# Re: Testing for the equivalence relation

Date: Fri, 01 Jul 2005 19:23:19 GMT

Message-ID: <HKgxe.134988$nN3.7127176_at_phobos.telenet-ops.be>

Dan wrote:

*>
*

> 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.
*

Why not simply take A as your domain? Why make strings of the symbols and then only those that consist of one symbol? And why put them in 1-tuples? This doesn't make much sense.

> 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.
*

No, there are two equivalence classes. Period. Saying that there are 4 is sloppy terminology.

> I can look at this relation and interpret whether it meets the criteria

*> for reflexivity in several ways
*

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.

> For me, one of the benfits of answering this question is that it goes a

*> long way ascertaining a distinction between type and domain.
*

Hm, you think there is a difference?

- Jan Hidders