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?

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*You're correct. Only the first two relations are reflexive; not the third.

*All three relations (even the third) are indeed anti-symmetric. The third one is vacuously anti-symmetric, since as you pointed out the hypothesis is always false. False hypotheses are a bit unintuitive, but they do happen fairly often in logic. As a convention, we interpret implications with false hypotheses to be true.
