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If $W=\left\{\begin{pmatrix} x\\\ y\end{pmatrix}\in\mathbb{R}^2\ |\ x,y\text{ are integers}\right\}$, which one of the following statements is TRUE?

$W$ is a subspace of $\mathbb{R}^2$

$W$ does not contain the zero vector

$W$ is not closed under scalar multiplication

$W$ is not closed under vector addition