On Jul 6, 12:11 pm, "Ross A. Finlayson" <ross.finlay...@gmail.com> wrote:
> So, not every function is a surjection onto its co-image or range.
In ordinary set theories, with the ordinary definition of 'function' as a set of ordered pairs that is single rooted, and 'co-image' as 'superset of the range', a function does not have a unique co-image, but has many co-images, but a function does have a unique range. Then evey function is a surjection onto its range; but not a surjection onto any of its co-images except the range itself.
(Of course, that does not apply where, alternatively, a function is understood not just as a set of ordered pairs that is single rooted, but rather as "an operation, a domain, and a range" all together.)
> That gets into definitions of "range", in classical, modern, standard, > and as well modern nonstandard definitions of function.
The definition of 'the range of a set' is pretty ordinary: range(x) = {y | Ez <z y> in x}.
Some authors give ranges only to relations and some, maybe, only to functions. But, as you see above, we can define 'the range' for any x whatsoever, as is common to do.
