Re: [math-fun] function "image" v. function "range"
An arbitrary function's image can't in general be determined, so a "range" (by which I mean here codomain) must be specified -- just as the domain must be -- as part of the definition of the concept of a function. It is unfortunate that authors of calculus books got "range" and "image" so confused that new terminology (codomain) had to be invented. (Though maybe it's good terminology from the point of view of category theory.) It took a few centuries to hammer out the correct definition of a function: A subset f of the cartesian product XxY of the domain X and codomain Y such that for each x in X there is exactly one y in Y such that (x,y) belongs to f. This should not be forgotten or tampered with. --Dan "Things are seldom what they seem." --W.S. Gilbert
On Tue, Nov 1, 2011 at 2:01 PM, Dan Asimov <dasimov@earthlink.net> wrote:
An arbitrary function's image can't in general be determined, so a "range" (by which I mean here codomain) must be specified -- just as the domain must be -- as part of the definition of the concept of a function.
It is unfortunate that authors of calculus books got "range" and "image" so confused that new terminology (codomain) had to be invented. (Though maybe it's good terminology from the point of view of category theory.)
It took a few centuries to hammer out the correct definition of a function: A subset f of the cartesian product XxY of the domain X and codomain Y such that for each x in X there is exactly one y in Y such that (x,y) belongs to f.
Well, that's the correct notion of extensional function. The process of computing the normal form of a lambda calculus term is a series of steps replacing one intensional function with another; we can identify the rewrite-equivalence class with an extensional function because of the Church-Rosser property, but in pi calculus the rewrites are not confluent, so that's not possible.
This should not be forgotten or tampered with.
--Dan
"Things are seldom what they seem." --W.S. Gilbert
_______________________________________________ math-fun mailing list math-fun@mailman.xmission.com http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
-- Mike Stay - metaweta@gmail.com http://www.cs.auckland.ac.nz/~mike http://reperiendi.wordpress.com
participants (2)
-
Dan Asimov -
Mike Stay