Set - Imported Knowlpedia Content

A set is an object $A$ for which statements of the form $x\in A$ (read “ $x$ is an element of $A$ ”) are meaningful, and whose identity is determined by extensionality: for sets $A,B$ ,

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

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 $a$ .