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Determine whether each of the following relations are reflexive, symmetric and transitive:
Relation $\mathrm{R}$ in the set $\mathrm{A}$ of human beings in a town at a particular time given by
$ \mathrm{R} =\{(\mathrm{x}, \mathrm{y}): \mathrm{x}$ and $ \mathrm{y}$ work at the same place $\}$
Solution
$\mathrm{R} =\{( \mathrm{x} , \mathrm{y} ): \mathrm{x}$ 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.
