Question

Given A = {x : ∈ N :< 6}, B = {3, 6, 9} and C = {x ∈ N : 2x - 5 ≤ 8}. show that: A ∪ (B ∩ C) = (A ∪  B) ∩ (A ∪ C)

Answer 1

A = {x : ∈ N :< 6}

B = {3, 6, 9}

C = {x ∈ N : 2x - 5 ≤ 8}

2x - 5 ≤ 8

⇒ 2x ≤ 8+5

⇒ 2x ≤ 13

⇒ x ≤ `13/2`

⇒ x ≤ 6.5

∴ C = {1, 2, 3, 4, 5, 6}

B ∩ C = {3, 6, 9} ∩ {1, 2, 3, 4, 5, 6}

= {3,6}

∴ A ∪ (B ∩ C) = {1, 2, 3, 4, 5} ∪ {3,6}

= {1, 2, 3, 4, 5, 6}

A ∪  B = {1, 2, 3, 4, 5} ∪ {3, 6, 9}

= {1, 2, 3, 4, 5, 6, 9}

A ∪ C = {1, 2, 3, 4, 5} ∪ {1, 2, 3, 4, 5, 6}

= {1, 2, 3, 4, 5, 6}

∴ (A ∪  B) ∩ (A ∪ C)

= {1, 2, 3, 4, 5, 6, 9} ∩ {1, 2, 3, 4, 5, 6}

= {1, 2, 3, 4, 5, 6}

∴ A ∪ (B ∩ C) = (A ∪ B) ∩ (A ∪ C)