Anti-Symmetric Relation

Anti-Symmetric Relation

## Definition / Concept

A relation $R$ on $A$ is anti-symmetric if two distinct elements cannot be related in both directions.

## Formula

$$aRb\text{ and }bRa\Rightarrow a=b$$

Equivalently:

$$a\ne b\Rightarrow\left[(a,b)\notin R\text{ or }(b,a)\notin R\right]$$

## Conditions / Special Cases

A relation can be both symmetric and anti-symmetric.

If a relation is both symmetric and anti-symmetric, then:

$$aRb\Rightarrow a=b$$

## Usage

- Used in identifying partial order relations.