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)…)}