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Re: Thinking about MINUS

From: J M Davitt <jdavitt_at_aeneas.net>
Date: Sun, 07 Jan 2007 15:36:17 GMT
Message-ID: <Rr8oh.7671$SJ3.6286_at_tornado.ohiordc.rr.com> 






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.

>>>




All this "boolean operations on sets" has me scratching my head.  There 
is, underneath it all, the presumption that one really means "is an 
element of," right?  I mean, what could the meaning of "NOT (strawberry 
OR apple) AND grape" be?



Here's a good example



>>> NOT operator.    NOT(A) = U MINUS A.





of something that requires significant presumptions that I don't think, 
if I may invent a design on the fly, can be extended to expressions like
"NOT product" or "NOT order INTERSECT customer" or "product INTERSECT 
NOT order INTERSECT customer."



Sure, we can think of JOIN as an analog to AND and UNION as an analog to 
OR, but they are very different operations.



>>>

>>> 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


Received on Sun Jan 07 2007 - 16:36:17 CET












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