Question

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

Answer 1

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

To show that B = C.

Let x ∈ B

Case I 

x ∈ A

Also, x ∈ B

∴ x ∈ A and x ∈ C

∴ x ∈ C

∴ B ⊂ C

Similarly, we can show that C ⊂ B.

∴ B = C