Antisymmetric

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.

Note that antisymmetry is not the opposite of symmetry? (aRb implies bRa), that is, it does not necessarily hold for an antisymmetric relation that aRb implies that not bRa. In fact, a binary relation can be antisymmetric and symmetric? at the same time if, and only if, no two different elements are related.