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

The correct option is Reflexive but not symmetric.

To determine the nature of the relation, let us analyze its properties step-by-step on the given set
$A=\{1, 2, 3\}$
and the relation
$R=\left\{\right(1,1), (2,2), (3,3), (1,2), (2,3), (1,3\left)\right\}$.

1. Reflexivity:
A relation $R$ on set $A$ is reflexive if for every element $a\in A$, the ordered pair $(a,a)$ belongs to $R$.
For set $A=\{1, 2, 3\}$, the pairs we must check are $(1,1)$, $(2,2)$, and $(3,3)$.
Looking at the relation:
$(1,1)\in R$,
$(2,2)\in R$, and
$(3,3)\in R$.
Since all identity pairs for elements in $A$ are present in $R$, the relation is reflexive.

2. Symmetry:
A relation $R$ on set $A$ is symmetric if $(a,b)\in R$ implies that $(b,a)\in R$ for all $a,b\in A$.
Let us check the pairs in relation $R$:
We have $(1,2)\in R$, but $(2,1)\notin R$.
Similarly, $(2,3)\in R$, but $(3,2)\notin R$.
Also, $(1,3)\in R$, but $(3,1)\notin R$.
Since there exists at least one pair $(a,b)\in R$ for which $(b,a)\notin R$, the relation is not symmetric.

Conclusion:
Since the relation $R$ is reflexive but not symmetric, it matches the option: Reflexive but not symmetric.