How can an inequality be reflexive and anti symmetric While going over my lecture notes for the preparation of exams, I stumbled upon  this;
 
Everything is making sense except the reflexive  and anti symmetric relation of natural numbers.
1) By the definition of inequality , we can say that that $(x, x)$ would not appear as an ordered pair.
2)How can it be anti symmetric if the hypothesis $(x,y) \wedge (y, x)$ does not hold? Wouldn't it be meaningless to say anything about the implication if the hypothesis is false?

The last relation is, indeed, not reflexive, as you have surmised. However, it is antisymmetric, since there are no $x,y$ such that $$(x\:R\:y)\wedge(y\:R\:x)\wedge(x\ne y).$$ After all, as you said, there are no $x,y$ such that $x\:R\:y$ and $y\:R\:x$ in the first place! So, one might say that it is vacuously antisymmetric.
