Set - Knowlpedia

In axiomatic set theory, set and the membership relation $\in$ are primitive notions. The axiom of extensionality determines equality by membership: for sets $A,B$ ,

A=B \iff \forall x\,\bigl(x\in A \Leftrightarrow x\in B\bigr).

Axiomatic role

Extensionality is not a complete axiomatization of set theory. Other axioms specify which sets exist and how sets may be formed.

Remarks

Many basic constructions in set theory are specified by describing their elements, such as union, intersection, and the power set.

Examples

The set of natural numbers $\mathbb{N}$ (see natural numbers).
For a real number $a$ , the singleton $\{a\}=\{x : x=a\}$ is the set containing exactly the element $a$ .