# User Contributed Dictionary

### Noun

- The target space into which a function maps elements of its domain. It always contains the range of the function, but can be larger than the range if the function is not surjective.

#### Synonyms

#### Antonyms

#### Translations

range of a function

- Czech: obor hodnot
- Swedish: målmängd

# Extensive Definition

In mathematics, the codomain,
or target of a function
f : X → Y is the set Y. Unlike the range,
which is a consequence of the definition of a function, the
codomain is part of the definition of a function. The range is a
subset of the codomain and depends upon (i.e. is a consequence of)
the definition of the domain, codomain, and map of a
function.

The domain
of f is the set X.

## Examples

As an example, let the function f be a function
on the real
numbers:

- f\colon \mathbb\rightarrow\mathbb

defined by

- f\colon\,x\mapsto x^2.

The codomain of f is \mathbb, but clearly f does
not map to any negative number. Thus the range of f is the set
\mathbb^+_0,i.e., the interval
[0,∞) where:

- 0\leq f(x)

We can define an alternative function g
thus:

- g\colon\mathbb\rightarrow\mathbb^+_0
- g\colon\,x\mapsto x^2.

While f and g map a given x to the same number,
they are not, in the modern view, the same function because they
have different codomains. To see why, suppose that we define a
third function h:

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

We must define the domain of h to be
\mathbb^+_0:

- h\colon\mathbb^+_0\rightarrow\mathbb.

Now let's define the compositions

- h \circ f,
- h \circ g.

As it turns out, h \circ f doesn't make sense.
Suppose (as we must, unless we explicitly state otherwise) that we
do not know what the range of f is; we only know that it can be
\mathbb. But then we are in trouble because the square root is not
defined for negative numbers. Now we have a possible contradiction
because function h, when composed on function f, might receive an
argument which it "can't handle."

This unclarity should be avoided in formal work.
Function composition therefore requires by definition that the
codomain of the function on the right side of a composition (not
its range, which is a consequence of the function and is said to be
indeterminate at the level of the composition) must be the same as
the domain of the function on the left side.

The codomain can affect whether a function is a
surjection. In our
example, g is a surjection while f is not. The codomain does not
affect whether a function is an injection.

A second example of the difference between
codomain and range can be seen by considering the matrix of a
linear transformation. By convention, the domain of a linear
transformation associated with a matrix
is Rn and its codomain is Rm, where the matrix is m \times n (has m
rows and n columns). But the range (the set of numbers obtained
when the matrix is right-multiplied
by every column
vector of length n) could be much smaller. For example, if the
matrix contains only 0s, then no matter how large it is, the range
is just the vector
0. But the dimension of the resulting vector is m. This is
important, because it is enough to change just one number in the
matrix to make its range non-zero.

codomain in Catalan: Codomini

codomain in German: Zielmenge

codomain in Finnish: Maalijoukko

codomain in French: Ensemble d'arrivée

codomain in Korean: 공역 (수학)

codomain in Croatian: Kodomena

codomain in Ido: Arivey-ensemblo

codomain in Italian: Codominio

codomain in Dutch: Codomein

codomain in Ukrainian: Область значень

codomain in Portuguese: Contradomínio

codomain in Chinese: 陪域