Question

If A = {x : x ∈ Z, -2 < x ≤ 4}, B = {x : x ∈ W, x ≤ 5}, C = {-4, -1, 0, 2, 3, 4}, then verify A ∪ (B ∩ C) = (A ∪ B) ∩ (A ∪ C).

Answer 1

A = {x : x ∈ Z, -2 < x ≤ 4} = {-1, 0, 1, 2, 3, 4}

B = {x : x ∈ W, x ≤ 5} = {0, 1, 2, 3, 4, 5}

C = {-4, -1, 0, 2, 3, 4}

A ∪ (B ∩ C)

B ∩ C = {0, 1, 2, 3, 4, 5} ∩ {-4, -1, 0, 2, 3, 4} = {0, 2, 3, 4}

A ∪ (B ∩ C) = {-1, 0, 1, 2, 3, 4} ∪ {0, 2, 3, 4} = {-1, 0, 1, 2, 3, 4} … (1)

(A ∪ B) ∩ (A ∪ C)

A ∩ B = {0, 1, 2, 3, 4}

A ∩ C = {-1, 0, 2, 3, 4}

(A ∩ B) ∪ (A ∩ C) = {0, 1, 2, 3, 4} ∪ {-1, 0, 2, 3, 4} = {-1, 0, 1, 2, 3, 4} … (2)

From (1) and (2), it is verified that

A ∪ (B ∩ C) = (A ∪ B) ∩ (A ∪ C)