A relation $\mathbf{R}$ is said to be $\textit{circular}$ if $a\mathbf{R}b$ and $b\mathbf{R}c$ together imply $c\mathbf{R}a$. Which of the following options is/are correct?

Let us test the properties of relation $\mathbf{S}$ in each option:

• For Option (C): If $\mathbf{S}$ is reflexive and circular:

  1. Symmetry: Suppose $a\mathbf{S}b$. Since $\mathbf{S}$ is reflexive, $b\mathbf{S}b$. Applying the circular property to $a\mathbf{S}b$ and $b\mathbf{S}b$, we get $b\mathbf{S}a$. Thus, $\mathbf{S}$ is symmetric.

  2. Transitivity: Suppose $a\mathbf{S}b$ and $b\mathbf{S}c$. By circularity, $c\mathbf{S}a$. Since $\mathbf{S}$ is symmetric, $c\mathbf{S}a \implies a\mathbf{S}c$. Thus, $\mathbf{S}$ is transitive.

  3. Since $\mathbf{S}$ is reflexive, symmetric, and transitive, it is an equivalence relation.

• Counterexamples for other options:

  • Option (A): A relation that is reflexive and symmetric is not necessarily transitive (e.g., compatibility/tolerance relation).

  • Options (B) and (D): The empty relation $\emptyset$ on a non-empty set is vacuously circular, symmetric, and transitive, but it is not reflexive, and hence not an equivalence relation.

Therefore, only C is correct.