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

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

or, equivalently, $a\ne b\to (\neg aRb\vee \neg bRa)$

Antisymmetry isn’t quite the compliment 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 of antisymmetric relations also include the successor relation, $\left\{\right(0,1),(1,2),(2,3),(3,4)...\}$, or this relation linking numbers to their prime factors $\{...(9,3),(10,5),(10,2),(14,7),(14,2)...)\}$