  ## Encyclopedia > Antisymmetric relation

Article Content

# 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).

All Wikipedia text is available under the terms of the GNU Free Documentation License

Search Encyclopedia
 Search over one million articles, find something about almost anything!

Featured Article
 Shoreham, New York ... of 2000, there are 417 people, 145 households, and 126 families residing in the village. The population density is 365.9/km² (938.3/mi²). There are 163 ...  