# Definition:Reflexive Transitive Closure

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## Definition

Let $\mathcal R$ be a relation on a set $S$.

### Smallest Reflexive Transitive Superset

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

### Reflexive Closure of Transitive Closure

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

- $\mathcal R^* = \left({\mathcal R^+}\right)^=$

### Transitive Closure of Reflexive Closure

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

- $\mathcal R^* = \left({\mathcal R^=}\right)^+$