Symmetry

Definition: Symmetric Relation

A relation $R\subseteq A\times A$ is symmetric if

$$xRy⟹yRx$$

for all $x,y\in A$.

Definition: Asymmetric Relation

A relation $R\subseteq A\times A$ is asymmetric if

$$((xRy)\wedge (yRx))⟹x=y$$

for all $x,y,\in A$