Definition:Transitive Relation/Definition 2

Definition

Let $\RR \subseteq S \times S$ be a relation in $S$.

$\RR$ is transitive if and only if:

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

where $\circ$ denotes composite relation.

Also see

• Equivalence of Definitions of Transitive Relation

• Results about relation transitivity can be found here.

Sources

• 1955: John L. Kelley: General Topology ... (previous) ... (next): Chapter $0$: Relations
• 1960: Paul R. Halmos: Naive Set Theory ... (previous) ... (next): $\S 10$: Inverses and Composites
• 1967: George McCarty: Topology: An Introduction with Application to Topological Groups ... (previous) ... (next): Chapter $\text{I}$: Sets and Functions: Relations