Do you mean that for all and ?I see that there is a n so that it holds for all x and for all y (n = 1)
Well, here is how to show antisymmetry. Assume and , i.e., and for some natural , . Then , so by the assumption in the problem statement or . If , then , so . If , then , so again .
If you are having difficulties with other parts, could you describe what they are?