Is an Anti-Symmetric Relation also Reflexive? According to the definition of an Anti-Symmetric Relation if xRy and yRx then x = y
Which means, effectively, x is in relation with itself. Does this mean that anti-symmetry implies reflexive property as well?

A relation that is antisymmetric but not reflexive is said to be "strongly antisymmetric" or "asymmetric". 
This implies :
$$(xRy) \implies (\neg(yRx))$$
As if $(xRy)$ and $(yRx)$, then $x=y$ but $x\not R x$ because $R$ is not reflexive (which mean you actually can't apply antisymetry to deduce equality).
$>$ and $<$ are the most common examples.
