U - A) ∪ (U - B)
A' = U - A
A' = {1, 2, 3, 4, 5, 6, 7, 8, 9} - {2, 4, 6, 8}
A' = {1, 3, 5, 7, 9}
B' = U - B
B' = {1, 2, 3, 4, 5, 6, 7, 8, 9} - {2, 3, 5, 7}
B' = {1, 4, 6, 8, 9}
A'∪ B' = {1, 3, 5, 7, 9} ∩ {1, 4, 6, 8, 9}
A'∪ B' = {1, 3, 4, 5, 6, 7, 8, 9}
L.H.S = R.H.S
Question-5 :- Draw appropriate Venn diagram for each of the following:
(i) (A ∪ B)'
(ii) A'∩ B'
(iii) (A ∩ B)'
(iv) A' ∪ B'
(i) (A ∪ B)'
(ii) A'∩ B'
(iii) (A ∩ B)'
(iv) A' ∪ B'
Question-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 ?
Solution :-
Given that U = {x : x is a triangle}.
A = {x : x is a triangle and has at least one angle different from 60°}
A' = U - A
A' = {x : x is a triangle} - {x : x is a triangle and has at least one angle different from 60°}
A' = {x : x is a triangle and has all angles equal to 60°}
A' = set of all equilateral triangles
Question-7 :- Fill in the blanks to make each of the following a true statement:
(i) A ∪ A'= ...
(ii) Φ′∩ A = ...
(iii) A ∩ A' = ...
(iv) U'∩ A' = ...
(i) A ∪ A'= U (ii) Φ′∩ A = U ∩ A = A (iii) A ∩ A' = Φ (iv) U'∩ A' = Φ ∩ A = Φ
