relations

A relation R is reflexive if

• forall x in X . x R x

A relation R is irreflexive if

• forall x in X . not (x R x)

A relation R is symmetric if

• forall x, y in X . x R y => y R x

A relation R is asymmetric if

• forall x, y in X . x R y => not (y R x)

A relation R is antisymmetric if

• forall x, y in X . x R y /\ y R x => x = y

A relation R is transitive if

• forall x, y, z in X . x R y /\ y R z => 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.