> For the complete documentation index, see [llms.txt](https://mandober.gitbook.io/math-debrief/llms.txt). Markdown versions of documentation pages are available by appending `.md` to page URLs; this page is available as [Markdown](https://mandober.gitbook.io/math-debrief/200-set-theory/topics/set-notation.md).

# Set notation

A set can be described intensionally or extensionally.

> An **intensional definition** gives the meaning of a term by specifying necessary and sufficient conditions for when the term should be used.
>
> An **extensional definition** gives the meaning of a term by specifying its extension, i.e. specifying every object that falls under the definition of the term in question.

Set notations:

* sets are implicitly denoted using capital letters
* sets are explicitly denoted using braces
* Intensional set definition
  * semantic declaration: "a set containing odd natural numbers"
  * roster notation: directly enumerating its elements, `{2,4,6,8,10}`
    * finite sets may use ellipsis and a terminal element
    * infinite set may use ellipsis
* Extensional set definition
  * set-builder notation: lists properties and conditions members must satisfy