Transitive Closure (Relation Theory)/Examples

Examples of Transitive Closures

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:

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