Asymmetric

Asymmetry

Asymmetry is the absence of, or a violation of, symmetry.-In organisms:Due to how cells divide in organisms, asymmetry in organisms is fairly usual in at least one dimension, with biological symmetry also being common in at least one dimension....

 often means, simply: not symmetric. In this sense an asymmetric relation is a binary relation

Binary relation

In mathematics, a binary relation on a set A is a collection of ordered pairs of elements of A. In other words, it is a subset of the Cartesian product A2 = . More generally, a binary relation between two sets A and B is a subset of...

 which is not a symmetric relation

Symmetric relation

In mathematics, a binary relation R over a set X is symmetric if it holds for all a and b in X that if a is related to b then b is related to a.In mathematical notation, this is:...

.

That is,.
or equivalently,.

In some texts the word is given the following stronger definition:

• 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:
.

In this sense, a relation is asymmetric if and only if it is both antisymmetric

Antisymmetric relation

In mathematics, a binary relation R on a set X is antisymmetric if, for all a and b in Xor, equivalently,In mathematical notation, this is:\forall a, b \in X,\ R \and R \; \Rightarrow \; a = bor, equivalently,...

 and irreflexive

Reflexive relation

In mathematics, a reflexive relation is a binary relation on a set for which every element is related to itself, i.e., a relation ~ on S where x~x holds true for every x in S. For example, ~ could be "is equal to".-Related terms:...

.
For nonempty relations, asymmetry in the second definition given here implies asymmetry in the first sense, but the reverse does not hold. Empty relations are, vacuously

Vacuous truth

A vacuous truth is a truth that is devoid of content because it asserts something about all members of a class that is empty or because it says "If A then B" when in fact A is inherently false. For example, the statement "all cell phones in the room are turned off" may be true...

, both asymmetric (in the second sense only) and symmetric.

See also

• Symmetry in mathematics
  Symmetry in mathematics
  Symmetry occurs not only in geometry, but also in other branches of mathematics. It is actually the same as invariance: the property that something does not change under a set of transformations....
  
  
• Symmetry
  Symmetry
  Symmetry generally conveys two primary meanings. The first is an imprecise sense of harmonious or aesthetically pleasing proportionality and balance; such that it reflects beauty or perfection...
  
  
• Antisymmetric relation
  Antisymmetric relation
  In mathematics, a binary relation R on a set X is antisymmetric if, for all a and b in Xor, equivalently,In mathematical notation, this is:\forall a, b \in X,\ R \and R \; \Rightarrow \; a = bor, equivalently,...