Re: Review of Mueckenheims book.

In article <1173209862.855639.297500@xxxxxxxxxxxxxxxxxxxxxxxxxxxx>,
"MoeBlee" <jazzmobe@xxxxxxxxxxx> wrote:

On Mar 5, 8:33 pm, David Marcus <DavidMar...@xxxxxxxxxxxxxx> wrote:

The normal definition of "function" includes specifying the domain and
codomain. So, a given function only has one codomain.

Not in set theory; not with the definition of 'codomain' that was
given:

c is a codomain of f <-> range(f) subset_of c.

So for NO set is there a UNIQUE codomain.

Sets per se do not have domains or codomains or ranges.

A function as a subset of a cartesian product has all of these.

If one chooses not to specify a cartesian product of which the function
is to be a subset (or a codomain for the function) then there are, in
general, many possible cartesian product supersets of the function which
will serve.
However, if one is concerned with whether a given function is surjective
or not, as one sometimes is, one needs a specific codomain.

For every set, there is a unique range, but not a unique codomain,
since there are lots and lots of sets that are supersets of any range.

If you want to have a different defintion of 'codomain' so that there
is a unique codomain for each function, then fine, but with the
definition that has been given in this thread, it is a theorem that
every set has more than one codomain.

Some "sets", not being functions at all, may not have any codomains at
all.

If you chose to regard functions as being merely somewhat specialized
sets of ordered pairs, feel free.

Do you consider relations as sets of ordered pairs equally amorphous as
to codomains? And what about the possible domains of a relation?
.

Relevant Pages

  • Re: Review of Mueckenheims book.
    ... I hope these resubmits tonight don't become duplicates to what ... One way we could formalize this is: ... given the definition of codomain: ... then, still, f has many codomains, but there is a unique codomain C ...
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  • Re: Review of Mueckenheims book.
    ... a given function only has one codomain. ... For every set, there is a unique range, but not a unique codomain, ... if you want to use your personal terminology instead of terminology ...
    (sci.math)
  • Re: Review of Mueckenheims book.
    ... a given function only has one codomain. ... Not in set theory; not with the definition of 'codomain' that was ... For every set, there is a unique range, but not a unique codomain, ...
    (sci.math)
  • Re: Why f:domain->codomain instead of f:domain->range?
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  • Re: Question Regarding the Definition of Cantors Set
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