Determine whether each of the following relations are reflexive, symmetric and transitive:
Relation $\mathrm{R}$ in the set $\mathrm{A}=\{1,2,3,4,5,6\}$ as $\mathrm{R} =\{(\mathrm{x}, \mathrm{y}): \mathrm{y}$ is divisible by $\mathrm{x}\}$
Determine whether each of the following relations are reflexive, symmetric and transitive:
Relation $\mathrm{R}$ in the set $\mathrm{A}=\{1,2,3,4,5,6\}$ as $\mathrm{R} =\{(\mathrm{x}, \mathrm{y}): \mathrm{y}$ is divisible by $\mathrm{x}\}$
$\mathrm{A}=\{1,2,3,4,5,6\}$
$\mathrm{R} =\{( \mathrm{x} , \mathrm{y} ): \mathrm{y} $ is divisible by $\mathrm{x} \}$
We know that any number $(\mathrm{x})$ is divisible by itself.
So, $(\mathrm{x}, \mathrm{x}) \in \mathrm{R}$
$\therefore \mathrm{R}$ is reflexive.
Now, $(2,4)\in \mathrm{R}$ $[$ as $4$ is divisible by $2]$
But, $(4,2) \notin \mathrm{R}$ . $[$ as $2$ is not divisible by $4]$
$\therefore \mathrm{R}$ is not symmetric.
Let $( \mathrm{x} , \mathrm{y} ),\,( \mathrm{y} , \mathrm{z} ) \in \mathrm{R} .$ Then, $\mathrm{y}$ is divisible by $\mathrm{x}$ and $\mathrm{z}$ is divisible by $\mathrm{y}$
$\therefore$ $ \mathrm{z}$ is divisible by $\mathrm{x}$ $\Rightarrow(\mathrm{x}, \mathrm{z}) \in \mathrm{R}$
$\therefore $ $\mathrm{R}$ is transitive.
Hence, $\mathrm{R}$ is reflexive and transitive but not symmetric.
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