Definition:Transitive Closure (Relation Theory)/Finite Chain

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Definition

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


The transitive closure of $\mathcal R$ is the relation $\mathcal R^+$ defined as follows:

For $x, y \in S$, $x \mathrel {\mathcal R^+} y$ if and only if for some $n \in \N_{>0}$ there exist $s_0, s_1, \dots, s_n \in S$ such that $s_0 = x$, $s_n = y$, and:

\(\ds s_0\) \(\mathcal R\) \(\ds s_1\)
\(\ds s_1\) \(\mathcal R\) \(\ds s_2\)
\(\ds \) \(\vdots\) \(\ds \)
\(\ds s_{n - 1}\) \(\mathcal R\) \(\ds s_n\)


That is:

$\forall k \in \N_n: s_k \mathrel {\mathcal R} s_{k+1}$


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