Equivalence of Definitions of Transitive Closure (Relation Theory)/Union of Compositions is Smallest

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

Let:
 * $\RR^n := \begin{cases}

\RR & : n = 0 \\ \RR^{n - 1} \circ \RR & : n > 0 \end{cases}$

where $\circ$ denotes composition of relations.

Finally, let:
 * $\ds \RR^+ = \bigcup_{i \mathop \in \N} \RR^i$

Then $\RR^+$ is the smallest transitive relation on $S$ that contains $\RR$.

$\RR^+$ is Transitive
By Relation contains Composite with Self iff Transitive, we can prove that $\RR^+$ is transitive by proving the following:
 * $\RR^+ \circ \RR^+ \subseteq \RR^+$

Let $\tuple {a, c} \in \RR^+ \circ \RR^+$.

Then:
 * $\exists b \in S: \tuple {a, b} \in \RR^+, \tuple {b, c} \in \RR^+$

Thus:
 * $\exists n \in \N: \tuple {a, b} \in \RR^n$
 * $\exists m \in \N: \tuple {b, c} \in \RR^m$

From Composition of Relations is Associative:
 * $\RR^{n + m} = \RR^n \circ \RR^m$

so:
 * $\tuple {a, c} \in \RR^{n + m} \subseteq \RR^+$

Since this holds for all $\tuple {a, c} \in \RR^+ \circ \RR^+$:
 * $\RR^+ \circ \RR^+ \subseteq \RR^+$

Thus $\RR^+$ is transitive.

$\RR^+$ contains $\RR$
$\RR \subseteq \RR^+$ by Set is Subset of Union of Family.

$\RR^+$ is Smallest
Let $\RR'$ be a transitive relation on $S$ such that $\RR \subseteq \RR'$.

We must show that $\RR^+ \subseteq \RR'$.

Let $\tuple {a, b} \in \RR^+$.

That is:
 * $a \mathrel \RR b$

Then:
 * $\exists n \in \N: \tuple {a, b} \in \RR^n$

Thus by the definition of composition of relations, there exists $x_{n - 1} \in S$ such that:
 * $a \mathrel {\RR^{n - 1} } x_{n - 1} \land x_{n - 1} \mathrel \RR b$

Likewise there exists $x_{n - 2} \in S$ such that:
 * $a \mathrel {\RR^{n - 2} } x_{n - 2} \land x_{n - 2} \mathrel \RR x_{n - 1}$

And so forth there exist elements $x_0, \dots, x_n \in S$ such that:
 * $x_0 = a$
 * $x_n = b$
 * $\forall k \in \N_n: x_k \mathrel \RR x_{k + 1}$

Since $\RR \subseteq \RR'$:
 * $\forall k \in \N_n: x_k \mathrel {\RR'} x_{k + 1}$

Since $\RR'$ is transitive:
 * $a \mathrel {\RR'} b$

That is:
 * $\tuple {a, b} \in \RR'$

Since this holds for all $\tuple {a, b} \in \RR^+$:
 * $\RR^+ \subseteq \RR'$

Since this holds for all transitive relations $\RR'$ that contain $\RR$:

$\RR^+$ is the smallest transitive relation containing $\RR$.