In set theory we have that a domain is the set of all inputs for which a function is defined and the image is the set of possible outputs given those inputs. You would think this would be enough, but no: we also have the concept of codomain, which is the set of all possible outputs and some.
This also leads to the distinction between "surjective" functions and non-"surjective" functions (functions that have the image being equal to the codomain and those that don't, respectively). This distinction seems to provide no useful information about the function itself.
Now, let's make an example, we have:
ƒ : {a, b, c, d} -> {1, 2, 3, 4}
ƒ a = 1
ƒ b = 2
ƒ c = 3
ƒ d = 4
This is a surjective well-defined function. The domain is {a, b, c, d} and the image/codomain is {1, 2, 3, 4}.
By just changing the "type" of the function definition, we have:
ƒ : {a, b, c, d} -> R
ƒ a = 1
ƒ b = 2
ƒ c = 3
ƒ d = 4
This now becomes a non-surjective function, which has:
- domain:
{a, b, c, d} - image:
{1, 2, 3, 4} - codomain:
{0, 1, 2, 3, ...}
The codomain seems to tell me precisely 0 additional information about how the function behaves or about anything else really.
So my question is, why was the concept of codomain even defined and why or how is it useful?