 Union of antisymmetric relations

The union of two antisymmetric relations need not be an antisymmetric relation. On $\{0,1\}$, take $R=\{(0,1)\}$ and $S=\{(1,0)\}$. Each is an antisymmetric relation, but $R\cup S$ contains both directions between distinct elements. This counterexample shows why testing each binary relation separately does not control cross-pairs in their union.