A relation R on A is symmetric if for all , whenever then . In a symmetric relation, the matrix is symmetric around the main diagonal. In an antisymmetric relation, the only symmetric entries are one the diagonal, i.e. the only back arcs are self-loops. If there is a path from a to c in a transitive relation, there must be a single arc from a to c . The transitive closure of a relation R adds all the arcs to R necessary to make it transitive.
Ancestor is the transitive closure of parent .
The universal relation . The empty relation . The identity relation is an equivalence relation if it is reflexive, symmetric, and transitive. Equivalence relations partition the elements into equivalence classes of identically-behaving elements. The universal and identity relations are both equivalence relations.