representing anti-simmetric relations with directed graphs

discrete-mathematics relations

Solution 1:

If the arrows go only one way, then there cannot be a case of distinct nodes $a,b$ where both $a$ points to $b$ and $b$ points to $a$.

In an antisymetric relation, all elements that are related to and from each other are the same element.   That is $\forall a\,\forall b~((a\mathrel Rb \land b\mathrel R a)\to a=b)~$ which is:

$$\forall a\,\forall b\,(a\neq b\to\lnot (a\mathrel R b\land b\mathrel R a))$$

So the relation over the nodes of "arrow points to" is by definition antisymmetric when "all the arrows only go one way".

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