## Encyclopedia > Antisymmetric relation

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# 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 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).

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