# 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 compliment of 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 $$\{...(9,3),(10,5),(10,2),(14,7),(14,2)...)\}$$

Parents: