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

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

Let:
 * $\mathcal R^n := \begin{cases}

\mathcal R & : n = 0 \\ \mathcal R^{n-1} \circ \mathcal R & : n > 0 \end{cases}$

where $\circ$ denotes composition of relations.

Finally, let
 * $\displaystyle \mathcal R^+ = \bigcup_{i \in \N} \mathcal R^i$.

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