Definition:Transitive Closure (Relation Theory)/Union of Compositions

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

Let:


 * $\RR^n := \begin{cases}

\RR & : n = 1 \\ \RR^{n-1} \circ \RR & : n > 1 \end{cases}$

where $\circ$ denotes composition of relations.

Finally, let:


 * $\ds \RR^+ = \bigcup_{i \mathop = 1}^\infty \RR^i$

Then $\RR^+$ is called the transitive closure of $\RR$.