# Antisymmetric relation

In mathematics, a binary relation on a set is **antisymmetric** if there is no pair of *distinct* elements of each of which is related by to the other. More formally, is antisymmetric precisely if for all

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indicates that the column's property is required by the definition of the row's term (at the very left). For example, the definition of an equivalence relation requires it to be symmetric. ✗ indicates that the property may, or may not hold. All definitions tacitly require the homogeneous relation be transitive: for all if and then and there are additional properties that a homogeneous relation may satisfy. |

or equivalently,

The definition of antisymmetry says nothing about whether actually holds or not for any . An antisymmetric relation on a set may be reflexive (that is, for all ), irreflexive (that is, for no ), or neither reflexive nor irreflexive. A relation is asymmetric if and only if it is both antisymmetric and irreflexive.