Let's analyze the and the correct based on the of relations: ### Definitions: - **Reflexive**: \(aRa\) for all \(a\). - **Symmetric**: \(aRb\) implies \(bRa\). - **Transitive**: \(aRb\) and \(bRc\) imply \(aRc\). - **Circular**: \(aRb\) and \(bRc\) imply \(cRa\). --- ### Statement Analysis: 1. **If a relation \(S\) is reflexive and symmetric, then \(S\) is an equivalence relation.** - **Incorrect**: Reflexivity and symmetry alone do not imply transitivity, which is necessary for an equivalence relation. 2. **If a relation \(S\) is circular and symmetric, then \(S\) is an equivalence relation.** - **Incorrect**: Circularity does not imply transitivity, so this is insufficient for equivalence. 3. **If a relation \(S\) is reflexive and circular, then \(S\) is an equivalence relation.** - **Correct**: Reflexivity + Circularity **implies transitivity** (since circularity combined with reflexivity ensures that the relation behaves like an equivalence relation). 4. **If a relation \(S\) is transitive and circular, then \(S\) is an equivalence relation.** - **Incorrect**: Transitivity + Circularity alone does not guarantee symmetry, which is required for equivalence. --- ### **Correct Answer:** > **If a relation \(S\) is reflexive and circular, then \(S\) is an equivalence relation.**