Asymmetric relation

In mathematics, an asymmetric relation is a binary relation on a set X where

• For all a and b in X, if a is related to b, then b is not related to a.

This can be written in the notation of first-order logic as

$\forall a,b\in X:aRb\rightarrow \lnot (bRa).$

A logically equivalent definition is $\forall a,b\in X:\lnot (aRb\wedge bRa).$ An example of an asymmetric relation is the "less than" relation < between real numbers: if x < y, then necessarily y is not less than x. The "less than or equal" relation ≤, on the other hand, is not asymmetric, because reversing e.g. x ≤ x produces x ≤ x and both are true. Asymmetry is not the same thing as "not symmetric": the less-than-or-equal relation is an example of a relation that is neither symmetric nor asymmetric. The empty relation is the only relation that is (vacuously) both symmetric and asymmetric.

Properties

• A relation is asymmetric if and only if it is both antisymmetric and irreflexive.
• Restrictions and converses of asymmetric relations are also asymmetric. For example, the restriction of < from the reals to the integers is still asymmetric, and the inverse > of < is also asymmetric.
• A transitive relation is asymmetric if and only if it is irreflexive: if aRb and bRa, transitivity gives aRa, contradicting irreflexivity.
• As a consequence, a relation is transitive and asymmetric if and only if it is a strict partial order.
• Not all asymmetric relations are strict partial orders. An example of an asymmetric non-transitive, even antitransitive relation is the rock-paper-scissors relation: if X beats Y, then Y does not beat X; and if X beats Y and Y beats Z, then X does not beat Z.
• An asymmetric relation need not have the connex property. For example, the strict subset relation ⊊ is asymmetric, and neither of the sets {1,2} and {3,4} is a strict subset of the other.

See also

• Tarski's axiomatization of the reals – part of this is the requirement that < over the real numbers be asymmetric.