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# Re: Testing for the equivalence relation

From: Jan Hidders <jan.hidders_at_REMOVETHIS.pandora.be>
Date: Fri, 01 Jul 2005 23:15:52 GMT
Message-ID: <I8kxe.135100\$Vs4.7201818@phobos.telenet-ops.be>

Dan wrote:
> "No, there are two equivalence classes. Period. Saying that there are 4
> is sloppy terminology."
>
> Equivalance classes, denoted with brackets:
> [a] = {a, b}
> [b] = {a, b}
> [c] = {c, d}
> [d] = {c, d}
>
> Distinct equivalence classes
> {a,b}
> {c,d}

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.

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.

> 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.

> 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?

```>>For me, one of the benfits of answering this question is that it goes a
>>long way ascertaining a distinction between type and domain.
```

>
> Jan: "Hm, you think there is a difference?"
>
> Actually, in one sense, yes and in another sense, no.

Very Zen.

• Jan Hidders
Received on Fri Jul 01 2005 - 18:15:52 CDT

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