Suppose the set $$S={1,2,3}.$$
I must show that the equivalence relation $$R=\{(1,1),(1,3),(2,2),(3,1),(3,3)\}$$ is on the set.
The reflexive property states that: $$(a,a) \in R \;\forall a \in S.$$
Observe that we have $\{(1,1),(2,2),(3,3)\}$ and that this subset $R$ of ordered pair obeys the reflexive property.
The symmetric property states that: $$(a,b) \in R \text{ implies } (b,a) \in R.$$
Suppose we have $R=\{(1,2),(1,3),(2,3)\}$, we see that $\{(2,1),(3,1),(3,2)\}$ does not hold.
Any help in correcting my train of thoughts is greatly appreciated.
You are correct in that if $R=\{(1,2),(1,3),(2,3)\}$ then it is not a symetric relation, however $R$ is not that relation.
$(a,b)\in R$ implies $(b,a)\in R$ means that we for each pair of $(a,b)$ in $R$ needs to check if $(b,a)$ is in $R$.
$(1,1) \in R$ We confirm that $(1,1)\in R$
$(1,3)\in R$ We confirm that $(3,1)\in R$.
$(2,2)\in R$ We confirm that $(2,2)\in R$.
$(3,1)\in R$ We confirm that $(1,3)\in R$.
$(3,3)\in R$ We confirm that $(3,3)\in R$.
Thus we have checked all pairs, which all hold. Hence the relation is symetric. You do not need to check $(2,3)$ since $(2,3)$ is not in $R$.