Definition:Transitive Relation
Definition
Let $\RR \subseteq S \times S$ be a relation in $S$.
Definition 1
$\RR$ is a transitive relation if and only if:
- $\tuple {x, y} \in \RR \land \tuple {y, z} \in \RR \implies \tuple {x, z} \in \RR$
that is:
- $\set {\tuple {x, y}, \tuple {y, z} } \subseteq \RR \implies \tuple {x, z} \in \RR$
Definition 2
$\RR$ is a transitive relation if and only if:
- $\RR \circ \RR \subseteq \RR$
where $\circ$ denotes composite relation.
Class Theory
In the context of class theory, the definition follows the same lines:
Let $V$ be a basic universe.
Let $\RR \subseteq V \times V$ be a relation in $V$.
$\RR$ is a transitive relation if and only if:
- $\tuple {x, y} \in \RR \land \tuple {y, z} \in \RR \implies \tuple {x, z} \in \RR$
that is:
- $\set {\tuple {x, y}, \tuple {y, z} } \subseteq \RR \implies \tuple {x, z} \in \RR$
Examples
Ancestor Relation
Let $P$ be the set of people.
Let $\sim$ be the relation on $P$ defined as:
- $\forall \tuple {x, y} \in P \times P: x \sim y \iff \text { $x$ is an ancestor of $y$}$
Then $\sim$ is a transitive relation.
Greater Than on Real Numbers
Let $>$ denote the greater than relation on the set of real numbers $\R$.
Then $>$ is a transitive relation, but neither reflexive nor symmetric.
Less Than on Real Numbers
Let $<$ be the usual ordering on the set of real numbers $\R$.
Then $<$ is a transitive relation, but neither reflexive nor symmetric.
Also see
- Results about transitive relation can be found here.