# Equivalence of Definitions of Reflexive Transitive Closure

## 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:

Transitive Closure of Reflexive Relation is Reflexive
Reflexive Closure of Transitive Relation is Transitive
Composition of Compatible Closure Operators

$\blacksquare$