Let $G=\mathbb{Q}$ be the rational numbers, with addition as the group operation. Let $x, y$ be non-zero elements of $G$, and let $N \leqslant G$ be the subgroup they generate. Show that $N$ is isomorphic to $\mathbb{Z}$.

Find non-zero elements $x, y \in \mathbb{R}$ which generate a subgroup that is not isomorphic to $\mathbb{Z}$.