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