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

$~$(aRb ∧ bRa) → a = b$~$

or, equivalently, $~$a ≠ b → (¬aRb ∨ ¬bRa)$~$

Antisymmetry isn't quite the [set_theory_compliment compliment] of [symmetric_relation Symmetry]. Due to the fact that $~$aRa$~$ is allowed in an antisymmetric relation, the equivalence relation, $~$\{(0,0), (1,1), (2,2)…\}$~$ is both symmetric and antisymmetric.

Examples of antisymmetric relations also include the successor relation, $~$\{(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)…)\}$~$