Definition:Transitive Closure (Relation Theory)/Smallest Transitive Superset

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Definition

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


The transitive closure of $\mathcal R$ is defined as the smallest transitive relation on $S$ which contains $\mathcal R$ as a subset.


The transitive closure of $\mathcal R$ is denoted $\mathcal R^+$.