> 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/_articles/_mappings.md).

# Types of functions

Types of mappings

* a relation
* a function
  * every function is a relation
  * a function is a relation that is functional and serial
  * functional relation: left-unique (wrt to the ordered pairs)
  * serial relation: right-serial (wrt to the ordered pairs)
* injection: one-to-one mapping
* surjection: onto mapping
* bijective: one-one and onto mapping
* identity function
* inverse function
  * inverse of a mapping is not necessarily a function (inverse relation)
* total function: usual definition of a function
* partial function allows for unassociated elements in the domain

## ajections

Mappings

* injection: one-to-one mapping
* surjection: onto mapping
* bijective: one-to-one and onto mapping

**Injection**: the main propery is that the mapping is *one-to-one*. Injection is a mapping `f: A -> B` associates a unique `a ∈ A` to a unique `b in B`:

* `∀a ∈ A. ∃!b ∈ B. f(a)=b`
* `∀a,a' ∈ A. f(a) = f(a') <-> a = a'`

Normally, a function can associate an element in a codomain to more then one element in a domain; that is, the same element `b` may be at the head of more then one arrow from A to B, `f(a)=b ∧ f(a')=b ∧ a ≠ a'`.

**Surjection** is a function with two properties, one "bad" and one "good". The good property ensures that *the entire codomain is involved in the mapping* - all elements in B are associated. The bad property is that some elements may be associated more then once.

**Bijection** takes an injection and the good property of surjection, so it is a mapping that is both one-to-one and onto (the entire codomain is associated).

## Functional relations

Functions define many interesting relations between sets. *Isomorphism* is the most important relation:

Sets D and C are *isomorphic*\
if there exist a pair of functions\
f:D->C and g:C->D\
such that\
g . f is the identity function for D\
and\
f . g is the identity function for C.\
The maps (functions) f and g are called *isomorphisms*.

D = {...} f:D->C I(D) = g . f C = {...} g:C->D I(C) = f . g

D=\[1,2,3,4], f:D->C, f(d)=c = `f(d)=d*2` : f(\[1,2,3,4]) => \[2,4,6,8] C=\[2,4,6,8], g:C->D, g(c)=d = `g(c)=c/2` : g(\[2,4,6,8]) => \[1,2,3,4]

I(D) = I(d)=d : \[1,2,3,4] => \[1,2,3,4] I(C) = I(c)=c : \[2,4,6,8] => \[2,4,6,8]

I(D) = g.f ? g(f(d)) : g(f(\[1,2,3,4])) => g(\[2,4,6,8]) => \[1,2,3,4] I(C) = f.g ? f(g(c)) : f(g(\[2,4,6,8])) => f(\[2,4,6,8]) => \[2,4,6,8]

...thus, f and g are isomorphisms and the sets D and C are isomorphic.

A function is an isomorphism if and only if it is *bijective* i.e. both, one-one and onto.

The inverse f^-1 of isomorphism f is also an isomorphism, as f^-1 . f and f . f^-1 are both identities.