Is an Anti-Symmetric Relation also Reflexive?

Question

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?

Answer 1

Your definition is wrong. The relation $R$ is antisymmetric if, whenever $x\mathrel{R}y$ and $y\mathrel R x$ it holds that $x=y$.

An example of a relation that is antisymmetric but not reflexive is $>$ on the set of integers.

Answer 2

Not really. For example the empty relation is anti-symmetric, but is not reflexive unless the underlying set is empty as well.

I hope this helps $\ddot\smile$.

Answer 3

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.