antisymmetric proof clarification

Question

i understand antisymmetric proof but i have a problem with an exercise. there 's a relation $xRy <==> x^2 <= y^2$

i have to check if it is antisymmetric or not over $Z$.

well of course when x and y are squared they will remain positive so even over $Z$, so $x^2$ and $y^2$ positive so if $x^2<=y^2$and $y^2<=x^2$ then $x^2 = y^2$

my problem is that i am confused wether if the proof is

$x^2<=y^2$and $y^2<=x^2$ then $x^2 = y^2$

or

$x^2<=y^2$and $y^2<=x^2$ then $x = y$

because the question is asked on $N$ and $Z$ so if it's the first, it will be antisymmetric on both $N$ and $Z$ and if it's the second, it won't be antisymmetric on $Z$.

Answer 1

Undisputably,

$$x^2\le y^2\land x^2\ge y^2\implies x^2=y^2.$$

(Just because $a\le b\land a\ge b\implies a=b$.)

Then

$$x^2=y^2\iff x=\pm y.$$

Answer 2

Welcome to MSE! Over the integers, you have $2R(-2)$ and $(-2)R2$ since $(-2)^2=2^2$. But $2\ne -2$.