1.

Let A, B and C be the sets such that A ∪ B = A ∪ C and A ∩ B = A ∩ C . Show, that B = C.

Answer»

Given A∪B=A∪C 

⇒(A∪B)∩C = (A∪C)∩C 

⇒(A∩C)∪(B∩C) = C 

⇒(A∩B)∪(B∩C) = C 

∵ A∩B = A∩C 

Now, A∪B = A∪C 

⇒ (A∪B)∩B = (A∪C)∩B 

⇒ (A∩B)u(B∩B) = (A∩B)∪(C∩B) 

⇒ B = (A∩B)∪(B∩C) B = C



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