Antisymmetric relation

An antisymmetric relation is a relation where no two distinct elements are related in both directions. In other words. $R$ is antisymmetric iff

$(aRb\wedge bRa)\to a=b$

and consequently, $a\ne b\to \neg (aRb\wedge bRa)$)

Antisymmetry is not the negation of Symmetry. Due to the fact that $aRa$ is allowed in an antisymmetric relation, the equivalence relation, $\left\{\right(0,0),(1,1),(2,2)...\}$ is both symmetric and antisymmetric.

Examples also include the successor relation ($\left\{\right(0,1),(1,2),(2,3),(3,4)...\}$), or a relation linking numbers to their prime factors ($\{...(9,3),(10,5),(10,2),(14,7),(14,2)...)\}$)