\#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.