Interpreting equivalence classes geometrically

[MathDude]

Homework Statement

For (x, y) and U, v) in R², define (x,y)~(u,v) if x²+y² = u²+v². Prove that ~ defines an equivalence relation on R² and interpret the equivalence classes geometrically.

Homework Equations

(none)

The Attempt at a Solution

The first part is easy. I proved transitivity, reflexivity, and symmetry as per the definition of an equivalence. I'm a little confused how to do so geometrically. Since x² + y² can represent the Pythagorean identity, I'm assuming the proof involves triangles. I understand that the properties of an equivalency (trans, reflect, and sym) can resemble triangle congruence and similarity, but I'm not quite so sure how to apply it. Any hints would be helpful.

 

[lanedance]

consider the distance a point (x,y) is from the origin...

 

[Office_Shredder]

Look at the set of all (x,y) so that x²+y²=1. Is this an equivalence class? What else is it?

 

[MathDude]

That makes sense conceptually, but I'm not quite sure how to apply it. If we're letting x²+y² equal to diameter one, what happens to our (u, v)? I'm confused how to get this to work. What do we define the relation as?

 

[lanedance]

say you have any (x,y) with x^2 + y^2 = 1

and any (u,v) with u^2 + v^2 = 1

then based on your definition
(x,y) ~ (u,v)

now on what geomertic object do they both lie?

 

[MathDude]

They both lie on a circle. I can prove they're equal through distance formula at any given point of origin.

How do you geometrically interpret reflexivity, symmetry, and transitivity? I mean, it's obvious to me visually, but how do you prove it?



Story of my life: "It's obvious conceptually, but how do I prove it?"

 

[MathDude]

Could you just draw a (x,y) and (u,v) circles and show their radii and origins are identical and say its obviously reflexive, symmetric, and transitive?

 

[lanedance]

sorry misread your question

it doesn't ask you to prove geometrically the equivalence realtion, just intepret what it is, which you have already done

each equivalence class, label it by r>0, corresponds to all points at a distance r form the origin, which as a set is the points contained in the circle of radius r, around the origin

 

[HallsofIvy]

Could you just draw a (x,y) and (u,v) circles and show their radii and origins are identical and say its obviously reflexive, symmetric, and transitive?

(x,y) and (u,v) are not circles- they are points on a circle. Geometrically, two points, (x,y) and (u,v) are equivalent if and only if they lie on the same circle about the origin.