Equivalence of Definitions of Transitive Relation

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Theorem

The following definitions of the concept of Transitive Relation are equivalent:

Definition 1

$\RR$ is transitive if and only if:

$\tuple {x, y} \in \RR \land \tuple {y, z} \in \RR \implies \tuple {x, z} \in \RR$

that is:

$\set {\tuple {x, y}, \tuple {y, z} } \subseteq \RR \implies \tuple {x, z} \in \RR$

Definition 2

$\RR$ is transitive if and only if:

$\RR \circ \RR \subseteq \RR$

where $\circ$ denotes composite relation.


Proof

Definition 1 implies Definition 2

Let $\RR$ be a relation which fulfills the condition:

$\tuple {x, y} \in \RR \land \tuple {y, z} \in \RR \implies \tuple {x, z} \in \RR$


Then:

\(\ds \tuple {x, z}\) \(\in\) \(\ds \RR \circ \RR\)
\(\ds \leadsto \ \ \) \(\ds \exists y \in \RR: \, \) \(\ds \tuple {x, y}\) \(\in\) \(\ds \RR\) Definition of Composition of Relations
\(\, \ds \land \, \) \(\ds \tuple {y, z}\) \(\in\) \(\ds \RR\)
\(\ds \leadsto \ \ \) \(\ds \tuple {x, z}\) \(\in\) \(\ds \RR\) Definition 1 of Transitive Relation
\(\ds \leadsto \ \ \) \(\ds \RR \circ \RR\) \(\subseteq\) \(\ds \RR\) Definition of Subset


Thus $\RR$ is transitive by definition 2.

$\Box$


Definition 2 implies Definition 1

Let $\RR$ be a relation that fulfills the condition:

$\RR \circ \RR \subseteq \RR$

Aiming for a contradiction, suppose $\RR$ does not fulfill the condition:

$\tuple {x, y} \in \RR \land \tuple {y, z} \in \RR \implies \tuple {x, z} \in \RR$

Then:

\(\ds \exists \paren {\tuple {x, y} \in \RR \land \tuple {y, z} \in \RR}: \, \) \(\ds \tuple {x, z}\) \(\notin\) \(\ds \RR\) Definition of Non-Transitive Relation
\(\ds \leadsto \ \ \) \(\ds \exists \tuple {x, z} \in \RR \circ \RR: \, \) \(\ds \tuple {x, z}\) \(\notin\) \(\ds \RR\) Definition of Composition of Relations
\(\ds \leadsto \ \ \) \(\ds \RR \circ \RR\) \(\nsubseteq\) \(\ds \RR\) Definition of Subset

This contradicts our statement that $\RR \circ \RR \subseteq \RR$.

Hence by Proof by Contradiction $\RR$ does fulfills the condition:

$\tuple {x, y} \in \RR \land \tuple {y, z} \in \RR \implies \tuple {x, z} \in \RR$


Thus $\RR$ is transitive by definition 1.

$\blacksquare$


Sources

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