> 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/260-function-theory/topics/function-statements.md).

# Functional statements

* function is a set
* function is a set the elements of which are ordered pairs
* function is a relation
* function is a binary relation that is functional and serial
* function is a binary relation that is right-unique and left-total
* function (relation) between two sets is an element of their dot product
* identity function is a mapping from a set back to itself
* identity function is identical to the reflexive closure

## misc/unsorted

* a relation that is functional means exactly that the relation is a function
* A relation that is injective and functional is precisely an injective function on its domain. It is a bijection of its domain with its range.
* relation need not be total; functions must be total (for now)
* R is a function X -> Y if for each x ∈ X, there is exactly one y ∈ Y such that xRy
* R is a bijection between X and Y if R and its inverse are both functions

## Synonyms and near terms

Function, \~AKA

* map
* mapping
* transformation
* correspondence
  * one-to-one correspondence
* adjection
  * surjection, onto
  * injection, one-to-one
  * bijection, one-to-one correspondence, inverse is a function
* association
* functional relation
* morphism (homomorphism, isomorphism)
* arrow
* routine, subroutine
* procedure, subprocedure