Restriction of Non-Transitive Relation is Not Necessarily Non-Transitive

Theorem

Let $S$ be a set.

Let $\RR \subseteq S \times S$ be a non-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 not necessarily a non-transitive relation on $T$.

Proof

Proof by Counterexample:

Let $S = \set {a, b}$.

Let $\RR = \set {\tuple {a, b}, \tuple {b, a}, \tuple {b, b} }$.

$\RR$ is a non-transitive relation, as can be seen by definition.

Now let $T = \set b$.

Then:

$\RR {\restriction_T} = \set {\tuple {b, b} }$

So:

$\forall x, y \in T: \tuple {x, y} \in \RR {\restriction_T} \land \tuple {y, z} \in \RR {\restriction_T} \implies \tuple {y, z} \in \RR {\restriction_T}$

as can be seen by setting $x = y = z = b$.

So $\RR {\restriction_T}$ is a transitive relation on $T$.

That is, $\RR {\restriction_T}$ is not a non-transitive relation on $T$.

$\blacksquare$

Also see

• Properties of Relation Not Preserved by Restriction‎ for other similar results.