What's the logic of this statement? "Define $xCy$ if $x^2<y^2$, or if $x^2 = y^2$ and $x<y$."

Question

I'm reading Munkre's Topology. There is an exercise which asks us to check that the relation defined in Example 7 is an order relation. But I don't understand what's the meaning of this relation. Can someone give some explanations? Thanks! Why did he write it in this way?

Answer 1

But I don't understand what's the meaning of this relation.

$C=\{(x,y)\in\Bbb R^2: x^2<y^2 ~\lor~(x^2=y^2\land x<y)\}$

It is an order comparison that compares the squares of the values, or if they are equal, compares the values themselves.

(Cf: Order people by weight, or when equal use their height. )

Can someone give some explanations? Thanks! Why did he write it in this way?

It is a self evaluation to see if you comprehend the topic of order relations by using a somewhat unfamiliar relation.   Is it an order relation?   Use what you have been taught about them to find out.

Answer 2

Hint:  $\,xCy\,$ is equivalent to $\,|x| \lt |y|\,$, or if $\,|x|=|y|\,$ and $\,x \lt y\,$. In other words, $\,xCy\,$ if $\,x\,$ is closer to the origin than $\,y\,$, or in case both are at the same distance from $\,0\,$, then if $\,x\,$ is negative and $\,y\,$ is positive.