Answers:
(i)
LHS,
(B ∩ C) = {x: x ∈ B and x ∈ C}
(B ∩ C) = {5, 6}
A ∪ (B ∩ C) = {x: x ∈ A or x ∈ (B ∩ C)}
A ∪ (B ∩ C) = {1, 2, 4, 5, 6}
RHS,
(A ∪ B) = {x: x ∈ A or x ∈ B}
(A ∪ B) = {1, 2, 4, 5, 6}.
(A ∪ C) = {x: x ∈ A or x ∈ C}
(A ∪ C) = {1, 2, 4, 5, 6, 7}
(A ∪ B) ∩ (A ∪ C) = {x: x ∈ (A ∪ B) and x ∈ (A ∪ C)}
(A ∪ B) ∩ (A ∪ C) = {1, 2, 4, 5, 6}
∴ LHS = RHS
Thus, Verified.
(ii)
LHS,
(B ∪ C) = {x: x ∈ B or x ∈ C}
(B ∪ C) = {2, 3, 4, 5, 6, 7}
(A ∩ (B ∪ C)) = {x: x ∈ A and x ∈ (B ∪ C)}
(A ∩ (B ∪ C)) = {2, 4, 5}
RHS,
(A ∩ B) = {x: x ∈ A and x ∈ B}
(A ∩ B) = {2, 5}
(A ∩ C) = {x: x ∈ A and x ∈ C}
(A ∩ C) = {4, 5}
(A ∩ B) ∪ (A ∩ C) = {x: x ∈ (A∩B) and x ∈ (A∩C)}
(A ∩ B) ∪ (A ∩ C) = {2, 4, 5}
∴ LHS = RHS
Thus, verified.