relations

A relation R is reflexive if

• \( \forall x \in X {\tiny\,\bullet\,} x~R~x\)

A relation R is irreflexive if

• \( \forall x \in X {\tiny\,\bullet\,} \lnot(x~R~x)\)

A relation R is symmetric if

• \( \forall x,y \in X {\tiny\,\bullet\,} x~R~y \implies y~R~x\)

A relation R is asymmetric if

• \( \forall x,y \in X {\tiny\,\bullet\,} x~R~y \implies \lnot(y~R~x)\)

A relation R is antisymmetric if

• \( \forall x,y \in X {\tiny\,\bullet\,} x~R~y \land y~R~x \implies x=y\)

A relation R is transitive if

• \( \forall x,y,z \in X {\tiny\,\bullet\,} x~R~y \land y~R~z \implies x~R~z\)

A relation \(R\) is an equivalence relation if it is reflexive, symmetric, and transitive. The set of all elements equivalent to \(x\) (related to \(x\) by the equivalence relation) is the equivalence class of \(x\) under that relation. The equivalence classes of a set \(X\) under a relation partition the set \(X\).