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✗ 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. | 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.

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

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.

The divisibility relation on the natural numbers is an important example of an antisymmetric relation. In this context, antisymmetry means that the only way each of two numbers can be divisible by the other is if the two are, in fact, the same number; equivalently, if and are distinct and is a factor of then cannot be a factor of For example, 12 is divisible by 4, but 4 is not divisible by 12.

The usual order relation on the real numbers is antisymmetric: if for two real numbers and both inequalities and hold then and must be equal. Similarly, the subset order on the subsets of any given set is antisymmetric: given two sets and if every element in also is in and every element in is also in then and must contain all the same elements and therefore be equal:

Partial and total orders are antisymmetric by definition. A relation can be both symmetric and antisymmetric (in this case, it must be coreflexive), and there are relations which are neither symmetric nor antisymmetric (for example, the "preys on" relation on biological species).

Antisymmetry is different from asymmetry: a relation is asymmetric if and only if it is antisymmetric and irreflexive.

- Reflexive relation – Binary relation that relates every element to itself
- Symmetry in mathematics – Symmetry in mathematics

- Weisstein, Eric W. "Antisymmetric Relation".
*MathWorld*. - Lipschutz, Seymour; Marc Lars Lipson (1997).
*Theory and Problems of Discrete Mathematics*. McGraw-Hill. p. 33. ISBN 0-07-038045-7. - nLab antisymmetric relation