Antisymmetric relation

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).