Question

Given an example of a relation. Which is Reflexive and symmetric but not transitive.

Answer 1

Let $A=\{4,6,8\}$

Define a relation $R$ on $A$ as:
$A=\left\{\right(4,4),(6,6),(8,8),(4,6),(6,4),(6,8),(8,6\left)\right\}$
Relation $R$ is reflexive since for every $\{a\in A,(a,a)\in Ri.e.,(4,4),(6,6),(8,8\left)\right\}\in R$

Relation $R$ is symmetric since $(a,b)\in R\Rightarrow (b,a)\in R$ for all

$a,b\in R$.

Relation $R$ is not transitive since $(4,6),(6,8)\in R$, but $(4,8)\notin R$.

Hence, relation $R$ is reflexive and symmetric but not transitive.