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

$(aRb ∧ bRa) → a = b$

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 $\{…(9,3),(10,5),(10,2),(14,7),(14,2)…)\}$