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
\(X\).