Re: x*x-1=0

From: Vadim Tropashko <vadimtro_at_yahoo.com>
Date: Sat, 20 Jan 2001 08:19:59 GMT
Message-ID: <94bhnd$3go$1@nnrp1.deja.com>

In article <94amg7$71k$1_at_news.tue.nl>,
hidders_at_win.tue.nl (Jan Hidders) wrote:
> wrote:
> Aha, now I am beginning to see what you mean. But I am not sure what
> your question exactly is. If you look at the equation you gave
>
> x^2 - 1 = 0
>
> then there is no problem. It is just an equation with two solutions.
> That can also happen in the relational algebra.

Classic Algebra folks study general systems of polynomial equations of multiple variables. Plus and minus in the above case are equally interested to them.

> 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>}) = {}
>
> Notation:
> TIMES = the cartesian product (similar to multiplication)
> UNION = the union (similar to addition)
> {<x>} = singleton set with a unary tuple containing the number x
> {} = the empty relation
>

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;-)

> 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:-)

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:

DUM (or '0') - is is a table with no rows an columns only (i), or any table with empty set of rows (ii)? In algebra, identities (equivalence classes) are quite common, for example, quotients 1/2 and 2/4 are the same thing. Realistically, we need be able to cleverly define some identities on the table literals as well. For DUM it seems to be more convenient to consider it as (ii) rather than (i); then, DUM*A=DUM holds. Somehow confusing is the fact that any table could be projected to the table with no rows. We would be fine with the definition (i) but modify the law to be PROJECT(DUM*A)=PROJECT(DUM)

The bottom line of this bullet is that it's quite hard to solve equations if we don't see that 1/2 and 2/4 are the same thing.

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.

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http://www.deja.com/ Received on Sat Jan 20 2001 - 02:19:59 CST