Definition:Symmetric Relation

Definition

Let $\mathcal R \subseteq S \times S$ be a relation in $S$.

Definition 1

$\mathcal R$ is symmetric if and only if:

$\tuple {x, y} \in \mathcal R \implies \tuple {y, x} \in \mathcal R$

Definition 2

$\mathcal R$ is symmetric if and only if it equals its inverse:

$\mathcal R^{-1} = \mathcal R$

Definition 3

$\mathcal R$ is symmetric if and only if it is a subset of its inverse:

$\mathcal R \subseteq \mathcal R^{-1}$

Also see

• Equivalence of Definitions of Symmetric Relation

• Definition:Symmetry (Relation)

• Definition:Asymmetric Relation
• Definition:Antisymmetric Relation
• Definition:Non-symmetric Relation

• Results about symmetry of relations can be found here.