Final answer:
The relation R on Z is an equivalence relation because it is reflexive, symmetric, and transitive; for any integers a, b, and c, if aRb (a² = b²) then the relation also implies bRa and if aRb and bRc, then aRc.
Step-by-step explanation:
The question asks whether the relation R on the set of integers Z, defined by aRb if a² = b², is an equivalence relation. To determine this, we must check if the relation is reflexive, symmetric, and transitive.
Reflexivity: For any integer a, we have a² = a², so aRa holds for all a in Z.
Symmetry: If aRb, then a² = b², which implies that b² = a², so bRa.
Transitivity: If aRb and bRc, then a² = b² and b² = c², which implies that a² = c², so aRc.
Since the relation R is reflexive, symmetric, and transitive, it is indeed an equivalence relation.