A binary relation $R$ over a set $X$ is symmetric when :
$\forall a,b \in X ( aRb \rightarrow bRa )$.
An example of symmetric relation : "... is married to _".
A binary relation $R$ on a set $X$ is asymmetric when :
$\forall a,b \in X ( aRb \rightarrow \lnot(bRa) )$.
An example of asymmetric relation : "... is father of _".
A binary relation $R$ on a set $X$ is antisymmetric if there is no pair of distinct elements of $X$ each of which is related by $R$ to the other; i.e. :
An example of antisymmetric relation : The usual order relation ≤ on the real numbers.
$\forall a,b \in X ((aRb \land bRa) \rightarrow a = b )$.
Every asymmetric relation is also an antisymmetric relation.
Your proposal for asymmetric is :
- if there is not any element where: if (x,y) (y,x);
this "sounds" like :
$\lnot \exists x,y ( xRy \rightarrow yRx )$.
Moving "inward" $\lnot$, it is equivalent to :
$\forall x,y (\lnot ( xRy \rightarrow yRx ))$.
Now we need the equivalence between $p \rightarrow q$ and $\lnot p \lor q$ and De Morgan, to get :
$\forall x,y (xRy \land \lnot (yRx) )$.
This is not the same as the formula in the definition.
According to the definition of asymmetric, if $a$ is father of $b$, then $b$ is not father of $a$, which is reasonable.
According to your "rewritten" condition, for every couple of individuals, we have that the first one is father of the second and the second is not father of the first one, which sound quite unreasonable.
Examples about reflexive and irreflexive (or anti-reflexive) :