- 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) (AUC)’
(iv) (AUB)’
(v)(A’)’
(vi)(B-C)’
Solution:
U ={1, 2, 3, 4, 5, 6, 7, 8, 9}
A = {1, 2, 3, 4}
B = {2, 4, 6, 8}
C = {3, 4, 5, 6}
(i) A’ = {5, 6, 7, 8, 9}
(ii)B’ = {1, 3, 5, 7, 9}
(iii) AUC = { 1, 2, 3, 4, 5, 6}
Thus, (AUC)’ = {7, 8, 9}
(iv) AUB = {1, 2, 3, 4, 6, 8}
Thus, (AUB)’ = {5, 7, 9}
(v) (A’)’ = A = {1, 2, 3, 4}
(vi) B – C = {2, 8}
Thus, (B – C)’ = {1, 3, 4, 5, 6, 7, 9}
- 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:
U = {a, b, c, d, e, f, g, h}
(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}
- 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}
Solution:
U = N: Set of natural numbers
(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 is a natural number that is not divisible by 3 or 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}
- 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) (AUB)’ = A’ ∩ B’
(ii) (A∩B)’ = A’ U B’
Solution:
U = {1, 2, 3, 4, 5, 6, 7, 8, 9} A = {2, 4, 6, 8}, B = {2, 3, 5, 7}
(i) (AUB) = {2, 3, 4, 5, 6, 7, 8}
(AUB)’ = {1, 9}
A’ = {1, 3, 5, 7, 9} ; B’ = {1, 4, 6, 8, 9}
A’ ∩ B’ = { 1, 9}
Thus, (AUB)’ = A’ ∩ B’
(ii) A∩B = {2}
(A∩B)’ = {1, 3, 4, 5, 6, 7, 8, 9}
A’ = {1, 3, 5, 7, 9} ; B’ = {1, 4, 6, 8, 9}
A’ U B’ = {1, 3, 4, 5, 6, 7, 8, 9}
(A∩B)’ = A’ U B’
- Draw appropriate Venn diagram for each of the following:
(i) (AUB)’
(II) A’∩ B’
(iii) (A∩B)’
(iv) A’UB’
Solution:
(i) (AUB)’
(II) A’∩ B’
(iii) (A∩B)’
(iv) A’UB’
- 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:
A’ is the set of all equilateral triangles.
7: Fill in the blanks to make each of the following a true statement:
(i) AUA’
(ii) Φ’∩ A =
(iii) A∩A’=
(iv) U’∩A =
Solution:
(i) AUA’ = U
(ii) Φ’∩ A = U∩A = A
Therefore, Φ’∩ A = A
(iii) A∩A’= Φ
(iv) U’∩A = Φ∩ A = Φ
Therefore U’∩ A = Φ
