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=                              Codomain                              =
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                            Introduction
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In mathematics, the codomain or target set 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 codomain is sometimes referred to
as the range, but that term is ambiguous because it may also refer to
the image.

The codomain is part of a function  if it is defined as described in
1954 by Nicolas Bourbaki, namely a triple , with  a functional subset
of the Cartesian product  and  is the set of first components of the
pairs in  (the 'domain'). The set  is called the 'graph' of the
function. The set of all elements of the form , where  ranges over the
elements of the domain , is called the image of . In general, the
image of a function is a subset of its codomain. Thus, it may not
coincide with its codomain. Namely, a function that is not surjective
has elements  in its codomain for which the equation  does not have a
solution.

An alternative definition of 'function' by Bourbaki [Bourbaki, 'op.
cit.', p. 77], namely as just a functional graph, does not include a
codomain and is also widely used. For example in set theory it is
desirable to permit the domain of a function to be a proper class , in
which case there is formally no such thing as a triple . With such a
definition functions do not have a codomain, although some authors
still use it informally after introducing a function in the form .


                              Examples
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For a function

:f\colon \mathbb{R}\rightarrow\mathbb{R}

defined by

:f\colon\,x\mapsto x^2, \text{ or equivalently }f(x)\ =\ x^2,

the codomain of  is \textstyle \mathbb R, but  does not map to any
negative number.
Thus the image of  is the set \textstyle \mathbb{R}^+_0; i.e., the
interval .

An alternative function  is defined thus:

: g\colon\mathbb{R}\rightarrow\mathbb{R}^+_0
: g\colon\,x\mapsto x^2.

While  and  map a given  to the same number, they are not, in this
view, the same function because they have different codomains. A third
function  can be defined to demonstrate why:

: h\colon\,x\mapsto \sqrt x.

The domain of  must be defined to be \textstyle \mathbb{R}^+_0:

: h\colon\mathbb{R}^+_0\rightarrow\mathbb{R}.

The compositions are denoted

:h \circ f,
:h \circ g.

On inspection,  is not useful. It is true, unless defined otherwise,
that the image of  is not known; it is only known that it is a subset
of \textstyle \mathbb R. For this reason, it is possible that , when
composed with , might receive an argument for which no output is
defined - negative numbers are not elements of the domain of , which
is the square root function.

Function composition therefore is a useful notion only when the
'codomain'  of the function on the right side of a composition (not
its 'image', which is a consequence of the function and could be
unknown at the level of the composition) is a subset of the domain of
the function on the left side.

The codomain affects whether a function is a surjection, in that the
function is surjective if and only if its codomain equals its image.
In the example,  is a surjection while  is not.  The codomain does not
affect whether a function is an injection.

A second example of the difference between codomain and image is
demonstrated by the linear transformations between two vector spaces -
in particular, all the linear transformations from \textstyle
\mathbb{R}^2 to itself, which can be represented by the  matrices with
real coefficients.  Each matrix represents a map with the domain
\textstyle \mathbb{R}^2 and codomain \textstyle \mathbb{R}^2.
However, the image is uncertain.  Some transformations may have image
equal to the whole codomain (in this case the matrices with rank ) but
many do not, instead mapping into some smaller subspace (the matrices
with rank  or ).  Take for example the matrix  given by
:T = \begin{pmatrix}
1 & 0 \\
1 & 0 \end{pmatrix}
which represents a linear transformation that maps the point  to .
The point  is not in the image of , but is still in the codomain since
linear transformations from \textstyle \mathbb{R}^2 to \textstyle
\mathbb{R}^2 are of explicit relevance.  Just like all  matrices,
represents a member of that set.  Examining the differences between
the image and codomain can often be useful for discovering properties
of the function in question.  For example, it can be concluded that
does not have full rank since its image is smaller than the whole
codomain.


                              See also
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* Range (mathematics)
* Domain of a function
* Surjective function
* Injective function
* Bijection


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