There are two relations defined in the example, the standard (usual) order relation on the real line, and the one such that $xCy$ iff $x^{2}<y^{2}$ , or $x^{2}=y^{2}$ and $x<y$ . So, we are asked about this second one. In other words, we can restate the same relation as $xCy$ iff $|x|<|y|$ , or $|x|=|y|$ and $x<y$ (the closer the point is to the origin, the lower it is, and if two different points are at the same distance from the origin, then the positive one is greater than the negative one: $0<-1<1<-100<100$ ).

Comparability: for every $x\neq y$ either $x^{2}>y^{2}$ ($yCx$ ), or $x^{2}<y^{2}$ ($xCy$ ), or $x^{2}=y^{2}$ , in which case either $x<y$ ($xCy$ ) or $y<x$ ($yCx$ ). Nonreflexivity: for every $x$ , $x^{2}=x^{2}$ but $x\not<x$ , so $xCx$ does not hold. Transitivity: $xCy$ and $yCz$ imply $x^{2}\le y^{2}\le z^{2}$ , and either $x^{2}<z^{2}$ ($xCz$ ), or there are two equalities, implying $x^{2}=z^{2}$ , and $x<y<z$ (again, $xCz$ ).