Union of antisymmetric relations
= 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.