EXERCISE 1.3
1. Look at each pair of sets to separate the disjoint and overlapping sets.
(i) A = {a, b, c, d, e}, B = {d, e, f, g, h}
(ii) L = {2, 4, 6, 8, 10}, M = {3, 6, 9, 12}
(iii) P = Set of Prime numbers, C = Set of Composite numbers
(iv) E = Set of Even numbers, O = Set of Odd numbers
Answers:
Disjoint Sets: Options (iii) and (iv) are disjoint sets
(Because there are some Common Elements in both sets.)
Overlapping Sets: Options (i) and (ii) are overlapping sets
(Because there are NO Common Elements in both sets.)
2. If U = {1, 2, 3, ...., 10}, A = {1, 2, 3, 4, 5}, B = {1, 3, 5, 7, 9}, C = {2, 4, 6,
8, 10} and D = { 3, 4, 5, 6, 7}, then find:
(i) A' (ii) B' (iii) C' (iv) D'
(i) A'
Solution:
U = {1, 2, 3, ...., 10}, A = {1, 2, 3, 4, 5}
A' = U \ A = {1, 2, 3, ...., 10} \ {1, 2, 3, 4, 5}
= { 6, 7, 8, 9, 10} .......Ans
(ii) B'
Solution:
U = {1, 2, 3, ...., 10}, B = {1, 3, 5, 7, 9}
B' = U \ B = {1, 2, 3, ...., 10} \ {1, 3, 5, 7, 9}
= { 2, 4, 6, 8, 10} .......Ans
(iii) C'
Solution:
U = {1, 2, 3, ...., 10}, C = {2, 4, 6, 8, 10}
C' = U \ C = {1, 2, 3, ...., 10} \ {2, 4, 6, 8, 10}
= { 1, 3, 5, 7, 9} .......Ans
(iv) D'
Solution:
U = {1, 2, 3, ...., 10}, D = { 3, 4, 5, 6, 7}
D' = U \ D = {1, 2, 3, ...., 10} \ { 3, 4, 5, 6, 7}
= { 1, 2, 8, 9, 10} .......Ans
3. If U = {a, b, c,...., i }, X = {a, c, e, g, i}, Y = {a, e, i}, and Z = {a, g, h}, then
find:
(i) X' (ii) Y' (iii) Z' (iv) U'
(i) X'
Solution:
U = {a, b, c,...., i }, X = {a, c, e, g, i}
A' = U \ A = {a, b, c,...., i } \ {a, c, e, g, i}
= {b, d, f, h } .......Ans
(ii) Y'
Solution:
U = {a, b, c,...., i }, Y = {a, e, i}
B' = U \ B = {a, b, c,...., i } \ {a, e, i}
= {b, c, d, f, g, h } .......Ans
(iii) Z'
Solution:
U = {a, b, c,...., i }, Z = {a, g, h}
C' = U \ C = {a, b, c,...., i } \ {a, g, h}
= {b, c, d, e, f, i } .......Ans
(iv) U'
Solution:
U = {a, b, c,...., i }
U' = U \ U = {a, b, c,...., i } \ {a, b, c,...., i }
= { } .......Ans
4. If U= {1, 2, 3, ..., 20}, A= {1, 3, 5, ... ,19} and B = {2, 4, 6, ... ,20}, then
prove that:
(i) B' = A (ii) A' = B (iii) A \ B = A (iv) B \ A = B
(i) B' = A
Solution:
U = {1, 2, 3, ..., 20 }, A= {1, 3, 5, ... ,19}, B = {2, 4, 6, ... ,20}
L.H.S = B' = U \ B
= {1, 2, 3, ..., 20 } \ {2, 4, 6, ... ,20}
= {1, 3, 5, ... ,19 }
= A = R.H.S
Hence Verified that
L.H.S = R.H.S
B' = A
(ii) A' = B
Solution:
U = {1, 2, 3, ..., 20 }, A= {1, 3, 5, ... ,19}, B = {2, 4, 6, ... ,20}
L.H.S = A' = U \ A
= {1, 2, 3, ..., 20 } \ {1, 3, 5, ... ,19}
= {2, 4, 6, ... ,20 }
= B = R.H.S
It is Verified that
L.H.S = R.H.S
A' = B
(iii) A \ B = A
Solution:
A= {1, 3, 5, ... ,19}, B = {2, 4, 6, ... ,20}
L.H.S = A \ B
= {1, 3, 5, ... ,19 } \ {2, 4, 6, ... ,20}
= {1, 3, 5, ... ,19 }
= A = R.H.S
It is Verified that
L.H.S = R.H.S
A \ B = A
(iv) B \ A = B
Solution:
A= {1, 3, 5, ... ,19}, B = {2, 4, 6, ... ,20}
L.H.S = B \ A
= {2, 4, 6, ... ,20} \ {1, 3, 5, ... ,19}
= {2, 4, 6, ... ,20}
= B = R.H.S
It is Verified that
L.H.S = R.H.S
B \ A = B
5. If U = set of integers and W = set of whole numbers, then find the
complement of set W.
Solution:
U = Set of integers = {0, ± 1, ± 2, ± 3,...},
W = set of whole numbers = {0, 1, 3, 5, .....}
W' = U \ W = {0, ± 1, ± 2, ± 3,...} \ {0, 1, 3, 5, .....}
= { -1, -2, -3,......} .......Ans
6. If U = set of natural numbers and P = set of prime numbers, then
find the complement of set P.
Solution:
U = Set of Natural Numbers = {1, 2, 3,.....},
P = set of Prime numbers = { 2, 3, 5, 7, 11, 13 .....}
P' = U \ P = {1, 2, 3,.....} \ { 2, 3, 5, 7, 11, 13 .....}
= { 1, 4, 6, 8, 9, 10, 12, 15 .....} .......Ans
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© 2022-23 Academic Skills and Knowledge (ASK)
Note: Write me in the comments box below for any query and also Share this information with your class-fellows and friends.
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