Let's evaluate each of the properties for the relation $R$.

Relation R : $R=\{(1,3),(4,2),(2,4),(2,3),(3,1)\}$

• A relation is a function if each element in the domain is related to exactly one element in the codomain. In this case, for example, 2 is related to both 4 and 3, so $R$ is not a function.




• A relation is transitive if for every pair of elements $(x,y)$ and $(y,z)$ in the relation, $(x,z)$ is also in the relation. In this case, for example, $(1,3)$ and $(3,1)$ are in the relation but $(1,1)$ is not, so $R$ is not transitive.




• A relation is symmetric if for every pair $(x,y)$ in the relation, $(y,x)$ is also in the relation. In this case, for example, $(1,3)$ is in the relation but $(3,1)$ is not, so $R$ is not symmetric.




• A relation is reflexive if every element is related to itself, i.e., if all pairs of the form $(x,x)$ are in the relation for all $x$ in the set. In this case, none of the pairs are of the form $(x,x)$, so $R$ is not reflexive.

Therefore, the correct answer is Option C : $R$ is not symmetric.