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)’
Solution.
Here 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
= {1, 2, 3, 4, 5, 6, 7, 8, 9} – {1, 2, 3, 4}
= {5, 6, 7, 8, 9}
(ii) B’=U – B
= {1, 2, 3, 4, 5, 6, 7, 8, 9} – {2, 4, 6, 8}
= {1, 3, 5, 7, 9}
(iii) A ∪ C = {1, 2, 3, 4} ∪ {3, 4, 5, 6}
= (1, 2, 3, 4, 5, 6}
(A∪C)’=U-(A∪C)
= {1, 2, 3, 4, 5, 6, 7, 8, 9} – {1, 2, 3, 4, 5, 6}
= {7, 8, 9}
(iv) A ∪ B = {1, 2, 3,4} ∪ {2, 4, 6, 8}
= {1, 2, 3, 4, 6, 8}
(A∪B)’ = U – (A∪B)
= {1, 2, 3, 4, 5, 6, 7, 8, 9} – {1, 2, 3, 4, 6, 8}
= {5, 7, 9}
(v) We know that A’ = {5, 6, 7, 8, 9}
(A’)’ =U – A’
= {1, 2, 3, 4, 5, 6, 7, 8, 9} – {5, 6, 7, 8, 9}
= {1, 2, 3, 4}
(vi) B – C = {2, 4, 6, 8} – {3, 4, 5, 6} = {2, 8}
(B-C)’=U – (B-C)
= {1, 2, 3, 4, 5, 6, 7, 8, 9} – {2, 8}
= {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}
Solution.
(i) A’ = U – A = {a, b, c, d, e, f, g, h} – {a, b, c}
= {d, e,f, g, h}
(ii) B’ = U – B = {a, b, c, d, e,f, g, h} – {d, e, f, g}
= {a, b, c, h}
(iii) C’ = U – C = {a, b, c, d, e, f, g, h} – {a, c, e, g}
= {b, d, f, h}
(iv) D’ = U – D = {a, b, c, d, e, f, g, h} – {f, g, h, a}
= {b, c, d, e}.
Question 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}
- (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 a perfect cube}
- (viii) {x: x + 5 = 8}
- (ix) (x: 2x + 5 = 9)
- (x) {x: x ≥ 7}
- (xi) {x: x ∈ W and 2x + 1 > 10}
Solution.
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’
Solution.
Question 5. Draw appropriate Venn diagram for each of the following:
- (i) (A ∪ B)’
- (ii) A’∩B’
- (iii) (A ∩ B)’
- (iv) A’ ∪ B’
Solution.
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 A’?
Solution.
Here U = {x : x is a triangle}
A = {x: x is a triangle and has at least one angle different from 60°}
∴ A’ = U – A = {x : x is a triangle} – {x : x is a triangle and has atleast one angle different from 60°}
= {x : x is a triangle and has all angles equal to 60°)
= 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 = .…….
Solution.
(i) A ∪ A’= U
(ii) φ’ ∩ A = U ∩ A = A
(iii) A ∩ A’ = φ
(iv) U’ ∩ A = φ ∩ A = φ