# Definition:Transitive Class

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## Definition

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

Then $A$ is **transitive** if and only if every element of $A$ is also a subclass of $A$.

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

- $x \in A \implies x \subseteq A$

or:

- $\forall x: \forall y: \paren {x \in y \land y \in A \implies x \in A}$

## Notation

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

- $\operatorname{Tr} A$

whose meaning is:

**$A$ 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 known as

A **transitive class** is also known as a **complete class**.

## Also see

- Results about
**transitive classes**can be found here.

## Also see

- Class has Subclass which is not Element: while there is nothing stopping a class to be such that all its elements are subclasses, it is not possible for all its subclasses to be among its elements.

## 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*: $\S 7.1$ - 2010: Raymond M. Smullyan and Melvin Fitting:
*Set Theory and the Continuum Problem*(revised ed.) ... (previous) ... (next): Chapter $2$: Some Basics of Class-Set Theory: $\S 2$ Transitivity and supercompleteness: Definition $2.1$