Question

Show that the relative $R$ in the set $\{1,2,3\}$ given by $R=\left\{\right(1,2),(2,1\left)\right\}$ is symmetric but neither reflexive nor transitive.

Answer 1

Let $A=\{1,2,3\}$
A relation $R$ on $A$ is defined as $R=\left\{\right(1,2),(2,1\left)\right\}$.
It is seen 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.