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 $\RR$.

Also see

• Equivalence of Definitions of Transitive Closure (Relation Theory)

• Results about transitive closures can be found here.

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