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)