Re: Proposal: 6NF

From: vldm10 <vldm10_at_yahoo.com>
Date: 20 Oct 2006 13:19:21 -0700
Message-ID: <1161375560.908400.124770@h48g2000cwc.googlegroups.com>

dawn wrote:
> JOG wrote:
> > On Oct 20, 11:12 am, "vc" <boston..._at_hotmail.com> wrote:
> > > Marshall wrote:
> > > > On Oct 19, 8:40 pm, "vc" <boston..._at_hotmail.com> wrote:
> > > > > Marshall wrote:
> > >
> > > > > > > "return new ModFour((val+a.val)%4)" is called a bug or cheating
> > > > > > > because you used a function defined on the entire domain NxN (where N
> > > > > > > is a subset of the natural numbers implemented by the computer) to
> > > > > > > generate a result which is undefined for the (2,3) pair.
> > >
> > > > > > It's not a bug, since the program produces the correct result, which is
> > > > > > 1. It's not "cheating" because WTF does cheating even mean in
> > > > > > this context. What it is, is the absence of closure over the subclass.
> > > > > > Which is no big deal.
> > >
> > > > > So producing an absurd result is no big deal ? You seem to have agreed
> > > > > that in the {2,3} subset there is no '1', how come your java
> > > > > implementation manages to extract '1' out of nowhere ?
> > >
> > > > Consider the below function:
> > >
> > > > Let S4 = {0, 1, 2, 3}
> > > > Let S23 = {2, 3}
> > > > f : S23, S23 -> S4
> > > The binary operation over S23 is g:S23:S23 -> S23, your f does not
> > > qualify. aAcommon example is '+'.
> >
> > Ok. So if "f : S23, S23 -> S4" does not qualify as a binary operation
> > from your point of view, you are saying there is a clear distinction
> > between a binary operation, and a binary function, the fomer being the
> > subset of the latter which exhibiti closure.
> >
> > In term's of the pure definition of 'operation' as opposed to
> > 'function' i think you may be correct.

>

> Whoever put in the wikipedia entry at
> http://en.wikipedia.org/wiki/Binary_operation might have helped us out.

>
>

> "a binary operation on a set S is a binary function from S and S to S,
> in other words a function f from the Cartesian product S × S to S.
> Sometimes, especially in computer science, the term is used for any
> binary function. That f takes values in the same set S that provides
> its arguments is the property of closure."

Above definition is good for the operations on one set. For example we can use it for one set of the attribute's values in a database. We can define operation on more then one set: it is mapping from set S1 x, ... ,x Sn to S.
For example: multiplying a vector by scalar. We can define a conditional operation, which is mapping from a subset of S1 x, ..., x Sn to S. For example the conditional operation is x - y, where x, y are natural numbers is the conditional operation. Here the condition is x < y.

All these definition are based on the set concept. However objects on which we are applying the operations (including the resulting objects) not necessary need to be gathered in the sets. They can be characterized by the properties. For example "to be a point", "to be a natural number" or "to be a set". (The property "to be a set" is non-collectivizing). Hope, that this is more light on the operations.

>

> I did not know that "operation" ever carried with it the added
> requirement of closure, but it sounds like that is the common
> mathematical def. Perhaps we can make a distinction here between a
> binary operation "on" the integers, for example, and a binary operation
> "from" the integers, or remove the confusion by eliminating the word
> "operation" and calling it a binary function on the set S, which is
> then a function on S x S. A third option might be to figure that since
> cdt is within computer science, we can use the term "binary operation"
> for any binary function. I think from this discussion we can safely
> say that there are two different defs for "binary operation" so we can
> clarify which one we are using and move on.

>

> In any case, a subclass is a subset. If it is a Duck, then it is also
> an Animal. If it is an Integer, then it is also a Real. If there is
> any function defined on Real, it is also a function defined on Integer.
> --dawn

Vladimir Odrljin Received on Fri Oct 20 2006 - 15:19:21 CDT