Determine whether each of the following relations are reflexive, symmetric and transitive:
Relation $\mathrm{R}$ in the set $\mathrm{Z}$ of all integers defined as $\mathrm{R} =\{(\mathrm{x}, \mathrm{y}): \mathrm{x}-\mathrm{y}$ is an integer $\}$
Determine whether each of the following relations are reflexive, symmetric and transitive:
Relation $\mathrm{R}$ in the set $\mathrm{Z}$ of all integers defined as $\mathrm{R} =\{(\mathrm{x}, \mathrm{y}): \mathrm{x}-\mathrm{y}$ is an integer $\}$
$\mathrm{R} =\{( \mathrm{x} , \mathrm{y} )\,:\, \mathrm{x} - \mathrm{y}$ 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.
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