$\mathrm{R} =\{( \mathrm{x} , \mathrm{y} )\,:\, \mathrm{x} – \mathrm{y}$ is an integer $\}$

Now, for every $\mathrm{x} \in \mathrm{Z} ,( \mathrm{x} ,\,\mathrm{x} ) \in \mathrm{R}$ as $\mathrm{x} – \mathrm{x} =0$ is an integer.

$\therefore $  $\mathrm{R}$ is reflexive.

Now, for every $\mathrm{x} ,\, \mathrm{y} \in \mathrm{Z} ,$ if $( \mathrm{x} ,\,\mathrm{ y} ) \in \mathrm{R} ,$ then $\mathrm{x} – \mathrm{y}$ is an integer.

$\Rightarrow-(\mathrm{x}-\mathrm{y})$ is also an integer.

$\Rightarrow(\mathrm{y}-\mathrm{x})$ is an integer.

$\therefore $     $( \mathrm{y} ,\, \mathrm{x} ) \in \mathrm{R}$

$\therefore \mathrm{R}$ is symmetric.

Now, Let $( \mathrm{x} , \,\mathrm{y} )$ and $( \mathrm{y} ,\, \mathrm{z} ) \in \mathrm{R} ,$ where $\mathrm{x} ,\, \mathrm{y} ,\, \mathrm{z} \in \mathrm{Z}$

$\Rightarrow $ $(\mathrm{x}-\mathrm{y})$ and $(\mathrm{y}-\mathrm{z})$ are integers.

$\Rightarrow $  $\mathrm{x}-\mathrm{z}=(\mathrm{x}-\mathrm{y})+(\mathrm{y}-\mathrm{z})$ is an integer.

$\therefore$         $( \mathrm{x} ,\, \mathrm{z} ) \in \mathrm{R}$

$\therefore $  $\mathrm{R}$ is transitive.

Hence, $\mathrm{R}$ is reflexive, symmetric, and transitive.