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Question
Let A and B be sets. If A ∩ X = B ∩ X = Φ 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
Given that A ∪ X = B ∪ X While X is any set.
⇒ A ∩ (A ∪ X) = A ∩ (B ∪ X) [A ⊂ A ∪ X, ∴ A ∩ (A ∪ X) = A]
⇒ A = A ∩ (B ⊂ X)
= (A ∩ B) ∪ (A ∩ X) [By distributive law]
= (A∩ B) U ϕ (∴ Given, A ∩ X = ϕ)
= A ∩ B
⇒ A ⊂ B …(i)
Again A ∪ X = B ∪ X
⇒ B ∩ (A ∪ X) = B ∩ (B ∪ X)
⇒ B ∩ (A ∪ X) = B [∴ B ⊂ B ∪ X]
⇒ (B ∩ A) ∪ (B ∩ X) = B [By distributive law]
⇒ (B ∩ A) ∪ ϕ = B [Given: B ∩ X = ϕ]
⇒ (B ∩ A) = B
⇒ B ⊂ A …..(ii)
Equation From (i) and (ii), we get that A = B.
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