CODOMAIN

In mathematics, the 'codomain' of a function ''$f$'' : ''$X$'' → ''$Y$'' is the set ''$Y$''.
The ''domain'' of ''$f$'' is the set ''$X$''.
The ''range'' of ''$f$'' is the set $f(X)$ defined as { $f(x)$ : ''$x$'' ∈ ''$X$'' }.
It follows from these definitions that the range of ''$f$'' is always a subset of the codomain of ''$f$''.

Contents

Examples

See also

Examples

As an example, let the function ''$f$'' be a function on the real numbers:
: $fcolon\; mathbb\{R\}$
ightarrowmathbb{R}
defined by
: $fcolon,xmapsto\; x^2.$
The codomain of ''$f$'' is $mathbb\{R\}$, but clearly ''f'' does not map to any negative number. Thus the range of ''f'' is the set $mathbb\{R\}$₀⁺,i.e., the interval [0,∞) where:
:
We can define an alternative function ''$g$'' thus:
: $gcolonmathbb\{R\}$
ightarrowmathbb{R}^+_0
: $gcolon,xmapsto\; x^2.$
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'':
: $hcolon,xmapsto\; sqrt\; x.$
We must define the domain of ''h'' to be $mathbb\{R\}^+\_0$:
: $hcolonmathbb\{R\}^+\_0$
ightarrowmathbb{R}.
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 $mathbb\{R\}$. 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 'R'^''n'' and its codomain is 'R'^''m'', where the matrix is $m\; imes\; n$ (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 $0$s, 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.

See also

★ Domain (mathematics)

★ Range (mathematics)