Axiom:Axiom of Extension/Set Theory

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Axiom

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:

every element of $A$ is also an element of $B$

and:

every element of $B$ is also an element of $A$.


This can be formulated as follows:


Formulation 1

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


Formulation 2

In set theories that define $=$ instead of admitting it as a primitive, the Axiom of Extension can be formulated as:

$\forall x: \paren {\paren {A = B \land A \in x} \implies B \in x}$


The order of the elements in the sets is immaterial.

Hence a set is completely and uniquely determined by its elements.


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.


Sources

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