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