Question

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

Answer 1

A, B and C are three given sets

To prove: A ∩ (B ∪ C) = (A ∩ B) ∪ (A ∩ C)

Let x ∈ A ∩ (B ∪ C)

⇒ x ∈ A and x ∈ (B ∪ C)

⇒ x ∈ A and (x ∈ B or x ∈ C)

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

⇒ x ∈ A ∩ B or x ∈ A ∩ C

⇒ x ∈ (A ∩ B) ∪ (A ∩ C)

⇒ A ∩ (B ∩ C) ⊂ (A ∩ B) ∪(A ∩ C)  ......(i)

Let y ∈ (A ∩ B) ∪ (A ∩ C)

⇒ y ∈ A ∩ B or x ∈ A ∩ C 

⇒ (y ∈ A or y ∈ B) or (y ∈ A or y ∈ C)

⇒ y ∈ A and (y ∈ B or y ∈ C)

⇒ y ∈ A and y ∈ (B ∪ C)

⇒ y ∈ A ∩ (B ∩ C) 

⇒ (A ∩ B) ∪ (A ∩ C) ⊂ A ∩ (B ∪ C)  ......(ii)

We know that:

P ⊂ Q and Q ⊂ P 

⇒ P = Q

From (i) and (ii)

A ∩ (B ∪ C) = (A ∩ B) ∪ (A ∩ C)

Hence Proved