# Definition:Transitive Class

## Contents

## Definition

Let $S$ denote a class, which can be either a set or a proper class.

Then $S$ is **transitive** if and only if every element of $S$ is also a subset of $S$.

That is, $S$ is **transitive** if and only if:

- $x \in S \implies x \subseteq S$

## Notation

In order to indicate that a class $S$ is transitive, this notation is often seen:

- $\operatorname{Tr} S$

whose meaning is:

**$S$ is (a) transitive (class or set).**

Thus $\operatorname{Tr}$ can be used as a propositional function whose domain is the class of all classes.

## Also see

## Sources

- 1960: Paul R. Halmos:
*Naive Set Theory*... (previous) ... (next): $\S 12$: The Peano Axioms - 1971: Gaisi Takeuti and Wilson M. Zaring:
*Introduction to Axiomatic Set Theory*: $\S 7.1$