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}

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

A ∩ (B ∪ C) = {1, 2, 3, 4, 5} ∩  {1, 2, 3, 4, 5, 6, 9}

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

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

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

A ∩  B = {1, 2, 3, 4, 5} ∩ {3, 6, 9}

= {3}

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

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

∴ A ∩ (B ∪ C) = (A ∩  B) ∪ (A ∩ C)