Transitive Closure (Relation Theory)/Examples/Arbitrary Example 2

Example of Transitive Closure
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:
 * $\RR^+ = \set {\tuple {1, 2}, \tuple {1, 3}, \tuple {1, 4}, \tuple {2, 3}, \tuple {2, 4}, \tuple {3, 4}, \tuple {5, 4} }$

Proof
By Construction of Transitive Closure of Relation:

That completes the phase of the construction which uses Rule $(1)$.

Then:

There are no further elements of $\RR$ for which there are corresponding elements of $\RR^*$ that satisfy Rule $(2)$.

Hence the result.