# Definition:Reflexive Transitive Closure

(Redirected from Definition:Transitive Reflexive Closure)

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## 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 $\RR$:

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