Question 53

A relation $R$ is defined to be circular if for all $a, b, c$:

Now evaluate Option C: 'If a relation $S$ is reflexive and circular, then $S$ is an equivalence relation.'

1. Reflexivity: $aRa$ holds for all $a$ by definition.
2. Symmetry: Suppose $aRb$. Since $S$ is reflexive, we also have $bRb$. Applying the circular property to $aRb$ and $bRb$:

Hence, $aRb \implies bRa$, so $S$ is symmetric.
3. Transitivity: Suppose $aRb$ and $bRc$. By circularity, this implies $cRa$. Because $S$ is symmetric, $cRa$ implies $aRc$. Hence $(aRb \land bRc) \implies aRc$, so $S$ is transitive.
Because $S$ is reflexive, symmetric, and transitive, $S$ is an equivalence relation. Thus option C is correct.

Counterexamples for other options:

• Option A: Reflexive and symmetric does not imply transitive (e.g., tolerance relations).
• Option B: Circular and symmetric does not imply reflexive (e.g., the empty relation $\emptyset$ on a non-empty set is circular and symmetric, but not reflexive).
• Option D: Transitive and circular does not imply reflexive (again, the empty relation $\emptyset$ is transitive and circular, but not reflexive).

Therefore, only option C is correct.