Exercise 8.6.3

$N$ possesses properties (f1), (f2), and (f5). The natural numbers do not contain 0 or any additive inverses. $Z$ additionally satisfies (f3) but still does not satisfy (f4) for multiplicative inverses (e.g. $2^{-1}$ is not in $Z$). $Q$ possesses all properties; in particular if $q\in Q$ then so are $-q$ and $1∕q$.