$\mathrm{R} =\{( \mathrm{x} , \mathrm{y} ): \mathrm{x}$ and $\mathrm{y}$  work at the same place $\}$

$\Rightarrow(\mathrm{x}, \mathrm{x}) \in \mathrm{R}$                   [as $\mathrm{x}$  and $\mathrm{x}$  work at the same place $]$

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

If $(\mathrm{x}, \mathrm{y}) \in \mathrm{R},$ then $\mathrm{x}$ and $\mathrm{y}$ work at the same place.

$\Rightarrow \mathrm{y}$ and $\mathrm{x}$ work at the same place.

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

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

Now, let $(\mathrm{x}, \mathrm{y}),\,(\mathrm{y}, \mathrm{z}) \in \mathrm{R}$

$\Rightarrow \mathrm{x}$ and $\mathrm{y}$ work at the same place and $\mathrm{y}$ and $\mathrm{z}$ work at the same place.

$\Rightarrow \mathrm{x}$ and $\mathrm{z}$ work at the same place.

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

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

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