Let A = {1, 2, 4, 5} B = {2, 3, 5, 6} C = {4, 5, 6, 7}. Verify the identities A ∪ (B ∩ C) = (A ∪ B) ∩ (A ∪ C)

A ∪ (B ∩ C) = (A ∪ B) ∩ (A ∪ C)

Firstly let us consider the LHS

(B ∩ C) = {x: x ∈ B and x ∈ C}

= {5, 6}

A ∪ (B ∩ C) = {x: x ∈ A or x ∈ (B ∩ C)}

= {1, 2, 4, 5, 6}

Now, RHS

(A ∪ B) = {x: x ∈ A or x ∈ B}

= {1, 2, 4, 5, 6}.

(A ∪ C) = {x: x ∈ A or x ∈ C}

= {1, 2, 4, 5, 6, 7}

(A ∪ B) ∩ (A ∪ C) = {x: x ∈ (A ∪ B) and x ∈ (A ∪ C)}

= {1, 2, 4, 5, 6}

LHS = RHS

Hence Verified.