Reflexive Reduction of Transitive Relation is Transitive

Theorem

Let $S$ be a set.

Let $\RR$ be a transitive relation on $S$.

Let $\RR^\ne$ be the reflexive reduction of $\RR$.

Then $\RR^\ne$ is transitive.

Proof

Let $a, b, c \in S$.

Let $a \mathrel {\RR^\ne} b$ and $b \mathrel {\RR^\ne} c$.

By the definition of reflexive reduction:

$a \ne b$ and $a \mathrel \RR b$

$b \ne c$ and $b \mathrel \RR c$

Since $\mathrel \RR$ is transitive:

$a \mathrel \RR c$

$\blacksquare$

Sources

• 1965: Seth Warner: Modern Algebra ... (previous) ... (next): Chapter $\text {III}$: The Natural Numbers: $\S 14$: Orderings: Exercise $14.7: 3^\circ$