Definition:Transitive Closure (Relation Theory)/Finite Chain

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

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

For $x, y \in S$, $x \mathrel{\mathcal R^+} y$ iff there exist $s_0, s_1, \dots, s_n \in S$ such that:
 * $s_0 = x$
 * $s_n = y$
 * $\forall k \in \N_n: s_k \mathrel{\mathcal R} s_{k+1}$.

That is:
 * $x \mathrel{\mathcal R} s_1$
 * $s_1 \mathrel{\mathcal R} s_2$
 * $\vdots$
 * $s_{n-1} \mathrel{\mathcal R} y$