Antisymmetric relation
In mathematics, a binary relation R over a set X is antisymmetric if it holds for all a and b in X that if a is related to b and b is related to a then a = b.
In notation, this is:
- <math>\forall a, b \in X,\ a R b \and b R a \; \Rightarrow \; a = b<math>
Strict inequality is antisymmetric; since a < b and b < a is impossible, the antisymmetry condition is vacuously true.
Note that antisymmetry is not the opposite of symmetry (aRb implies bRa). There are relations which are both symmetric and antisymmetric (equality), there are relations which are neither symmetric nor antisymmetric (divisibility on the integers), there are relations which are symmetric and not antisymmetric (congruence modulo n), and there are relations which are not symmetric but anti-symmetric ("is less than" ).
As noted above the condition of antisymmetry of "is less than" is vacuously true. The relation "is less than or equal to" is not symmetric but is antisymmetric, and the antisymmetric condition is not vacuous.
An antisymmetric relation that is also transitive and reflexive is a partial order.
