The relation R on R^2 is defined such that (x₁, y₁)R(x₂, y₂) if x₁² + y₁² = x₂² + y₂². To prove that R is an equivalence relation, it must satisfy three properties: reflexivity, symmetry, and transitivity. Reflexivity holds as (x, y)R(x, y) since x² + y² = x² + y². Symmetry is satisfied because if (x₁, y₁)R(x₂, y₂), then (x₂, y₂)R(x₁, y₁) follows directly. Transitivity is confirmed as if (x₁, y₁)R(x₂, y₂) and (x₂, y₂)R(x₃, y₃), then (x₁, y₁)R(x₃, y₃) holds true, completing the proof that R is indeed an equivalence relation.
PREREQUISITESMathematics students, educators, and anyone interested in understanding the properties of equivalence relations and their applications in various mathematical contexts.