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From: "Mikito Harakiri"
Newsgroups: comp.databases.theory
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Subject: Re: Extending my question. Was: The relational model and relationalalgebra - why did SQL become the industry standard?
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Date: Thu, 13 Feb 2003 16:28:15 -0800
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"Steve Kass" wrote in message
news:b2h96p$ma2$1@slb9.atl.mindspring.net...
> Mikito,
>
> It's pointless to talk about the limit of a set S without setting forth
> some context. If you are talking about sets, the limit of {0,x} as the
> real number x goes to zero is {0}. Why is the change of cardinality
> "nasty"? There is no mathematical reason to think that the
> cardinality function must be continuous.
Steve,
Let me try to provide an example as a context.
Consider a database of all polynomial roots. In multiset model we have
table Polynomial (
id integer,
power integer,
coeff number
)
CONSTRAINT c1 UNIQUE KEY (id, power)
CONSTRAINT c2 CHECK ( power >= 0 )
table Roots (
polynomial id number,
root number
)
In the set model, I would have to make all tuples in the Roots relation to
be distinct, and, therefore, add some extra column. When sets advocates talk
about a tuple reflecting some real-world relationship, how would they
interpret this extra column? If numbers are real, it might be the
enumeration induced by the total ordering of the roots. What if the numbers
are complex?
> You could, if you chose, define the limit of a multiset, and you would
> have a different discussion. If you defined things in the most natural
way,
> the limit of {0,x} as x goes to zero would be {0,0}. Discussing
continutity
> is pointless without context. If f is a function from the collection of
> multisets
> of reals to the collection of multisets of reals, then you need a topology
> on multisets of reals to be able to discuss continuity (relative to that
> topology). Without thinking it all through, I suspect a good topology
> is the
> smallest one containing all open intervals (a,b) (these being sub_sets_ of
> reals.
I don't follow. A set of all open intervals (a,b) defines just a standard
topology on R. How would it work for sets, or multisets of reals?
> To make the limit {0} make sense for multisets, you'd need every open se
t
> around {0} to contain {e, e} for some e <> 0, and probably for it to
> contain {e,e,e,e,...} as well. Maybe it has nice properties, too. But
> the {0}
> limit is exactly the correct one for sets, of course, under the usual
> topology
> on the real numbers.
Again, in which topology? We are talking about the space of all sets (or,
alternatively, multisets) of real numbers, not just the space of reals.
> Change of cardinality on projection is nothing at all
> difficult. Project a line onto a plane in the direction of the line,
> and you've
> gone from an uncountable set to a finite one. Not changing cardinality is
> more trouble.
Agreed.
> Be careful not to make claims about good and bad properties of
operations
> on multisets (or sets) without specifying the context in which you are
> making
> these statements.
Granted, but does, the Relational Theory itself always provide such a
context? For example, the Alice Book defines relation in the "Name vs.
Unnamed Perspective", then in "Conventional vs. Logic Programming
Perspective". Not exactly presize definition. What is a relation? Is the
column ordering important? If not, is the naming function important?
>
> Mikito Harakiri wrote:
>
> >Suppose we have a set {0,1}. Let's move the "second" set member a little
> >bit: {0,1/2}. So far so good, the set still has 2 elements. Let's move it
> >one more time: {0,1/4}. We still have set cardinality 2. In the limit,
> >however, we have the set with only one element {0}. This change of
> >cardinality is very nasty from math perspective, because it presents a
break
> >of continuity. This is why matematians would consider the limit to be
> >multiset {0,0} rather than just a set {0}. Mathematical definitions do
> >contain multiset concept explicitly within their definitions, for
example,
> >"a spectrum is a *multiset* of eigenvalues". Break of continuity
manifests
> >itself in relational theory as a change of row set cardinaly after
> >projection is applied. On the contrary, in multiset model, projection
> >doesn't change number of rows. (Selection -- being dual to projection --
> >doesn't change number of columns in both set and bag models).