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.