Question

If A, B and C are three sets such that A ∩ B = A ∩ C and A ∪ B = A ∪ C, then
(a) A = C 

(b) B = C

(c) A ∩ B = ϕ 

(d) A = B

Answer 1

(b) : Let x ∈ C
Suppose x ∈ A ⇒ x ∈ A ∩ C
⇒ x ∈ A ∩ B (Q A ∩ C = A ∩ B)
Thus x ∈ B
Again suppose x ∉ A ⇒ x ∈ C ∪ A
⇒ x ∈ B ∪ A ⇒ x ∈ B
Thus in both cases x ∈ C ⇒ x ∈ B
Hence C ⊆ B ....(1)
Similarly we can show that B ⊆ C ....(2)
Combining (1) and (2) we get B = C.