Re: Why f:domain->codomain instead of f:domain->range?

kj <socyl@xxxxxxxxxxxxxxxxx> wrote:
>
>
>
>
> I never understood why the notation
>
> f:X->Y
>
> instead of
>
> f:X->f(X)
>
> In other words, I don't understand the utility of the notion of a
> codomain. Why not make the definition of "function" be so that
> every function is surjective? I'm sure there are very good reasons
> for this, but I don't see them.
>
> The only explanation I can think of is that there are often times
> when it is much easier to describe a function's codomain than than
> its range. Is this it? Or are there more fundamental reasons
> behind this practice?

That is already quite a good reason.

>
> I suspect the answer has to do with category theory (or its
> antecedents), but this is a wild, ignorant guess.

Your guess is probably quite good. Specifying the codomain explicitely
certainly makes the bookkeeping easier once you start being interested
in composing maps, because the information is static:

(1) two maps f and g are composable iff the codomain of f equals the
codomain of g.

(2) if f and g are composable then their composite inherits its domain
from f and its codomain from g.

Now, while you could rephrase (1) in terms of range and domain, the range
of a possible composite is not determined by the range of the second map.
So the result would in fact lack simplicity and symmetry.

>
> Does anyone know who generally gets the credit (fairly or not) for
> realizing that the codomain was more suitable than the range in
> the definition of concept of a function?

I remember having read somewehere (unfortunately I cannot remember where)
that people working in algebraic topology started this in the thirties
while it took longer to be established in algebra. Perhaps someone else
can com up with good references.

Marc
.

Relevant Pages

  • Re: Why f:domain->codomain instead of f:domain->range?
    ... Or are there more fundamental reasons ... Specifying the codomain explicitely ... > two maps f and g are composable iff the codomain of f equals the ... > of a possible composite is not determined by the range of the second map. ...
    (sci.math)
  • Re: Why f:domain->codomain instead of f:domain->range?
    ... Or are there more fundamental reasons ... Specifying the codomain explicitely ... >two maps f and g are composable iff the codomain of f equals the ... >of a possible composite is not determined by the range of the second map. ...
    (sci.math)
  • Re: Why f:domain->codomain instead of f:domain->range?
    ... >>> codomain. ... > fundamental reasons ... >>two maps f and g are composable iff the codomain ... >>if f and g are composable then their composite ...
    (sci.math)
  • Re: Why f:domain->codomain instead of f:domain->range?
    ... >>> when it is much easier to describe a function's codomain than than ... Or are there more fundamental reasons ... The official definition of a function as a set of ordered pairs ... The categorical usage, on the other hand, has many things to recommend ...
    (sci.math)
  • Re: Why f:domain->codomain instead of f:domain->range?
    ... > I never understood why the notation ... > codomain than than ... > fundamental reasons ... There is indeed a very good categorical reason for not considereing only surjective functions. ...
    (sci.math)