Antisymmetric relation

A relation $R$ is antisymmetric when $xRy$ and $yRx$ together imply $x=y$.

The property of a binary relation that opposite directed relationships between two elements can hold only when those elements are equal. It does not require every pair to be comparable and does not prohibit a longer directed cycle whose reverse edges are initially absent. Taking reflexive transitive closure can therefore destroy this property.

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.