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 $\mathcal R$:

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


Also see