Definition:Transitive Closure (Relation Theory)/Finite Chain

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

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

For $x, y \in S$, $x \mathrel {\RR^+} y$ 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:

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

That is:

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

Also see

  • Results about transitive closures can be found here.