If A, B and C are three sets such that ..

If A, B and C are three sets such that A ∩ B = A ∩ C and A ∪ B = A ∪ C, then (1) A = C (2) B = C (3) A ∩ B = ϕ (4) A = B

Solution: (B)

Given that A ∪ B = A ∪ C

⇒ (A ∪ B) ∩ C = (A ∪ C) ∩ C

⇒ (A ∩ C) U (B ∩ C) = C [(A ∪ C) ∩ C = C ]

⇒ (A ∩ B) U (B ∩ C) = C ……….(1) [(A ∩ C) = A ∩ B ]

Take A ∪ B = A ∪ C

(A ∪ B) ∩ B = (A ∪ C) ∩ B

B = (A ∩ B) U (C ∩ B)

= (A ∩ B) U (B ∩ C) ………..(2)

From equations 1 & 2,

Therefore, B = C.

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