^{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.

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