Re: The naive test for equality

From: VC <boston103_at_hotmail.com>
Date: Sun, 7 Aug 2005 22:47:54 -0400
Message-ID: <WqadnZ2dnZ2OY_SHnZ2dnUdaa9-dnZ2dRVn-zp2dnZ0@comcast.com>

"Paul" <paul_at_test.com> wrote in message news:42f4cf30$0$91514$ed2e19e4_at_ptn-nntp-reader04.plus.net...
> vc wrote:

>>> <Paul> wrote:
>>> "well, the equivalence class can be thought of as a set of possible
>>> representations for the "value" that "is" the equivalence class "
>>
>> I do not see how 'possible representations' (whatever they are), or
>> 'literals', are relevant to the simple notion of equivalence class.
>

> maybe you're readng more into it than I mean.

>

> Probably a concrete example might best explain what I'm trying to say.

>

> Consider simple fractions. You have several ways of writing the number
> 0.5, for example 1/2, 2/4, 3/6, etc. (infinitely many in fact). I'm just
> saying that all these are possible ways of representing the same number
> or "value".

As I wrote before, the commonly accepted terminology is to call the integer pairs (5,10), (1,2), (2,4), (3,4) members of the same equivalence class, or sometimes *representatives* of the same equivalence class. Why you insists on using your private vocabulary [representations], rather than the straigtforward and unambiguos "pairs of integers" is a complete mystery to me. Whether one prefers to use the (5,10) pair or the (1,2) pair is dictated solely by convenience.

>

> You gave the details of how the rationals are constructed mathematically
> using equivalence relations. In practice, you aren't going to write the
> rational number 0.5 as the set (1/2, 2/4, 3/6, ...), you will pick one
> example and use that. In think the standard notation used is square
> brackets e.g. [1/2] to denote the equivalence class to which 1/2
> belongs. Or you could just as well use [2/4].

As I said, it just does not matter which specific member of the equivalence class you pick. Sometimes 0.5 (5, 10) is more convenient. sometimes (1,2) is more preferable.

>

> I've kind of lost track of what started this thread in the first place
> now! I think it was just to say I didn't think there was any real
> difference between equality and equivalence relations. Each one defines
> the other.

>

> When we write 1/2 or 2/4 it is just shorthand for "[1/2]" or "the
> equivalence class containing 1/2" so 1/2 and 2/4 are actually identical
> at some level

To rephrase it correctly, the equivalence relation for pairs of integers (where the second component must be non-zero) defines the equality relation between the equivalence classes (sort of obvious).

.>But clearly at the level of marks on paper or bytes on a
> computer they are different.

Who cares, at the logical level, about the specific language (marks on paper) or specific implementation (bytes) ?

> So something can be
> equal at the logical level but different at the physical level.

OK, so what ?

>

> Slightly confusing the issue is the fact that we are using the word
> "relation" in a mathematical rather than database sense here.

There is not much substantial difference.

>
> Paul.
Received on Sun Aug 07 2005 - 21:47:54 CDT