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6. Let \( A=\{x \in W \mid x<2\}, B=\{x \in N \mid 1<x \leq 4\} \) and \( C=\{3,5\} \). Verify that (i) \( A \times(B \cup C)=(A \times B) \cup(A \times C) \) (ii) \( A \times(B \cap C)=(A \times B) \cap(A \times C) \) (iii) \( (A \cup B) \times C=(A \times C) \cup(B \times C) \)

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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.

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