Final answer:
The 'lives in the same city as' relation is an equivalence relation because it is reflexive, symmetric, and transitive. In contrast, 'is descendant of' is not an equivalence relation because it lacks symmetry.
Step-by-step explanation:
The question asks which of the listed properties (lives in the same city as or is a descendant of) forms an equivalence relation. An equivalence relation is characterized by three properties: it must be reflexive, symmetric, and transitive. Let's analyze each option:
- Reflexive: An individual lives in the same city as themselves, and a person is a descendant of their own lineage.
- Symmetric: If person A lives in the same city as person B, then person B lives in the same city as person A. However, the descendant relationship is not symmetric; if A is a descendant of B, B is not necessarily a descendant of A.
- Transitive: If person A lives in the same city as person B, and person B lives in the same city as person C, then person A lives in the same city as person C. In the case of being a descendant, if A is a descendant of B and B is a descendant of C, A is also a descendant of C.
Based on these criteria, living in the same city as is an equivalence relation because it satisfies the reflexive, symmetric, and transitive properties. In contrast, is descendant of is not an equivalence relation because it is not symmetric.