Check if $(x,y)\rho(a,b)\Leftrightarrow sgn(y-\pi x)=sgn(b-\pi a)$ is equivalence relation on $\mathbb{R^2}$, find the set of equivalence classes and $C_{(1,\pi)}$. Give geometric representation.
$1)$ $\rho$ is reflexive: $(\forall (x,y)\in \mathbb{R^2})(x,y)\rho (x,y)\Leftrightarrow sgn(y-\pi x)=sgn(y-\pi x)$
$2)$ $\rho$ is symmetric: $(\forall (x,y),(a,b)\in \mathbb{R^2})(x,y)\rho (a,b) \Rightarrow (a,b)\rho (x,y)$
$(x,y)\rho (a,b) \Leftrightarrow sgn(y-\pi x)=sgn(b-\pi a)$
$(a,b) \rho (x,y)\Leftrightarrow sgn(b-\pi a)=sgn(y-\pi x)$
$3)$ $\rho$ is transitive: $(\forall (x,y),(a,b),(u,v)\in \mathbb{R^2})(x,y)\rho (a,b) \land (a,b)\rho(u,v)\Rightarrow (x,y)\rho (u,v)$
$(x,y)\rho (a,b)\land (a,b)\rho(u,v)\Leftrightarrow sgn(y-\pi x)=sgn(b-\pi a) \land sgn(b-\pi a)=sgn(v-\pi u)\Rightarrow sgn(y-\pi x)=sgn(v-\pi u)\Rightarrow (x,y)\rho(u,v)$
From $1),2)$ and $3)$, $\rho$ is equivalence relation.
Equivalence class $C_{(1,\pi)}$ is $C_{(1,\pi)}=\{(x,y): y-\pi x=0\}$
How to find complete set of equivalence classes and represent it geometrically?