Antisymmetric relation

In mathematics, a binary relation R on a set X is antisymmetric if, for all a and b in X, if a is related to b and b is related to a, then a = b.

In mathematical notation, this is:

$\forall a, b \in X,\ a R b \and b R a \; \Rightarrow \; a = b$

Strict inequality is antisymmetric; 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 antisymmetric (equality), relations which are neither symmetric nor antisymmetric (divisibility on the integers), relations which are symmetric and not antisymmetric (congruence modulo n), and relations which are not symmetric but are anti-symmetric ("is less than").

As noted above the condition of antisymmetry of "is less than" is vacuously true. The relation "is less than or equal to" is not symmetric but is antisymmetric, and the antisymmetric condition is not vacuous.

An antisymmetric relation that is also transitive and reflexive is a partial order.

A relation R on X is asymmetric if, for all a and b in X, if a is related to b, then b is not related to a.

In mathematical notation, this is:

$\forall a, b \in X,\ a R b \; \Rightarrow \lnot(b R a)$ .

Asymmetric is the same as antisymmetric and non-reflexive.

Categories: Set theory