The relation $R=\{(1,1),(2,2),(3,3),(1,2),(2,3),(1,3)\}$ on the set $A=\{1,2,3\}$ is :

Reflexive but not Symmetric

$R=\{(1,1),(2,2),(3,3),(1,2),(2,3),(1,3)\}$

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

for every $a \in A \quad(a, a)$ is present in $R$

i.e. $(1,1)(2,2)(3,3)$ are present hence its reflexive.

for every $(a, b) \in R,(b, a) \in R$

This is not satisfied here eg: $(1,2) \in R$ but $(2,1) \notin R$

Hence it is not symmetric

for $(a, b),(b, c) \in R \quad(a, c)$ must be preselect to be transitive relation

in this case $(1,2)$ and $(2,3) \in R$

$(1,3) \in R$ as well

So this is transitive as wall

Reflexive but not symmetric