Question

Let A, B and C be sets. Then show that A ∪ (B ∩ C) = (A ∪ B) ∩ (A ∪ C)

Answer 1

We first show that A ∪ (B ∩ C) ⊂ (A ∪ B) ∩ (A ∪ C)

Let x ∈ A ∪ (B ∩ C).

Then x ∈ A or x ∈ B ∩ C

⇒ x ∈ A or (x ∈ B and x ∈ C)

⇒ (x ∈ A or x ∈ B) and (x ∈ A or x ∈ C)

⇒ (x ∈ A ∪ B) and (x ∈ A ∪ C)

⇒ x ∈ (A ∪ B) ∩ (A ∪ C)

Thus, A ∪ (B ∩ C) ⊂ (A ∪ B) ∩ (A ∪ C)  ......(1)

Now we will show that (A ∪ B) ∩ (A ∪ C) ⊂ (A ∪ C)

Let x ∈ (A ∪ B) ∩ (A ∪ C)

⇒ x ∈ A ∪ B and x ∈ A ∪ C

⇒ (x ∈ A or x ∈ B) and (x ∈ A or x ∈ C)

⇒ x ∈ A or (x ∈ B and x ∈ C)

⇒ x ∈ A or (x ∈ B ∩ C)

⇒ x ∈ A ∪ (B ∩ C)

Thus, (A ∪ B) ∩ (A ∪ C) ⊂ A ∪ (B ∩ C)   ......(2)

So, from (1) and (2)

We have A ∩ (B ∪ C) = (A ∪ B) ∩ (A ∪ C)