Re: Testing for the equivalence relation

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