An antisymmetric relation is a relation where no two distinct elements are related in both directions. In other words. is antisymmetric iff

$(a R b \land b R a) \to a = b$

or, equivalently, $a \neq b \to (\neg a R b \lor \neg b R a)$

Antisymmetry isn't quite the compliment of Symmetry. Due to the fact that $a R a$ 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 ${. . . (9, 3), (10, 5), (10, 2), (14, 7), (14, 2) . . .)}$

Antisymmetric relation