Show that the relation $R$ in the set $\{1,2,3\}$ given by $R =\{(1,2),(2,1)\}$ is symmetric but neither reflexive nor transitive.
Show that the relation $R$ in the set $\{1,2,3\}$ given by $R =\{(1,2),(2,1)\}$ is symmetric but neither reflexive nor transitive.
Let $A=\{1,2,3\}$
A relation $R$ on $A$ is defined as $R =\{(1,2),\,(2,1)\}$
It is clear that $(1,1),\,(2,2),\,(3,3) \notin R$
$\therefore R$ is not reflexive.
Now, as $(1,2)\in R$ and $(2,1)\in R$, then $R$ is symmetric.
Now, $(1,2) $ and $(2,1)\in R$
However, $(1,1)\notin R$
$\therefore R$ is not transitive.
Hence, $R$ is symmetric but neither reflexive nor transitive.
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