Definition:Reflexive Transitive Closure

Definition

Let $\RR$ be a relation on a set $S$.

Smallest Reflexive Transitive Superset

The reflexive transitive closure of $\RR$ is denoted $\RR^*$, and is defined as the smallest reflexive and transitive relation on $S$ which contains $\RR$.

Reflexive Closure of Transitive Closure

The reflexive transitive closure of $\RR$ is denoted $\RR^*$, and is defined as the reflexive closure of the transitive closure of $\RR$:

$\RR^* = \paren {\RR^+}^=$

Transitive Closure of Reflexive Closure

The reflexive transitive closure of $\RR$ is denoted $\RR^*$, and is defined as the transitive closure of the reflexive closure of $\RR$:

$\RR^* = \paren {\RR^=}^+$

Also see

• Results about reflexive transitive closures can be found here.