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

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According to the question,

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 and x ∈ B) or (x ∈ A and 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 and y ∈ B) or (y ∈ A and 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 equations (i) and (ii), we have,

A ∩ (B ∪ C) = (A ∩ B) ∪ (A ∩ C)

Hence Proved

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