# Definition:Transitive Closure (Relation Theory)/Finite Chain

## 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:

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