Given relation

$R=\{(1,3),(4,2),(2,4),(2,3),(3,1)\}$ on $A=\{1,2,3,4\}.$

We check each option one by one.

Option A: Is $R$ a function?

For a relation on $A$ to be a function from $A$ to $A$, each element of $A$ must have exactly one image.

From the relation:

• $1 \mapsto 3$
• $2 \mapsto 4$ and $2 \mapsto 3$
• $3 \mapsto 1$
• $4 \mapsto 2$

Since the element $2$ has two images ($4$ and $3$), $R$ is not a function.

So, A is false.

Option B: Is $R$ transitive?

A relation is transitive if whenever $(a,b) \in R$ and $(b,c) \in R$, then $(a,c) \in R$ must also be in $R$.

Check a counterexample:

• $(1,3) \in R$
• $(3,1) \in R$

Then transitivity would require: $(1,1) \in R$ But $(1,1) \notin R$.

Hence, $R$ is not transitive.

So, B is false.

Option C: Is $R$ not symmetric?

A relation is symmetric if whenever $(a,b) \in R$, then $(b,a) \in R$ also belongs to $R$.

Check the pairs:

• $(1,3) \in R$ and $(3,1) \in R$ ✔️
• $(4,2) \in R$ and $(2,4) \in R$ ✔️
• $(2,3) \in R$, but $(3,2) \notin R$ ❌

Since one pair fails symmetry, the relation is not symmetric.

Therefore, C is true.

Option D: Is $R$ reflexive?

A relation on $A=\{1,2,3,4\}$ is reflexive if $(1,1),(2,2),(3,3),(4,4) \in R.$

None of these ordered pairs are present in $R$.

Hence, $R$ is not reflexive.

So, D is false.

Final conclusion

The only correct option is: $\boxed{\text{C: not symmetric}}$

Comparison with stored answer

Stored correct answer: C

Our derived answer is also C, so they agree.