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
Toronto Raptors

... However Carter, a coach without playoff experience, and the lack of overall playoff experiences of the team precipitated in a sweep by the Knicks. Progresive tea ...