Re: Thinking about MINUS

"Bob Badour" <bbadour_at_pei.sympatico.ca> wrote in message news:Zt9oh.41743$cz.613288_at_ursa-nb00s0.nbnet.nb.ca...
> Walt wrote:
>
> > "paul c" <toledobythesea_at_oohay.ac> wrote in message
> > news:ooZnh.560288$5R2.4691_at_pd7urf3no...
> >
> >>Bob Badour wrote:
> >>
> >>>paul c wrote:
> >>>
> >>>
> >>>>Walt wrote:
> >>>>
> >>>>
> >>>>>The discussion over in "Curious SQL question" started me thinking
> >>>>>(after I
> >>>>>got over my embarassment at posting a wrong solution).
> >>>>>
> >>>>>Suppose you started with these two primitive concepts:
> >>>>>
> >>>>>A universal set, called U (whatever that is)
> >>>>>
> >>>>>and
> >>>>>
> >>>>>MINUS(A,B), a function that removes from set A any elements common
> >>>>>to A and
> >>>>>B.
> >>>>>
> >>>>>Can you derive the rest of it? Here's my first attempt:
> >>>>>
> >>>>>Infix notation: A MINUS B = MINUS (A,B). Just a notational
> >>>>>convenience
> >>>>>for me. This should be trivial. I apologize to any readers who find
> >>>>>this
> >>>>>inconvenient.
> >>>>>
> >>>>>Empty Set: PHI = U MINUS U. Note that PHI is somehow "bound" to U.
> >>>>>Whether the PHIs of different universes are or are not the same PHI
is
> >>>>>something I'll let the rest of you discuss.
> >>>>>
> >>>>>NOT operator. NOT(A) = U MINUS A.
> >>>>>
> >>>>>Left association. (A MINUS B) MINUS C = (A MINUS C) MINUS B (proof
> >>>>>omitted)
> >>>>>
> >>>>>INTERSECTION. A INTERSECT B = A MINUS (A MINUS B)
> >>>>> = B
> >>>>>MINUS
> >>>>>(B MINUS A)
> >>>>>
> >>>>>UNION. A UNION B = NOT (NOT (A) INTERSECT NOT (B))
> >>>>>
> >>>>>From here, it looks like we can bootstrap our way up to the rest of
> >
> > set
> >
> >>>>>theory and Boolean algebra.
> >>>>>Or am I seeing something wrong (again)?
> >>>>
> >>>>
> >>>>I guess NAND would then be U MINUS A MINUS B, and since NAND is said
> >>>>to be enough to "bootstrap" (I seem to remember), then I'd say you are
> >>>>right but where it leads as far as database is concerned, I don't
know.
> >>>
> >>>
> >>>NAND = U MINUS ( A MINUS ( A MINUS B ) )
> >>> = U MINUS ( B MINUS ( B MINUS A ) )
> >>>
> >>>NOR = ( U MINUS A ) MINUS B
> >>
> >>Thanks and Oops, right you are (I think now). I should have say NOR
> >>instead of NAND.
> >>
> >>p
> >
> > Either NOR or NAND are enough to bootstrap to the rest of it, provided
you
> > make one little extension to them:
> >
> > NOR or NAND with only one operand is NOT.
>
> That's not really an extension. That's just the natural outcome of the
> functions themselves:
>
> NOT A = A NAND A
> NOT B = B NOR B
>
> This derives from the identity:
>
> A AND A = A
> B OR B = B
>
>
> > I don't know where it leads regarding database, either. Just a random
> > thought.
>
> What I find interesting is we can express all of the base logical
> operations in the relational compromise for NOT.
>
> Of course, so far, we are only looking at relations of the same type,
> and MINUS doesn't get us PROJECT or JOIN etc. I wonder whether one can
> define some reasonable extension to MINUS for relations of different
> types that gives us PROJECT and/or JOIN etc.

Thanks for the NOT A= A NAND A and NOT B = B NOR B.

PROJECT and JOIN would indeed be interesting. Received on Mon Jan 08 2007 - 19:59:04 CET