NCERT Class XI Mathematics - Sets - Solutions
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Question : 68
Total: 73
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)
(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.
⇒ 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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