Let ${\mathrm{R}}_1=\left\{\right(\mathrm{a},\mathrm{b})\in \mathrm{N}\times \mathrm{N}:|\mathrm{a}-\mathrm{b}|⩽13\}$ and ${\mathrm{R}}_2=\left\{\right(\mathrm{a},\mathrm{b})\in \mathrm{N}\times \mathrm{N}:|\mathrm{a}-\mathrm{b}|\ne 13\}$. Then on N :

$\begin{array}{l}{\mathrm{R}}_1=\left\{\right(\mathrm{a},\mathrm{b})\in \mathrm{N}\times \mathrm{N}:|\mathrm{a}-\mathrm{b}|\le 13\}\\ {\mathrm{R}}_2=\left\{\right(\mathrm{a},\mathrm{b})\in \mathrm{N}\times \mathrm{N}:|\mathrm{a}-\mathrm{b}|\ne 13\}.\end{array}$

For R₁ :

i) Reflexive relation

     $(\mathrm{a},\mathrm{a})\in \mathrm{N}\times \mathrm{N}:|\mathrm{a}-\mathrm{a}|\le 13$

  $R_1$   is reflexive

ii) Symmetric relation

     $$\begin{gathered} (\mathrm{a},\mathrm{b})\in {\mathrm{R}}_1 \\ then |\mathrm{a}-\mathrm{b}|\le 13\Rightarrow |\mathrm{b}-\mathrm{a}|\le 13 \\ \Rightarrow (\mathrm{b},\mathrm{a})\in {\mathrm{R}}_1 \end{gathered}$$

iii) Transitive relation

     $\begin{array}{l}(\mathrm{a},\mathrm{b})\in {\mathrm{R}}_1,(\mathrm{b},\mathrm{c})\in {\mathrm{R}}_1\Rightarrow (\mathrm{a},\mathrm{c})\in {\mathrm{R}}_1:\\ (1,3)\in {\mathrm{R}}_{1,}(3,16)\in {\mathrm{R}}_1\text{ but }(1,16)\notin {\mathrm{R}}_1\end{array}$

For R₂ :

i) Reflexive relation

     $(\mathrm{a},\mathrm{a})\in \mathrm{N}\times \mathrm{N}:|\mathrm{a}-\mathrm{a}|\ne 13$

ii) Symmetric relation

     $(\mathrm{a},\mathrm{b})\in \mathrm{N}\times \mathrm{N}:|\mathrm{a}-\mathrm{b}|\ne 13\Rightarrow |\mathrm{b}-\mathrm{a}|\ne 13\Rightarrow (\mathrm{b},\mathrm{a})\in \mathrm{N}\times \mathrm{N}$

iii) Transitive relation

     $\begin{array}{l}(\mathrm{a},\mathrm{b})\in {\mathrm{R}}_2, (\mathrm{b},\mathrm{c})\in {\mathrm{R}}_2, (\mathrm{a},\mathrm{c})\in {\mathrm{R}}_2\\ (1,3)\in {\mathrm{R}}_{2,} (3,14)\in {\mathrm{R}}_{2,}\text{ but }(1,14)\notin {\mathrm{R}}_2\end{array}$