# 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 $X$.