as the title says I am trying to prove if the relation in quesiton is anti-symmetric or not with respect ot the domain of integers. Can the following procedure be applied?

$R$ is antisymmetric if for every integer $x,y$, if $(x,y)\in R$ and $(y,x)\in R$, then $x=y$.

We could try to give a direct proof of this:

if $(x,y)\in R$, then $x\geq y^{2}$ so $x\geq y$. Since $(y,x)\in R$ by assumption, then $y\geq x^{2}$ so $y\geq x$.

But if $y\geq x$ and $x\geq y$, then $x=y$. So the relation is anti-symmetric

## Best Answer

As noted in the comments, this is correct.

Mostly just posting this to get this out of the unanswered queue. Posting as Community Wiki in particular since I have nothing further to add.