Axiom:Axiom of Extension/Set Theory

Axiom
Let $A$ and $B$ be sets.

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

That is, :
 * 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 2
In set theories that define $=$ instead of admitting it as a primitive, the axiom of extension can be formulated as:

The order of the elements in the sets is immaterial.

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

Also see

 * Definition:Set Equality
 * Definition:Equals


 * Axiom:Axiom of Extension (Classes)