# Axiom:Axiom of Extension

## Axiom

The **axiom of extension** is the fundamental definition of the nature of a collection: it is completely determined by its elements.

### Set Theory

Let $A$ and $B$ be sets.

The **axiom of extension** states that $A$ and $B$ are equal if and only if they contain the same elements.

That is, if and only if:

and:

This can be formulated as follows:

- $\forall x: \paren {x \in A \iff x \in B} \iff A = B$

### Class Theory

The **axiom of extension** in the context of class theory has the same form:

Let $A$ and $B$ be classes.

Then:

- $\forall x: \paren {x \in A \iff x \in B} \iff A = B$

## Also known as

The **axiom of extension** is also known as:

- the
**axiom of extensionality** - the
**axiom of extent**.

## Also see

## Linguistic Note

The nature of the **axiom of extension**, or **axiom of extensionality** as it is frequently called, suggests that the **axiom of extent**, ought in fact to be the preferred name, as it gives a precise definition of the **extent** of a collection.

However, the word **extensionality** is a term in logic which determines equality of objects by its external features, as opposed to **intensionality**, which is more concerned with internal structure.