Equivalence of Definitions of Reflexive Transitive Closure
Jump to navigation
Jump to search
Theorem
Let $\RR$ be a relation on a set $S$.
The following definitions of the concept of Reflexive Transitive Closure are equivalent:
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 $\RR$:
- $\RR^* = \paren {\RR^=}^+$
Proof
The result follows from:
$\blacksquare$