Main Index
Foundation of mathematics
Relations
Subject Index
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Main Index
Foundation of mathematics
Relations
Subject Index
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A function 
from a set 
to a set 
associates to each element 
an element 
, in prefix notation 
. Often the parentheses around 
are omitted, for instance 
. Also the so-called reverse Polish notation 
, 
, is used, for instance in factorial notation 
.
In other words, a function from X to Y is a single-valued, total relation between X and Y, that is, for every 
there exists a unique element 
such that 
.
Since function concept is a special case of the concept of a relation, many notions defined for relations can be applied to functions.
The set 
is called the domain of 
, often denoted by 
. The target set 
is called the codomain of 
, denoted by 
.
A specific input 
in a function is called an argument and the corresponding unique 
in the codomain is called the function value at 
, or the image of 
under 
.
The concept of the image can be extended to the image of a set. If 
, then 
is the subset of the range consisting of all images of elements of 
and is called the image of 
under 
, denoted 
.
The set 
is called the range of 
. The range of 
is also called the image of 
, denoted by 
. In other words, 
.
The preimage (or inverse image) of a subset 
of the codomain 
under a function 
is the subset of the domain 
defined by
The notation 
is used to indicate that 
is a function with domain X and codomain Y. Another forms of notations are 
, or 
.
If 
and 
are two functions, then a composite function is a function 
defined by 
for all 
. The notation 
and read 
composed with 
. If function 
and 
are considered as relations then the composition 
of relations 
is again a function.
The composition of functions is always associative.
If 
, then the composite function 
, denoted 
, is again a function from 
to 
. Repeated composition of a function with itself is called function iteration. The following notation is used 
for every positive integer 
. Clearly, 
. By convention, 
is the identity function 
on the domain 
, defined by 
for each 
.
The identity function 
is neutral in the following sense: if 
then 
and 
.
If 
is a function from 
to 
then an inverse function for 
, denoted by 
, is a function from 
to 
such that if 
then 
. Because every function is a relation, the inverse relation to a function is well defined, but the inverse relation to a function may be not a function. Therefore not every function has an inverse. Moreover, inverse function is uniquely determined. If 
has an inverse then 
is called invertible.
If 
and 
are invertible functions and the inverse function 
exists then 
.
If 
is invertible then 
, 
, is defined by 
.
A restriction 
of a function 
is the result of trimming its domain 
to a subset 
. In other words, the restriction of 
to 
is the function 
from S to Y such that 
for all 
. If 
is any restriction of 
, we say that 
is an extension of 
.
A partial function 
is a partial binary relation from 
to 
that associates each element of domain 
with at most one (possible no) element of codomain 
. This means that (contrary to the definition of a function) not every element of the domain has to be associated with an element of the codomain.
A function 
is called (following Bourbaki)
implies 
; a function 
that is not injective is called many-to-one.
there is an 
such that 
,
is both one-to-one and onto.Consider functions 
and 
, and the composition 
:
and 
are injective, then 
is injective
and 
are surjective, then 
is surjective
and 
are bijective, then 
is bijectiveLet 
, 
be two functions with common domain 
and common codomain 
which is a ring with addition 
and multiplication 
. The we can define two new functions
by 
by 
.This turns the set of all such functions into a ring. Instead of 
we can take any other algebraic structure 
to turn the set of function 
into an algebraic structure of the same type as 
.
For the history of the concept of a function visit .
Cite this web-page as:
Štefan Porubský: Function.