Axiom:Axiom of Extension

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Axiom

Two sets are equal if and only if they contain the same elements:

$\forall x: \left({x \in A \iff x \in B}\right) \iff A = B$

The order of the elements in the sets is immaterial.


Also defined as

For set theories that define $=$ instead of admitting it as a primitive, the Axiom of Extension becomes:

$\forall x: \left({ \left({ A = B \land A \in x }\right) \implies B \in x }\right)$

where $A$ and $B$ denote sets.


Notes

This is the fundamental definition of what a set is: a set is determined by its elements.


Also known as

Otherwise known as the Axiom of Extensionality or Axiom of Extent.


Also see


Sources