We have A = A = {x ϵ w|x < 2}

⇒ A = {0, 1}

B = {x ϵ N|1 <x ≤ 4}

⇒ B = {2, 3, 4} and C = {3, 5}

**(i) **A X (B ⋃ C) = {0, 1} X {2, 3, 4, 5}

= {(0, 2), (0, 3), (0, 4), (0, 5), (1, 2), (1, 3), (1, 4), (1, 5)}

**Also, **(A X B) ⋃ (A X C) = {(0, 2), (0, 3), (0, 4), (1, 2), (1, 4)} ⋃ {(0, 3), (0, 5), (1, 3), (1, 5)}

= {(0, 2), (0, 3), (0, 4), (0, 5), (1, 2), (1, 3),(1, 4), (1, 5)}

**= A X (B ⋃ C)**

**Hence Verified.**

**(ii) **A X (B ∩ C) = {0, 1} X {3}

= {(0, 3), (1, 3)} And

(A X B) ∩ (A X C) = {(0, 2), (0, 3), (0, 4), (1, 2), (1, 3), (1, 4)} ∩ {(0, 3), (0, 5), (1, 3), (1, 5)}

= {(0, 3), (1, 3)} =** A X (B ∩ C)**

**Hence Verified.**

**(iii)** (A ⋃ B) X C = {0, 1, 2, 3, 4} X {3, 5}

= {(0, 3), (1, 3), (2, 3), (3, 3), (4, 3), (0, 5), (1, 5), (2, 5), (3, 5), (4, 5)}

And (A X C) ⋃ (B X C) = {(0, 3), (0, 5), (1, 3), (1, 5)} ⋃ {(2, 3), (2, 5), (3, 3), (3, 5), (4, 3), (4, 5)}

= {(0, 3), (1, 3), (2, 3), (3, 3), (4, 3), (0, 5), (1, 5), (2, 5), (3, 5), (4, 5)}

**= (A ⋃ B) X C**

**Hence Verified.**