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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Job & Exam Mathematics Rare (JEMrare)

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Exercise 1.3
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Exercise 1.2  ⇚ 7th Mathematics Main Manu ⇛  Exercise 1.4