Are all sets generated by each <n> from $\displaystyle \mathbb{Z}_n$ subgroups? (I know that they are not all generators, and some subgroups contain others)
are you asking if <k> = $\displaystyle \mathbb{Z}_n$ for every k in $\displaystyle \mathbb{Z}_n$, or are you asking if every subgroup of $\displaystyle \mathbb{Z}_n$ is cyclic (and thus equal to <k> for some k)?
the answer to the first is no, the answer to the second is yes.