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

Definition

Let $\RR$ be a relation on a set $S$.

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

The transitive closure of $\RR$ is denoted $\RR^+$.

Also see

• Equivalence of Definitions of Transitive Closure (Relation Theory)

• Results about transitive closures can be found here.

Sources

• 1979: John E. Hopcroft and Jeffrey D. Ullman: Introduction to Automata Theory, Languages, and Computation ... (previous) ... (next): Chapter $1$: Preliminaries: $1.5$ Relations: Closures of Relations