# 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:

$\displaystyle \mathcal R^+ = \bigcup_{i \mathop \in \N} \mathcal R^i$

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