1. In general, f(A ∩ B) = f(A) ∩ f(B). Suppose y ∈ f(A) ∩ f(B). This implies that y = f(A) for some a ∈ A and y = f(B) for some b ∈ B. From this, we cannot conclude that y = f(x) for some x ∈ A ∩ B. Hence the equality is not valid in (ii).
2 In this section, the reader might feel that certain obvious results are proved in a roundabout manner. But the point is that although the results are obvious and intuitively clear, their proofs need to be rigorous.