# Equivalence of Definitions of Reflexive Transitive Closure

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## 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:

$\blacksquare$