Definition:Transitive Relation/Class Theory
< Definition:Transitive Relation(Redirected from Definition:Transitive Relation (Class Theory))
Jump to navigation
Jump to search
Definition
Let $V$ be a basic universe.
Let $\RR \subseteq V \times V$ be a relation in $V$.
$\RR$ 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:
- $\set {\tuple {x, y}, \tuple {y, z} } \subseteq \RR \implies \tuple {x, z} \in \RR$
Also see
- Results about relation transitivity can be found here.
Sources
- 2010: Raymond M. Smullyan and Melvin Fitting: Set Theory and the Continuum Problem (revised ed.) ... (previous) ... (next): Chapter $4$: Superinduction, Well Ordering and Choice: Part $\text I$ -- Superinduction and Well Ordering: $\S 1$ Introduction to well ordering