We say $X$ is a Vitali set if there exists a countably dense subgroup, $\Gamma$, of the additive group $\mathbb{R}$, such that $X$ is a selector of the partition of $\mathbb{R}$ canonically associated with the equivalence relation $x \in \mathbb{R}$ & $y \in \mathbb{R}$ & $x - y \in \Gamma$.

Let $V$ be a Vitali set and let $r \in \mathbb{R}$. Is $V \cup (V \oplus r)$ a Vitali set where $V \oplus r$ = {$x + r : x \in V$ }?

A slightly easier question, perhaps, is the special case $V$ is the Vitali set with respect to the countably dense subgroup $\mathbb{Q}$ and with the restriction $r \in \mathbb{Q}$.