Antisymmetric relation

Binary relation such that if A is related to B and is different from it then B is not related to A

In mathematics, a binary relation ${\displaystyle R}$ on a set ${\displaystyle X}$ is antisymmetric if there is no pair of distinct elements of ${\displaystyle X}$ each of which is related by ${\displaystyle R}$ to the other. More formally, ${\displaystyle R}$ is antisymmetric precisely if for all ${\displaystyle a,b\in X,}$

$${\displaystyle {\text{if }}\,aRb\,{\text{ with }}\,a\neq b\,{\text{ then }}\,bRa\,{\text{ must not hold}},}$$

$${\displaystyle {\text{if }}\,aRb\,{\text{ and }}\,bRa\,{\text{ then }}\,a=b.}$$

${\displaystyle aRa}$ ${\displaystyle a}$ ${\displaystyle R}$ ${\displaystyle X}$reflexive ${\displaystyle aRa}$ ${\displaystyle a\in X}$irreflexive ${\displaystyle aRa}$ ${\displaystyle a\in X}$asymmetric

Examples

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 ${\displaystyle n}$ and ${\displaystyle m}$ are distinct and ${\displaystyle n}$ is a factor of ${\displaystyle m,}$ then ${\displaystyle m}$ cannot be a factor of ${\displaystyle n.}$ For example, 12 is divisible by 4, but 4 is not divisible by 12.

The usual order relation ${\displaystyle \,\leq \,}$ on the real numbers is antisymmetric: if for two real numbers ${\displaystyle x}$ and ${\displaystyle y}$ both inequalities ${\displaystyle x\leq y}$ and ${\displaystyle y\leq x}$ hold, then ${\displaystyle x}$ and ${\displaystyle y}$ must be equal. Similarly, the subset order ${\displaystyle \,\subseteq \,}$ on the subsets of any given set is antisymmetric: given two sets ${\displaystyle A}$ and ${\displaystyle B,}$ if every element in ${\displaystyle A}$ also is in ${\displaystyle B}$ and every element in ${\displaystyle B}$ is also in ${\displaystyle A,}$ then ${\displaystyle A}$ and ${\displaystyle B}$ must contain all the same elements and therefore be equal:

$${\displaystyle A\subseteq B{\text{ and }}B\subseteq A{\text{ implies }}A=B}$$

Properties

Symmetric and antisymmetric relations

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.

See also

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

References

• 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