# Homework Help: Interpreting equivalence classes geometrically

1. Nov 9, 2009

### MathDude

1. The problem statement, all variables and given/known data

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.

2. Relevant equations

(none)

3. 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.

2. Nov 9, 2009

### lanedance

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

3. Nov 9, 2009

### Office_Shredder

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

4. Nov 9, 2009

### 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?

5. Nov 9, 2009

### lanedance

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

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

(x,y) ~ (u,v)

now on what geomertic object do they both lie?

6. Nov 9, 2009

### 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?"

7. Nov 9, 2009

### 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?

8. Nov 10, 2009