Mathematics
If A = {5, 6, 7, 8, 9}, B ={x : 3 < x < 8 and x ∈ W} and C = {x : x ≤ 5 and x ∈ N}. Find :
(i) A ∪ B and (A ∪ B) ∪ C
(ii) B ∪ C and A ∪ (B ∪ C)
(iii) A ∩ B and (A ∩ B) ∩ C
(iv) B ∩ C and A ∩ (B ∩ C)
Is (A ∪ B) u C = A ∪ (B ∪ C) ?
Is (A ∩ B) ∩ C = A ∩ (B ∩ C) ?
Sets
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Answer
A = {5, 6, 7, 8, 9}
B ={x : 3 < x < 8 and x ∈ W}
B = {4, 5, 6, 7}
C = {x : x ≤ 5 and x ∈ N}
C = {1, 2, 3, 4, 5}
(i) A ∪ B - contains all elements in both set A and B.
A ∪ B = {5, 6, 7, 8, 9} ∪ {4, 5, 6, 7}
A ∪ B = {4, 5, 6, 7, 8, 9}
(A ∪ B) ∪ C - contains all elements in set A, B and C.
(A ∪ B) ∪ C = {4, 5, 6, 7, 8, 9} ∪ {1, 2, 3, 4, 5}
(A ∪ B) ∪ C = {1, 2, 3, 4, 5, 6, 7, 8, 9}
(ii) B ∪ C - contains all elements in both set B and C.
B ∪ C = {4, 5, 6, 7} ∪ {1, 2, 3, 4, 5}
B ∪ C = {1, 2, 3, 4, 5, 6, 7}
A ∪ (B ∪ C) - contains all elements in set A, B and C.
A ∪ (B ∪ C) = {5, 6, 7, 8, 9} ∪ {1, 2, 3, 4, 5, 6, 7}
A ∪ (B ∪ C) = {1, 2, 3, 4, 5, 6, 7, 8, 9}
(iii) A ∩ B - contains common elements in both set A and B.
A ∩ B = {5, 6, 7, 8, 9} ∩ {4, 5, 6, 7}
A ∩ B = {5, 6, 7}
(A ∩ B) ∩ C - contains common elements in set A, B and C.
(A ∩ B) ∩ C = {5, 6, 7} ∩ {1, 2, 3, 4, 5}
(A ∩ B) ∩ C = {5}
(iv) B ∩ C - contains common elements in both set B and C.
B ∩ C = {4, 5, 6, 7} ∩ {1, 2, 3, 4, 5}
B ∩ C = {4, 5}
A ∩ (B ∩ C) - contains common elements in set A, B and C.
A ∩ (B ∩ C) = {5, 6, 7, 8, 9} ∩ {4, 5}
A ∩ (B ∩ C) = {5}
As we have seen in (i) and (ii), all elements in (A ∪ B) u C and A ∪ (B ∪ C) are same.
(A ∪ B) u C = A ∪ (B ∪ C)
As we have seen in (iii) and (iv), all elements in (A ∩ B) ∩ C and A ∩ (B ∩ C) are same.
(A ∩ B) ∩ C = A ∩ (B ∩ C)
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