Restriction of Transitive Relation is Transitive
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Theorem
Let $S$ be a set.
Let $\RR \subseteq S \times S$ be a transitive relation on $S$.
Let $T \subseteq S$ be a subset of $S$.
Let $\RR {\restriction_T} \subseteq T \times T$ be the restriction of $\RR$ to $T$.
Then $\RR {\restriction_T}$ is a transitive relation on $T$.
Proof
Suppose $\RR$ is transitive on $S$.
Then by definition:
- $\set {\tuple {x, y}, \tuple {y, z} } \subseteq \RR \implies \tuple {x, z} \in \RR$
So:
\(\ds \set {\tuple {x, y}, \tuple {y, z} }\) | \(\subseteq\) | \(\ds \RR {\restriction_T}\) | ||||||||||||
\(\ds \leadsto \ \ \) | \(\ds \set {\tuple {x, y}, \tuple {y, z} }\) | \(\subseteq\) | \(\ds \paren {T \times T} \cap \RR\) | Definition of Restriction of Relation | ||||||||||
\(\ds \leadsto \ \ \) | \(\ds \tuple {x, z}\) | \(\in\) | \(\ds \paren {T \times T} \cap \RR\) | $\RR$ is transitive on $S$ | ||||||||||
\(\ds \leadsto \ \ \) | \(\ds \tuple {x, z}\) | \(\in\) | \(\ds \RR {\restriction_T}\) | Definition of Restriction of Relation |
Therefore, if $x, y, z \in T$, it follows that:
- $\set {\tuple {x, y}, \tuple {y, z} } \subseteq \RR {\restriction_T} \implies \tuple {x, z} \in \RR {\restriction_T}$
and so by definition $\RR {\restriction_T}$ is a transitive relation on $T$.
$\blacksquare$
Also see
- Properties of Restriction of Relation‎ for other similar properties of the restriction of a relation.
Sources
- 1965: Seth Warner: Modern Algebra ... (previous) ... (next): Chapter $\text {III}$: The Natural Numbers: $\S 14$: Orderings