Proving that an equivalence relation is a bijection

[number0]

Homework Statement

Let (a, b), (c, d) be in R x R. We define (a, b) ~ (c, d) iff a^2 + b^2 = c^2 + d^2.

Let R* = all positive real numbers (including 0).

Prove that there is a bijection between R* and the set of all equivalence classes for this equivalence relationship.

Homework Equations

The Attempt at a Solution

I do not know how the formula for the relationship ~ looks like. I tried mapping from R* x R* -> R* where f: x^2 + y^2 but I got stuck. Any help please?

 

[LCKurtz]

Homework Statement

Let (a, b), (c, d) be in R x R. We define (a, b) ~ (c, d) iff a^2 + b^2 = c^2 + d^2.

Let R* = all positive real numbers (including 0).

Prove that there is a bijection between R* and the set of all equivalence classes for this equivalence relationship.

Homework Equations

The Attempt at a Solution

I do not know how the formula for the relationship ~ looks like. I tried mapping from R* x R* -> R* where f: x^2 + y^2 but I got stuck. Any help please?

Doesn't (a, b) ~ (c, d) mean the two points are on the same circle centered at the origin? Is that what you mean by "what it looks like"?

 

[number0]

Doesn't (a, b) ~ (c, d) mean the two points are on the same circle centered at the origin? Is that what you mean by "what it looks like"?

I knew what it looks like... but I do not know the equation is supposed to be set up. For example, I did f(x,y) = x^2 + y^2 but I could not find injection.

 

[LCKurtz]

I knew what it looks like... but I do not know the equation is supposed to be set up. For example, I did f(x,y) = x^2 + y^2 but I could not find injection.

Well, it's true that the radius is common to all the elements in an equivalence class. So I would say you are on the right track. Try the map that maps the equivalence class for (a,b) into the common radius or radius squared. If you denote the equivalence class for (a,b) by
[(a,b)] your map could map that to a²+b². Then you have to show your function is well defined and has all the necessary properties to be a bijection.

 

[number0]

Well, it's true that the radius is common to all the elements in an equivalence class. So I would say you are on the right track. Try the map that maps the equivalence class for (a,b) into the common radius or radius squared. If you denote the equivalence class for (a,b) by
[(a,b)] your map could map that to a²+b². Then you have to show your function is well defined and has all the necessary properties to be a bijection.

Okay, this may sound really bad... but I do not know how to prove bijection in equivalence classes (I know how to do it for functions but my book never showed me examples about proving bijection in equivalence classes). I tried googling examples but I cannot find any. My finals is tomorrow! Can anyone help me?