Re: Proposal: 6NF
Date: 19 Oct 2006 16:53:30 -0700
Message-ID: <1161302010.279018.183110_at_f16g2000cwb.googlegroups.com>
JOG wrote:
> On Oct 19, 6:42 pm, "vc" <boston..._at_hotmail.com> wrote:
> > JOG wrote:
> > > vc wrote:
> > > > Jan Hidders wrote:
> > > > > vc wrote:
> > > > > > Jan Hidders wrote:
> > > > > > [...]
> >
> > > > > > A much simpler example. Let {0, 1, 2, 3} be a set of four integers
> > > > > > with addition modulo 4. Then, none of its subsets, except {0} and
> > > > > > {0, 2}, retains the addition mod 4 operation which makes the idea of
> > > > > > 'subtype as subset' utterly silly, [....].
> >
> > > > > You keep on making the same mistake. The expression a +[mod 4] b has a
> > > > > well defined result if a and b are from any subset of {0, 1, 2, 3}.
> >
> > > > Consider the subset {2, 3}. What is the result of (2+3) mod 4 ? If
> > > > you say it's '1', what is '1'? There is no such element in {2, 3}.
> > > > [snip]
> >
> > > Hi vc. PMFJI, but does this argument not rely on the assumption of the
> > > set needing to satisfy the closure property in respect to the
> > > operation?
> >
> > > Mathematically, surely the modulo addition operation as described above
> > > above can be applied to the subset {2, 3}, while happily having a
> > > closure of {0,1,2} for instance?It's not a 'closure'. The closure property, in this context, means
> > that the result of a binary operation must belong to the same set the
> > operands come from What are '0' and '1' ? They do not exist in {2,
> > 3}, so {2, 3} is not closed under '+ mod 4'. Very much in the same
> > fashion, the natural numbers are not closed under subtraction, or odd
> > numbers under addition, etc.
> >
> > > After all, any operation just maps one
> > > set of values to another - why the self-closure requirement?Any math binary operation should map pairs of values from some set to a
> > value in the same set (SxS -> S), that's the essential property of the
> > math structures known from one's school years, such as the natural
> > numbers, the set of ntegers, the set of rationals, the set of reals,
> > the complex numbers, etc,
> >
> >
> >
> > > I stand ready for correction if I have made a misinterpretation.
>
> As far as I understood from my school years, an operation just mapped
> one set of values to another. I remain unconvinced of the need for an
> operation upon a set to map to itself. Where is such a definition? As I
> said, I am open to convincing, but I was not aware of such a
> pre-requisite for closure?
Functions can be defined on a set where the result is not in the set, such as mapping hats to professions in pre-school, but then the set (of hats) is not "closed under that function," as I understand it. I just checked http://en.wikipedia.org/wiki/Closure_(mathematics) which at first glance seems to use the terminology this way too.
--dawn Received on Fri Oct 20 2006 - 01:53:30 CEST