Find the number of equivalence relations on $X=\left\{1,2,3\right\}$.

Let $R$ be the relation on the set $\mathrm{ℝ}$ of real numbers such that $aRb$ iff $a-b$ is an integer. Test whether $R$ is an equivalence relation. If so find the equivalence class of $1$ and $\frac{1}{2}$ with respect to this equivalence relation.