# Definition:Transitive Closure (Relation Theory)

## Definition

### Definition 1

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

The transitive closure of $\RR$ is defined as the smallest transitive relation on $S$ which contains $\RR$ as a subset.

### Definition 2

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

The transitive closure of $\RR$ is defined as the intersection of all transitive relations on $S$ which contain $\RR$.

### Definition 3

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:

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

### Definition 4

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

Let:

$\RR^n := \begin{cases} \RR & : n = 1 \\ \RR^{n-1} \circ \RR & : n > 1 \end{cases}$

where $\circ$ denotes composition of relations.

Finally, let:

$\ds \RR^+ = \bigcup_{i \mathop = 1}^\infty \RR^i$

Then $\RR^+$ is called the transitive closure of $\RR$.

## Also known as

Some authors refer to the transitive closure of $\mathcal R$ as the ancestral of $\mathcal R$.

The symbolism varies: some authors use $\mathcal R^t$. The literature is inconsistent, so any notation needs explanation when used on $\mathsf{Pr} \infty \mathsf{fWiki}$.