

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