Definition:Transitive Closure (Relation Theory)/Finite Chain

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

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

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

That is:


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

Also see

 * Equivalence of Definitions of Transitive Closure (Relation Theory)