antisymmetric

A relation $\mathcal{R}$ on $A$ is antisymmetric iff $\forall x,y\in A$ , $(x\mathcal{R}y\land y\mathcal{R}x)\rightarrow(x=y)$ . For a finite set $A$ with $n$ elements, the number of possible antisymmetric relations is $2^{n}3^{\frac{n^{2}-n}{2}}$ out of the $2^{n^{2}}$ total possible relations.

Antisymmetric is not the same thing as “not symmetric”, as it is possible to have both at the same time. However, a relation $\mathcal{R}$ that is both antisymmetric and symmetric has the condition that $x\mathcal{R}y\Rightarrow x=y$ . There are only $2^{n}$ such possible relations on $A$ .

An example of an antisymmetric relation on $A=\{\circ,\times,\star\}$ would be $\mathcal{R}=\{(\star,\star),(\times,\circ),(\circ,\star),(\star,\times)\}$ . One relation that isn’t antisymmetric is $\mathcal{R}=\{(\times,\circ),(\star,\circ),(\circ,\star)\}$ because we have both $\star\mathcal{R}\circ$ and $\circ\mathcal{R}\star$ , but $\circ\not=\star$

Title	antisymmetric
Canonical name	Antisymmetric
Date of creation	2013-03-22 12:15:50
Last modified on	2013-03-22 12:15:50
Owner	aoh45 (5079)
Last modified by	aoh45 (5079)
Numerical id	14
Author	aoh45 (5079)
Entry type	Definition
Classification	msc 03E20
Synonym	antisymmetry
Related topic	Reflexive
Related topic	Symmetric
Related topic	ExteriorAlgebra
Related topic	SkewSymmetricMatrix