Re: Thinking about MINUS

From: Bob Badour <bbadour_at_pei.sympatico.ca>
Date: Sun, 07 Jan 2007 17:09:23 GMT
Message-ID: <7P9oh.41760$cz.613377@ursa-nb00s0.nbnet.nb.ca>

J M Davitt wrote:

> 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?

Containment is inherent in sets. Given the definitions for U, A and B, A and B are both subsets of U which means U contains both A and B.

If you wanted to, you could define predicates P and Q such that:

A = { x in U | P(x) }
B = { y in U | Q(y) }

The predicates P and Q would be eqivalent to "is an element of A" and "is an element of B" respectively.

We could express the functions in terms of those predicates:

A MINUS B = { x in U | P(x) & ~Q(x) }
NOT B = U MINUS B = { x in U | 1 & ~Q(x) } = { x in U | ~Q(x) }

etc.

As Lennart already pointed out, the meaning of your expression above is: The set of all things that are neither strawberries nor apples but are grapes. Since no grape is a strawberry or an apple (at least in common understanding of those words), the expression simplifies to the "set of all grapes."

> 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."

I don't see why it would need any extension at all. You have simply omitted a concrete specification for U. However, I don't see how that even constrains us much. Given our previous discussion, product, order and customer are all subsets of some universe and the universe is some superset of product, order and 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.

Right now, we are at the level of using INTERSECT as an analog to AND. Join is an extension of INTERSECT, and I am not yet clear how our discussion extends to JOIN yet.

>>>> 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 - 11:09:23 CST