Greeting I need help to figure out this question In part 2 A

Greeting, I need help to figure out this question:

In part 2, {A_r : r } is family of subsets of ×. prove it is a partition, describe the partition geometrically, and give the corresponding equivalence relation.

2. For each  r , A_r = {(x,y): x² + y² = r²}.

Solution

Let us define the relation , S

(x,y)S(u,v)

If and only if: x^2+y^2=u^2+v^2

ie the points lie on the same circle

We show it is an equivalence relation

1. (x,y)S(x,y)

HEnce, S is reflexive

2. Let, (x,y)S(u,v)

So, x^2+y^2=u^2+v^2

SO, (u,v)S(x,y)

So, S is reflexive

3. Let,

(x,y)S(u,v)

(u,v)S(r,s)

So,x^2+y^2=u^2+v^2=r^2+s^2

So,(x,y)S(r,s)

HEnce, S is transitive

HEnce S is an equivalence class and the equivalence classes form the partitino for RxR

For each r in R we have a unique equivalence class which is the circle centered at origion of radius r

given byL

x^2+y^2=r^2