Re: Testing for the equivalence relation

"VC" <boston103_at_hotmail.com> wrote in message news:DYudnQMXGOzb-lrfRVn-gg_at_comcast.com...
>
> "Dan" <guntermann_at_verizon.net> wrote in message
> news:1120240628.808945.261440_at_g47g2000cwa.googlegroups.com...
>> Thank you Jan.
>>
>> Another counter example that now seems so obvious in hindsight.
>>
>> So this leads me to my real objective. I now am wondering how we apply
>> this to a relational model of data, or to what degree it is applicable
>> without modification? I have quite a few question, but I'll try to
>> focus on only one for now.
>>
>> Suppose I define an alphabet A to be the the set of symbols {'a', 'b',
>> 'c', 'd', 'e'}, and the set of strings S over A consisting of the set
>> of 1-tuples such that formal language is defined as {'a', 'b', 'c',
>> 'd', 'e'}. We can also refer to S as a domain S in some universe of
>> discourse.
>>
>> Further suppose that a binary relation is defined over S, is given as
>> R(u: S, v: S); and we claim that it is an equivalence relation, and it
>> has the following extensional value:
>>
>> u v
>> -- --
>> a a
>> b b
>> c c
>> d d
>> a b
>> b a
>> c d
>> d c
>>
>> Here there are 4 equivalence classes and two distinct equivalence
>> classes.
>
> There are just two equivalence classes. What's a 'distinct' equivalence
> class ? Is it some kind of 'computer'-math speak ?
>
Erratum

Change the words "4 equivalence classes" to "four elements with equivalence classes, denoted [a], [b], [c], [d]."

I hope that helps.

>
> Saw your later comment. Since an equivalence class is a subset of the
> original set over which the equivalence relation is defined, how the
> {a,b} subset is different from the {a, b} subset ?
>
Yes, it is a partition and the sets are the same. Perhaps if one applies a name to both sets, we now have a basis for distinguishing the first from the second, even if they were to be proven equal? This shouldn't be surprising -- we do the same with relations of the same predicate and arity, or when we ask whether set A is equal to set B. They might be exactly the same, but we still distinguish them because they might not be the same.

Incidentally, this is somewhat tangential to the thrust of my questions. Abiteboul, Hull, and Vianu mention active domains in the context of query languages as well as some other concepts, and that information helped me answer my own intuitions and natural curiousity a great deal (in the context of my questions posed in the thread). I should have known better than to ask.

Regards,

• Dan