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Codomain
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In mathematics, the codomain, range or target set, of a function, described symbolically as ' : ' ? ', is the set ' into which all of the output of the function is constrained to fall.
All the output that the function can possibly produce from its given domain, , is the image. The function's image will not necessarily fill the entire codomain ', even though the output must all land inside of the codomain: there can be points in the codomain that are "not used."
The codomain (or target) is part of the definition of a function.
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Encyclopedia
In mathematics, the codomain, range or target set, of a function, described symbolically as ' : ' ? ', is the set ' into which all of the output of the function is constrained to fall.
All the output that the function can possibly produce from its given domain, , is the image. The function's image will not necessarily fill the entire codomain ', even though the output must all land inside of the codomain: there can be points in the codomain that are "not used."
The codomain (or target) is part of the definition of a function. The image (or range) is a consequence of the definition of a function: the image is a subset of the codomain and depends upon (i.e. is a consequence of) how the definition of the function prescribes the domain, codomain, and map or formula.
(The domain of ' is the set '.)
Examples
As an example, let the function ' be a function on the real numbers:
defined by
, or equivalently .
The codomain of ' is , but clearly f does not map to any negative number.
Thus the image of f is the set
,i.e., the interval [0,8) where:
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We can define an alternative function ' thus:
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While ' and ' 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:
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We must define the domain of h to be :
- .
Now let's define the compositions
,
.
As it turns out, doesn't make sense. Suppose (as we must, unless we explicitly state otherwise) that we do not know what the image of ' is; we only know that it can be . 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 image, 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 affects whether a function is a surjection. In our 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 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 (has ' rows and ' columns) and the image is
usually called the range. But the range (the set of numbers obtained when the matrix is right-multiplied by every column vector of length ') could be much smaller. For example, if the matrix contains only s, then no matter how large it is, the range is just the vector 0. But the dimension of the resulting vector is '. This is important, because it is enough to change just one number in the matrix to make its range non-zero.
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