Re: Review of Mueckenheims book.

In article <1173132878.685879.156140@xxxxxxxxxxxxxxxxxxxxxxxxxxxx>,
"MoeBlee" <jazzmobe@xxxxxxxxxxx> wrote:

On Mar 2, 7:09 pm, Virgil <vir...@xxxxxxxxxxx> wrote:
In article <1172858408.942752.235...@xxxxxxxxxxxxxxxxxxxxxxxxxxx>,
"MoeBlee" <jazzm...@xxxxxxxxxxx> wrote:
The range of a function is usually defined as the set of all values that
the fuction takes,

Right, but the definition may be more general, so that it is not just
for functions or even relations:

range(s) = {y | Ex <x y>es}

and can be a proper subest of the codomain, which is
merely a set in which all those values lie but which may contain other
members which are not values of the function.

Right, that is how many authors use 'codomain'. And since there is not
a unique codomain, I don't use the expresssion 'THE codomain'.

In the category of sets and functions, different codomains always imply
different functions. In analysis, one need not always be so precise.

See domain, range and codomain as explained
inhttp://en.wikipedia.org/wiki/Function_%28mathematics%29

or seehttp://mathworld.wolfram.com/Codomain.html

Oh, come on, Virgil, as to the extent that that is addressed to me, I
really don't need that. I don't get my definitions from WIKIPEDIA, for
godssakes or usually from other Internet sites, but rather from good
textbooks. If I do consult an Internet site for a definition, I'll use
the Springer encyclopedia online.

For those accustomed to dealing mostly with real functions, as in
calculus, distinguishing between functions which have the same domain
and and range but differing codomains may seem excessively picky, but
there are areas, e.g., category theory, in which it is important.

Okay. And there's nothing I posted about this that fails to hold up to
whatever pickiness you want to apply.

In the category of sets and functions, equality of functions requires
equality of codomains. In other contexts, people are sometimes a bit
more casual.
.

Relevant Pages

  • Re: Proof difficulty with inverse image of a function
    ... inverse image of the codomain (a set that contains all the images of ... codomain, so you cannot apply the theorem. ... How does one prove equality of sets? ... --- Calvin ...
    (sci.math)
  • Re: Review of Mueckenheims book.
    ... the fuction takes, ... that is how many authors use 'codomain'. ... godssakes or usually from other Internet sites, ...
    (sci.math)
  • Re: Review of Mueckenheims book.
    ... that a function has more than one codomain (as 'codomain' was defined ... in this thread) and see that there is no contradiction between that ... equality of f and g as functions by proving that f and g have the same ... theory definition of 'is a function' in a discussion about set theory! ...
    (sci.math)
Loading