Understanding Equivalence Classes in the Plane

[QuantumP7]

Homework Statement

Define two points $$(x_{0}, y_{0})$$ and $$(x_{1}, y_{1})$$ of the plane to be equivalent if $$y_{0} - x_{0} ^2 = y_{1} -x_{1}^2$$. Check that this is an equivalence relation and describe the equivalence classes.

Homework Equations

The Attempt at a Solution

I can understand how to check that it is an equivalence relation. But apparently, the equivalence classes are the sets of points on the parabolas $$y = x^2 + c$$. I don't really understand why? Would the equivalence classes not be all z such that $$y - x^2$$, which would be a paraboloid?

 

[Tedjn]

Equivalence classes are each subsets of the plane. Each equivalence class is a parabola. By adding in the z coordinate you are just layering the equivalence classes vertically -- if you project downward onto the plane, you get a picture of the equivalence classes all in the plane.

 

[QuantumP7]

Wow!

Thank you SO much! I get it now!

Physics Forums need some kind of point system for thanks.