# Definition:Transitive Relation

## Definition

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

### Definition 1

$\RR$ 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:

$\set {\tuple {x, y}, \tuple {y, z} } \subseteq \RR \implies \tuple {x, z} \in \RR$

### Definition 2

$\RR$ is transitive 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 transitive if and only if:

$\tuple {x, y} \in \RR \land \tuple {y, z} \in \RR \implies \tuple {x, z} \in \RR$

that is:

$\set {\tuple {x, y}, \tuple {y, z} } \subseteq \RR \implies \tuple {x, z} \in \RR$

## 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.

### 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.