![Codomain2](https://upload.wikimedia.org/wikipedia/commons/6/64/Codomain2.SVG)
In
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, the image of a
function
Function or functionality may refer to:
Computing
* Function key, a type of key on computer keyboards
* Function model, a structured representation of processes in a system
* Function object or functor or functionoid, a concept of object-oriente ...
is the set of all output values it may produce.
More generally, evaluating a given function
at each
element of a given subset
of its
domain
Domain may refer to:
Mathematics
*Domain of a function, the set of input values for which the (total) function is defined
**Domain of definition of a partial function
**Natural domain of a partial function
**Domain of holomorphy of a function
* Do ...
produces a set, called the "image of
under (or through)
". Similarly, the inverse image (or preimage) of a given subset
of the
codomain
In mathematics, the codomain or set of destination of a function is the set into which all of the output of the function is constrained to fall. It is the set in the notation . The term range is sometimes ambiguously used to refer to either the ...
of
is the set of all elements of the domain that map to the members of
Image and inverse image may also be defined for general
binary relations
In mathematics, a binary relation associates elements of one set, called the ''domain'', with elements of another set, called the ''codomain''. A binary relation over sets and is a new set of ordered pairs consisting of elements in and in ...
, not just functions.
Definition
The word "image" is used in three related ways. In these definitions,
is a
function
Function or functionality may refer to:
Computing
* Function key, a type of key on computer keyboards
* Function model, a structured representation of processes in a system
* Function object or functor or functionoid, a concept of object-oriente ...
from the
set
Set, The Set, SET or SETS may refer to:
Science, technology, and mathematics Mathematics
*Set (mathematics), a collection of elements
*Category of sets, the category whose objects and morphisms are sets and total functions, respectively
Electro ...
to the set
Image of an element
If
is a member of
then the image of
under
denoted
is the
value
Value or values may refer to:
Ethics and social
* Value (ethics) wherein said concept may be construed as treating actions themselves as abstract objects, associating value to them
** Values (Western philosophy) expands the notion of value beyo ...
of
when applied to
is alternatively known as the output of
for argument
Given
the function
is said to "" or "" if there exists some
in the function's domain such that
Similarly, given a set
is said to "" if there exists
in the function's domain such that
However, "" and "" means that
for point
in
's domain.
Image of a subset
Throughout, let
be a function.
The under
of a subset
of
is the set of all
for
It is denoted by
or by
when there is no risk of confusion. Using
set-builder notation
In set theory and its applications to logic, mathematics, and computer science, set-builder notation is a mathematical notation for describing a set by enumerating its elements, or stating the properties that its members must satisfy.
Defining ...
, this definition can be written as
This induces a function
where
denotes the
power set
In mathematics, the power set (or powerset) of a set is the set of all subsets of , including the empty set and itself. In axiomatic set theory (as developed, for example, in the ZFC axioms), the existence of the power set of any set is po ...
of a set
that is the set of all
subset
In mathematics, Set (mathematics), set ''A'' is a subset of a set ''B'' if all Element (mathematics), elements of ''A'' are also elements of ''B''; ''B'' is then a superset of ''A''. It is possible for ''A'' and ''B'' to be equal; if they are ...
s of
See below for more.
Image of a function
The ''image'' of a function is the image of its entire
domain
Domain may refer to:
Mathematics
*Domain of a function, the set of input values for which the (total) function is defined
**Domain of definition of a partial function
**Natural domain of a partial function
**Domain of holomorphy of a function
* Do ...
, also known as the
range
Range may refer to:
Geography
* Range (geographic), a chain of hills or mountains; a somewhat linear, complex mountainous or hilly area (cordillera, sierra)
** Mountain range, a group of mountains bordered by lowlands
* Range, a term used to i ...
of the function. This last usage should be avoided because the word "range" is also commonly used to mean the
codomain
In mathematics, the codomain or set of destination of a function is the set into which all of the output of the function is constrained to fall. It is the set in the notation . The term range is sometimes ambiguously used to refer to either the ...
of
Generalization to binary relations
If
is an arbitrary
binary relation
In mathematics, a binary relation associates elements of one set, called the ''domain'', with elements of another set, called the ''codomain''. A binary relation over Set (mathematics), sets and is a new set of ordered pairs consisting of ele ...
on
then the set
is called the image, or the range, of
Dually, the set
is called the domain of
Inverse image
Let
be a function from
to
The preimage or inverse image of a set
under
denoted by
is the subset of
defined by
Other notations include
and
The inverse image of a
singleton set
In mathematics, a singleton, also known as a unit set or one-point set, is a set with exactly one element. For example, the set \ is a singleton whose single element is 0.
Properties
Within the framework of Zermelo–Fraenkel set theory, the ...
, denoted by