\#2 Michael, the absurd function may not count as surjective, since to be surjective asks for every element of the co-domain to be mapped to by at least one member of the domain, but the empty set has not one element to spare! Also, since the domain is not S, I don't think you could get a useful notion of partition of S from the preimages of the absurd function.

Remark 1.16 in the book is the first time we meet the relation between surjective functions and partitions in Chapter 1.

Remark 1.16 in the book is the first time we meet the relation between surjective functions and partitions in Chapter 1.