Table of Contents - 5.8. Reflexive and transitive closures of relations
A relation, , has the reflexive property if holds whenever
is an element which could be related by the relation, namely, an
element of its domain or range. Eliminating dummy variables, we see that a
segment of the identity relation must be a subset of the relation, or
. See idref.
A relation, , has the transitive property if holds whenever
there exists an intermediate value such that both and
hold. This can be expressed without dummy variables as
. See cotr.
The transitive closure of a relation, , is the smallest
superset of the relation which has the transitive property. Likewise, the
reflexive-transitive closure, , is the smallest superset
which has both the reflexive and transitive properties.
Not to be confused with the transitive closure of a set, trcl, which is a
closure relative to a different transitive property, df-tr.