function.math

::[function.on.A.to.B]

## Functions

Functions are a fundamental concept of mathematics that describe mappings
between mathematical structures.  The distinctive characteristic of a function
is that it maps an input to a *unique* output.

::
[\function:on{A}:to{B}]
Describes: f(x)
when: 'A, B is \set'
specifies:
. 'x [.in.]: A'
. 'f(x) [.in.]: B'
satisfies:
. forAll: a
  where: 'a [.in.]: A'
  then:
  . existsUnique: b
    where: 'b [.in.]: B'
    suchThat: 'f(a) = b'
Documented:
. written: "A? \rightarrow B?"
. called: "function on A? to B?"
. overview: "::function.on.A.to.B"
------------------------------------------
Id: "dd66298e-4e01-400b-897b-712c4c0892a1"


::[injective.function]

## Injective Functions

In general, a function can map two distinct inputs to the same output.  If a
particular function always maps distinct inputs to distinct outputs it is
called an *injective function*.

Note: These types of functions are also said to be *one-to-one*.

::
[\injective.function:on{A}:to{B}]
Describes: f(x)
when: 'A, B is \set'
extends: 'f is \function:on{A}:to{B}'
satisfies:
. forAll: x1, x2
  where: 'x1, x2 [.in.]: A'
  then:
  . if: 'f(x1) = f(x2)'
    then: 'x1 = x2'
Documented:
. written:
  . "\textrm{injective function on } A? \textrm{ to } B?"
  . "\textrm{one-to-one function on } A? \textrm{ to } B?"
. called: "injective function on $A?$ to $B?$"
. overview: "::injective.function"
------------------------------------------
Id: "84e7fd56-af43-441b-bc58-8b6093b14443"


::[surjective.function]

Similarly, in general, for the function $f$ where $f$ is a function on $A$ to
$B$ it is possible for there to be an element $b \in B$ such that there doesn't
exist any $a \in A$ that maps to $b$, i.e. $f(a) = b$.

If a particular function has the property that every output is mapped to by
at least one input, it is called a **surjective function**.

A surjective function on $A$ to $B$ satisfies the property
that for every element $b \in B$ there exists at least one element in
$A$ that maps to $b$.  Note, though, that there may be more than one
element.  Surjective functions are also said to be *onto*.

::
[\surjective.function:on{A}:to{B}]
Describes: f(x)
when: 'A, B is \set'
extends: 'f is \function:on{A}:to{B}'
satisfies:
. forAll: b
  where: 'b [.in.]: B'
  then:
  . exists: a
    where: 'a [.in.]: A'
    suchThat: 'f(a) = b'
Documented:
. written:
  . "\textrm{surjective function on } A? \textrm{ to } B?"
  . "\textrm{onto function on } A? \textrm{ to } B? "
. called: "surjective function on $A?$ to $A?$"
. overview: "::surjective.function"
------------------------------------------
Id: "0e38e388-ca17-482f-aec3-8c5c4ff87e12"


::
Functions that are both one-to-one and onto are special because they can be
"undone", and are called bijective functions.
::


[\bijective.function:on{A}:to{B}]
Describes: f(x)
when: 'A, B is \set'
extends:
. 'f is \injective.function:on{A}:to{B}'
. 'f is \surjective.function:on{A}:to{B}'
Documented:
. written: "\textrm{bijective function on } A? \text{ to } B?"
. called: "bijective function on $A?$ to $B?$"
------------------------------------------
Id: "d7b1e64f-1769-4385-a075-fcb4ac6d90a1"


::
The inverse of a function is a function that "undoes" that function.
::


[\function.inverse:of{f}]
Defines: g(x)
using: A, B
when: 'f is \function:on{A}:to{B}'
means: 'g is \function:on{B}:to{A}'
expresses:
. forAll: a
  where: 'a [.in.]: A'
  then: 'g(f(a)) = a'
. forAll: b
  where: 'b [.in.]: B'
  then: 'f(g(b)) = b'
Documented:
. written: "f+?^{-1}"
. called: "inverse of $f?$"
------------------------------------------
Id: "bc3d179e-0d30-4a25-98f9-0a4200d68806"


::
Function composition is a basic method for building functions out of other
functions.
::


[f \.function.compose./ g]
Defines: h(x)
using: A, B, C
when:
. 'g is \function:on{A}:to{B}'
. 'f is \function:on{B}:to{C}'
means: 'h is \function:on{A}:to{C}'
expresses: 'h(x) := f(g(x))'
Documented:
. written: "f+? \circ g+?"
. called: "function composition"
------------------------------------------
Id: "85db5d32-e2a8-4951-94b3-be9093ccf3b0"


::
Some types of functions need more structure, such as a group or ring structure,
to define, but the identify function on a set $A$ does not need any such
structure.
::


[\identity.function:on{A}]
Defines: f(x)
when: 'A is \set'
means: 'f is \function:on{A}:to{A}'
expresses: 'f(x) := x'
Documented:
. written: "\textrm{id}_{A?}"
. called: "identify function on $A?$"
------------------------------------------
Id: "a28941e4-23f3-44e9-bc22-eb7cd53bb819"