Symmetric Relation

Definition

Given a set $A$, a relation $\sim :A\to A\to \mathrm{Prop}$ is symmetric iff $\forall x,y:A.x\sim y\to y\sim x.$

Asymmetry

A relation $\sim$ is asymmetric iff $\forall x,y:A.x\sim y\to \neg (y\sim x).$ Asymmetry implies irreflexivity: no element can be related to itself.

Related Concepts

• Reflexive Relation
• Transitive Relation
• Equivalence Relation
• Anti-Symmetric Relation
• Predicate

References

• munkres2000-topology