I would really appreciate some help with the following problem:
Prove that each of the following relations defined on Z x Z is an equivalence relation. Determine the equivalence classes for each relation.
(x,y) ~ (v,w) if x^2 + y^2 = v^2 + w^2
(x,y) ~ (v,w) if y - x^2 = w - v^2
(x,y) ~ (v,w) if xy = vw
(x,y) ~ (v,w) if x + 2y = v + 2w