(i) A ∩ B = A ∩ C need not imply B = C.
Consider, A = {1, 2}
B = {2, 3}
C = {2, 4}
Now,
A ∩ B = {2}
A ∩ C = {2}
Thus, A ∩ B = A ∩ C, here B is not equal to C
(ii) A ⊂ B ⇒ C – B ⊂ C – A
Given as: A ⊂ B
To prove: C–B ⊂ C–A
Consider x ∈ C– B
⇒ x ∈ C and x ∉ B [by definition C–B]
⇒ x ∈ C and x ∉ A
⇒ x ∈ C–A
Hence x ∈ C–B ⇒ x ∈ C–A. This is true for all x ∈ C–B.
∴ A ⊂ B ⇒ C – B ⊂ C – A