Re: Thinking about MINUS
Date: Sun, 07 Jan 2007 12:25:35 GMT
Message-ID: <3F5oh.734$3L1.322_at_trndny03>
"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.
I don't know where it leads regarding database, either. Just a random
thought.