Re: Review of Mueckenheims book.

On Mar 5, 3:06 pm, Virgil <vir...@xxxxxxxxxxx> wrote:
In article <1173132878.685879.156...@xxxxxxxxxxxxxxxxxxxxxxxxxxxx>,

"MoeBlee" <jazzm...@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.

I understand your point. However, I would not call that a matter of
precision as opposed to a matter of further specification. As you
know, if we like, in set theory, we can also specify as to the these
things about domains and ranges and subsets of domains and subsets of
ranges and supersets of domains and supersets ranges.

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

Again, I only object to describing this as being 'casual'. The set
theoretic definitions are precise. And it is precision, not
casualness, to regard a function as a exactly a certain kind of set of
ordered pairs.

Meanwhile, what strikes me as casual is people saying "the codomain"
instead of "a codomain". Yes, in set theory, we could define, e.g.: a
funmorph is <f d s> for some f, d, and s such that f is a function, d
is the domain of f, and s is a superset of the range of f. Then every
funmorph does have a unique codomain, the third member of the triple,
of which we can speak of 'the codomain'. And, aside from that
definition, if we are just generally talking about some superset of a
range of function, then we can say 'the codomain' as brief for 'the
particular codomain that is under discussion in this context'. And
that is okay, but it is actually more casual than any terminology I've
used here.

MoeBlee

.

Relevant Pages

  • Re: Review of Mueckenheims book.
    ... a unique codomain, I don't use the expresssion 'THE codomain'. ... ranges and supersets of domains and supersets ranges. ... casualness, to regard a function as a exactly a certain kind of set of ... funmorph is for some f, d, and s such that f is a function, d ...
    (sci.math)
  • Re: Review of Mueckenheims book.
    ... a unique codomain, I don't use the expresssion 'THE codomain'. ... ranges and supersets of domains and supersets ranges. ... casualness, to regard a function as a exactly a certain kind of set of ... Functions are often defined as special sorts of relations in set theory, ...
    (sci.math)
  • Re: Review of Mueckenheims book.
    ... David Marcus wrote: ... casualness, to regard a function as a exactly a certain kind of set of ... what strikes me as casual is people saying "the codomain" ... but mathematics isn't just set theory. ...
    (sci.math)
  • Re: Review of Mueckenheims book.
    ... casualness, to regard a function as a exactly a certain kind of set of ... what strikes me as casual is people saying "the codomain" ... but mathematics isn't just set theory. ...
    (sci.math)
  • Re: Review of Mueckenheims book.
    ... casualness, to regard a function as a exactly a certain kind of set of ... what strikes me as casual is people saying "the codomain" ... but mathematics isn't just set theory. ...
    (sci.math)
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