Write out the definitions of the equivalence relations induc

Write out the definitions of the equivalence relations induced by the partitions {Li}iR and {Cr}r0.

{Li}iR - set L of vertical lines yield these notations for L: {Li}iR

{Cr}r0 - Cr={(x,y)| x2+y2=r}, the circle with radius r and center (0, 0)

Solution

1. Equivalence relation defined by Li ,denoted by R

Two points: (x,y) and (u,v) are related ie (x,y)R(u,v) if and only if they both lie on a vertical line: x=a for some real number a

2. Equivalence relation defined by Cr,denoted by S

Two points: (x,y) and (u,v) are related ie (x,y)S(u,v) if and only if they both lie on a circle: x^2+y^2=r for some r>=0