Re: Thinking about MINUS

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.

p Received on Sun Jan 07 2007 - 02:18:18 CET