Re: Thinking about MINUS
From: Bob Badour <bbadour_at_pei.sympatico.ca>
Date: Sun, 07 Jan 2007 02:29:07 GMT
Message-ID: <TVYnh.41577$cz.610561_at_ursa-nb00s0.nbnet.nb.ca>
>> The discussion over in "Curious SQL question" started me thinking
>> (after I
>> got over my embarassment at posting a wrong solution).
>>
>> Suppose you started with these two primitive concepts:
>>
>> A universal set, called U (whatever that is)
>>
>> and
>>
>> MINUS(A,B), a function that removes from set A any elements common to
>> A and
>> B.
>>
>> Can you derive the rest of it? Here's my first attempt:
>>
>> Infix notation: A MINUS B = MINUS (A,B). Just a notational
>> convenience
>> for me. This should be trivial. I apologize to any readers who find this
>> inconvenient.
>>
>> Empty Set: PHI = U MINUS U. Note that PHI is somehow "bound" to U.
>> Whether the PHIs of different universes are or are not the same PHI is
>> something I'll let the rest of you discuss.
>>
>> NOT operator. NOT(A) = U MINUS A.
>>
>> Left association. (A MINUS B) MINUS C = (A MINUS C) MINUS B (proof
>> omitted)
>>
>> INTERSECTION. A INTERSECT B = A MINUS (A MINUS B)
>> = B
>> MINUS
>> (B MINUS A)
>>
>> UNION. A UNION B = NOT (NOT (A) INTERSECT NOT (B))
>>
>> From here, it looks like we can bootstrap our way up to the rest of set
>> theory and Boolean algebra.
>> Or am I seeing something wrong (again)?
Date: Sun, 07 Jan 2007 02:29:07 GMT
Message-ID: <TVYnh.41577$cz.610561_at_ursa-nb00s0.nbnet.nb.ca>
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 Received on Sun Jan 07 2007 - 03:29:07 CET