# Definition:Transitive Closure (Relation Theory)

## 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

- Equivalence of Definitions of Transitive Closure (Relation Theory)
- Recursive Construction of Transitive Closure

- Results about
**transitive closures**can be found**here**.

## Sources

- 1967: Saunders Mac Lane and Garrett Birkhoff:
*Algebra*: $\S 1.3$: Exercise $4$