Re: Proposal: 6NF
Date: 11 Oct 2006 09:18:50 -0700
Message-ID: <1160583530.590341.41840_at_e3g2000cwe.googlegroups.com>
Brian Selzer wrote:
> "David Cressey" <dcressey_at_verizon.net> wrote in message
> news:8VWWg.1111$P92.756_at_trndny02...
> >
> > "Hugo Kornelis" <hugo_at_perFact.REMOVETHIS.info.INVALID> wrote in message
> > news:96jdi251552m2rb54f8edjrmhitcld1kni_at_4ax.com...
> >> On Fri, 06 Oct 2006 12:59:15 GMT, David Cressey wrote:
> >>
> >> >There is one point I'm confused on: what is the domain of the empty
> >> >set?
> >> >does it even have a domain? To me, the empty set of character strings
> > is
> >> >not "the same thing" as the empty set of integers. But I may be
> >> >thinking
> >> >like a computer person and not like a mathematician.
> >>
> >> Hi David,
> >>
> >> Here are some thoughts from someone who is far from a mathematician and
> >> who is more a database practictioner than a database theorist, so take
> >> them with whatever amount of salt you see fit.
> >>
> >> When I worked with sets during the Dutch equivalent of highschool, I
> >> usually had to use a two-part notation. I can't replicate the symbols
> >> here and I don't recall all the correct names, but it consisted of a
> >> definition of a domain and a listing or formula to define the values. So
> >> you could have a set that was defined as a subset of the domain of
> >> positive integers consisting of the numbers 2, 4, and 7; but you could
> >> also have a set defined as a subset of the domain of real values
> >> consisting of the numbers 2, 4, and 7.
> >> Later, after highschool, I started to see a simplified notation for sets
> >> that exposes only the values in the set but not the domain.
> >>
> >> Are the two sets above equal? I guess that you could defend both answers
> >> here - the sets have the same members, but not the same domain
> >> definition. I also guess that the notation used can sometimes be an
> >> indication of how the answer would be in any give UoD.
> >>
> >> For a general answer, I'm tempted to say that there have to be two
> >> equality operators for set arithmetic, one looking at the values of the
> >> set members only, the other also looking at the domain.
> >>
> >> Anyway, whatever you favor as an answer to the question of equality of
> >> the two sets above - once you've chosen an answer, the answer to
> >> equality of two empty sets logicallly follows.
> >>
> >> Best, Hugo
> >
> > Thanks for your reply, Hugo. I'm also more of a practitioner than a
> > theoretician, and certainly no mathematician.
> > So I'll take what you say with a grain of salt, and you can do the same
> > with
> > what I say.
> >
> > It's not clear to me that the real number 2 is the same thing as the
> > integer
> > 2. Just for clarity, let me represent the real number 2 as 2.0. It
> > seems
> > to me that {2.0, 4.0, 7.0} is not equal to {2, 4, 7} the elements are
> > counterparts, but they aren't the same.
> >
>
> Hi David
>
> The domain of integers is a proper subset of the domain of reals.
Strictly speaking, Z is not a subset of R or even of Q (some stucture is acquired some is lost when one moves between R, Q and Z), but the subtlety is rarely important so one can say that Z is a subset of R (or Q). A more correct way would be to say that Z is isomorphic to a certain subset of R (or Z is ebedded in R). Computationally, of course, the floating point number is a creature very different from R
> I think
> the concept is called "specialization by constraint."
What's that ?
> That means that every
> integer is also a real number; therefore, {2, 4, 7} is identical to {2.0,
> 4.0, 7.0}. 2.0 and 2 are just possible representations of the same number,
Obviously, it's not "the same number" only because, as one might have discovered in the secondary school, sqrt(2) and sqrt(2.0) would yield rather different results.
> which belongs not only to the set of all real numbers but also to the set of
> all integers.
>
> --Brian
Received on Wed Oct 11 2006 - 18:18:50 CEST