In mathematics, a binary relation

*R*over a set

*X*is

**antisymmetric**if it holds for all

*a*and

*b*in

*X*that if

*aRb*and

*bRa*then

*a*=

*b*. Many interesting binary relations such as partial orders and total orders have this property.

The relation < on the integers is also anti-symmetric; 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 anti-symmetric (equality), there are relations which are neither symmetric nor anti-symmetric (divisibility on the integers), there are relations which are symmetric and not anti-symmetric (congruence modulo *n*), and there are relations which are not symmetric but anti-symmetric (less-than on the integers).