Question

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

Answer 1

Set $=\{1,2,3\}$

$R=\left\{\right(1,2\left)\right(2,1\left)\right\}$

If relation is reflexive.

then (a,a) $ϵ$ R for all a $ϵ$ {1,2,3}

as (1,1) $\notin$ R

so , R is not reflexive

If relation is symmetric

then (a, b) & (b, a) $ϵ$ R

Here (1,2) & (2,1) $ϵ$ R

So, R is symmetric

If relation is transitive

then (a ,b) $ϵ$ R & (b,c) $ϵ$ r $\to$ (a,c) $ϵ$R

Here (1, 2) & (2, 1) $ϵ$ R but (1,1) $\notin$ R

So, R is not transitive.