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# Re: Testing for the equivalence relation

From: Dan Guntermann <guntermann_at_verizon.net>
Date: Sat, 02 Jul 2005 02:41:38 GMT
Message-ID: <C9nxe.1662\$H64.468@trnddc07>

"Jan Hidders" <jan.hidders_at_REMOVETHIS.pandora.be> wrote in message news:I8kxe.135100\$Vs4.7201818_at_phobos.telenet-ops.be...
> Dan wrote:

>
> I have 4 sisters. There is the daughter of my father, the daughter of my
> mother, the grand-daughter of my grand-mother and the grand-daughter of my
> grand-father. Since all these girls happen to be identical I have only one
> distinct sister.

Yes, like rational numbers. That is exactly the point.

[1/2] = {1/2, 2/4, 3/6, ....}.
[2/4] = {1/2, 2/4, 3/6, ... }

>
> Wouldn't you agree that the first sentence is a bit misleading? What I
> have 4 of is not sisters, but descriptions of my sister. What you have 4
> of is not equivalence classes, but definitions of equivalence classes.

I've given a definition of equivalence class that I think is pretty solid. The math reference I gave previously also goes into some detail somewhat formally about the distinction between equivalence class and distinct equivalence class. If you feel that a different formalism is better or more appropriate, I'm sure I'll agree.

>
>> Jan: "Er, actually, you cannot. The notion of 'reflexivity' is defined
>> wrt. (1) a domain and (2) a binary relation over that domain. The notion
>> is
>> not really defined if you only have (2). Of course you could introduce
>> a related notion that only needs (2), but that would be a different
>> notion."
>>
>> So why wouldn't the set S suffice as being the basis of (1)?
>
> It would.

OK. So R couldn't be an equivalence relation since reflexivity is not met using the set S. If we were to use a domain of VARCHAR(), I am assuming the same would apply. So my next question is, would some unary relation with a predicate that constrains the set S, T(w), suffice as a basis of (1)?

>
>> And yes, I am curious whether another notion applies here.
>
> Sure. That's Dan-reflexivity and it's defined as follows. A binary
> relation is said to be Dan-reflexive if it is reflexive wrt. the union of
> its left-column and right column. So where do we go from here?

Nice, Jan. In an equivalence relation, I imagine one wouldn't really need a union and could just project away a single attribute and get the same result with less cost, but I get your point.

Your sarcasm actually reflects a lot about what I am asking: in the logical data model world, what is really the domain by which we evaluate reflexivity -- is the set represented by type? or is it a subset, and therefore a different domain by definition of set? Does the extension have to match the intension?

What I really am curious to know is whether for some primitive type, such as INTEGER for example, if a relation appears to be an reflexive, symmetric, and transitive over a subset of INT as a binary relation, whether the relation should be tested in terms of the INT type of in terms of the subset.

I haven't ruled out taking the union of the left and right columns yet as being the set upon which to determine reflexivity.

• Dan
Received on Fri Jul 01 2005 - 21:41:38 CDT

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