Re: x*x-1=0

Vadim Tropashko wrote:
> In article <94amg7$71k$1_at_news.tue.nl>,
> hidders_at_win.tue.nl (Jan Hidders) wrote:
> > wrote:
> > But I suspect what you wanted to talk about was the equation
> >
> > x^2 + 1 = 0
> >
> > which under the usual interpretation does not have a solution, but
> > leads to interesting theories if you assume that it does.
> > Let's see
> > what happens. First, we try to translate this into rel. algebra:
> >
> > (X TIMES ({<1>} UNION {<2>})) UNION ({<3>} TIMES {<3>}) = {}
>
> Let me verify here that you are translating linear equation
> A*x + B = 0
> , not the quadratic one
> x*x + 1 = 0
> (Because we need to be able to solve linear equations first, before
> moving onto more complex cases;-)

Oh dear, you are right, of course. A more correct translation would be:

  (X TIMES X) UNION ({<1>} UNION {<1>}) = {}

> > 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.

> 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.

> 2. Column name renaming. Some equations wouldn't have any solutions
> simply because column name signatures are different, so we need some
> identities here as well. Here operator RENAME is very confusing. Are
> we allowed to insert this operator into any part of the equation to
> "sweeten" the solution (othervise it might bail out on trivial
> signature mismatch?) BTW, is RENAME a true relational operator that
> should be added to grand five? I'm personally more happy if it's
> possible to define some table identities based on table header
> signatures alone, and drop RENAME altogether.

RENAME is not really neccesary if you simply assume that your tables contain simple tuples like <a,"harry","12-3-52"> without column names.

-- 
  Jan Hidders