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NCERT Class XI Mathematics - Sets - Solutions
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Question : 68 of 73
Marks:
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Let A and B be sets. If A ∩ X = B ∩ X = f and A ∪ X = B ∪ X for some set X, show that A = B. (Hints A = A ∩ (A ∪ X), B = B ∩ (B ∪ X) and use Distributive law)
Solution:
Here A ∪ X = B ∪ X for some set X ⇒ A ∩ (A ∪ X) = A ∩ (B ∪ X) ⇒ A = (A ∩ B) ∪ (A ∩ X) [Since A ∩ (A ∪ X) = A] ⇒ A = (A ∩ B) ∪ f ⇒ A = A ∩ B ... (i) Also A ∪ X = B ∪ X ⇒ B ∩ (A ∪ X) = B ∩ (B ∪ X) ⇒ (B ∩ A) ∪ (B ∩ X) = B [Since B ∩ (B ∪ X) = B] ⇒ (B ∩ A) ∪ ϕ = B [Since B ∩ X = ϕ] ⇒ B ∩ A = B ... (ii) From (i) and (ii), we have, A = B.
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