# 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?

• Two familiar order relations on say the reals are $\lt$ and $\le$. The first is not reflexive, while the second is. – André Nicolas Sep 22 '15 at 16:08

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.