# Re: x*x-1=0

Date: 22 Jan 2001 09:38:29 GMT

Message-ID: <94gv2l$c38$1_at_news.tue.nl>

Vadim Tropashko wrote:

*>
*

> > > > Assuming that this equation is solveable leads to the peculiar

*> > > > property that there will be sets that you can add a non-empty set
**> > > > to such that the result will be an empty. You might call them
**> > > > "negative sets" if you will.
**> > >
**> > > This is a discovery of negative tables/sets, right? (We are in
**> > > the very beginning, therefore, of the classic sequence
**> > > negative->rational->complex numbers:-)
**> >
**> > Sort of. Perhaps you could treat them as some kind of generalized bag
**> > where a bag can contain an element -3 times. I doubt if that is
**> > really a new idea.
**>
**> OK. We assign a weight to each element. Is there a definition of joint
**> operation upon such things?
*

Yes, there is. I will answer that below.

> > > Still something is not quite right here, and until things would be

*> > > cleaned up we cant expect those to be good concepts. Things that
**> > > bother me:
**> > >
**> > > 1. DUM (or '0') - is is a table with no rows an columns only (i), or
**> > > any table with empty set of rows (ii)?
**> >
**> > It is both because both tables are simply the empty set.
**> >
**> I'm kind of uneasy about set reductionism. This reminds me unfamous
**> 0 = empty set
**> 1 = {0}
**> 2 = {0,1}
**> sorry for been off topic here.
*

I don't really see anything wrong with that as long as you keep in mind that you are just defining a model of the numbers. Statements like 1 is really just singleton 0 make me always slightly cringe. :-)

> > RENAME is not really neccesary if you simply assume that your tables

*> > contain simple tuples like <a,"harry","12-3-52"> without column names.
**>
**> A newbie question here: how do we join tables without column names?
**> (first rule of relational -- verify everything against joints:-)
*

You don't need the joins because they can be simulated with the (relational) cartesian product followed by a selection and a projection. But things get, from an algebraic perspective, a little more complicated because the cartesian product does not commute as the join does.

And that also tells how a join can be defined on bags because it will be similar to the cartesian product on bags. So if a tuple occurs n times in a table and is joined with another tuple in another table that appears there m times, then the joined tuple will appear n*m times in the result. And that gives an idea of what will happen if you also assume that your equation has a solution: you will get "complex bags" that may contain a tuple i times where i is the complex number such that i^2 = -1.

-- Jan HiddersReceived on Mon Jan 22 2001 - 10:38:29 CET