Determine whether each of the following relations are reflexive, symmetric and transitive :

Relation $\mathrm{R}$ in the set $\mathrm{A}=\{1,2,3, \ldots, 13,14\}$ defined as $\mathrm{R}=\{(x, y): 3 x-y=0\}$

$\mathrm{A}=\{1,2,3 \ldots 13,14\}$

$\mathrm{R}=\{(x, y): 3 x-y=0\}$

$\therefore  $ $\mathrm{R} =\{(1,3),\,(2,6),\,(3,9),\,(4,12)\}$

$\mathrm{R}$ is not reflexive since $(1,1),(2,2) \ldots(14,\,14) \notin \mathrm{R}$

Also, $\mathrm{R}$ is not symmetric as $(1,3) \in \mathrm{R},$ but $(3,1) \notin \mathrm{R}$ . $[3(3)-1 \neq 0]$

Also, $\mathrm{R}$ is not transitive as $(1,3),\,(3,9) \in \mathrm{R},$ but $(1,9) \notin \mathrm{R}$ . $[3(1)-9 \neq 0]$

Hence, $\mathrm{R}$ is neither reflexive, nor symmetric, nor transitive.