Definition:Transitive Closure (Relation Theory)

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Definition

Smallest Transitive Superset

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.


Intersection of Transitive Supersets

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 $\RR$.


Finite Chain

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:

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


Union of Compositions

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 $\RR$ as the ancestral of $\RR$.

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


Examples

Arbitrary Example $1$

Let $S = \set {1, 2, 3}$ be a set.

Let $\RR$ be the relation on $S$ defined as:

$\RR = \set {\tuple {1, 2}, \tuple {2, 2}, \tuple {2, 3} }$

The transitive closure $\RR^+$ of $\RR$ is given by:

$\RR^+ = \set {\tuple {1, 2}, \tuple {2, 2}, \tuple {2, 3}, \tuple {1, 3} }$


Arbitrary Example $2$

Let $S = \set {1, 2, 3, 4, 5}$ be a set.

Let $\RR$ be the relation on $S$ defined as:

$\RR = \set {\tuple {1, 2}, \tuple {2, 3}, \tuple {3, 4}, \tuple {5, 4} }$

The transitive closure $\RR^+$ of $\RR$ is given by:

$\RR^+ = \set {\tuple {1, 2}, \tuple {1, 3}, \tuple {1, 4}, \tuple {2, 3}, \tuple {2, 4}, \tuple {3, 4}, \tuple {5, 4} }$


Also see

  • Results about transitive closures can be found here.


Sources

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