# Re: equivalence relation

From: Henry Story <henry.story@bblfish.net>
Date: Mon, 7 Feb 2005 14:57:21 +0100
Message-Id: <5a386870ef639f9b131911e896d8e0df@bblfish.net>

```
I just realized that not everyone may know (including me btw.)
what I mean by an equivalence relation .

I think this is the concept I am getting at:

<http://en.wikipedia.org/wiki/Equivalence_relation>

Henry Story

On 7 Feb 2005, at 09:02, Henry Story wrote:

>
> I am looking for a way to state that a relation is an equivalence
> relation [1]. I want to know this so that I can starting from a graph
> such as
>
> _blank ---relation---> <http://bblfish.net/>
>   |------owner-------> "Henry Story"
>
> deduce the graph
>
> <http://bblfish.net> ----owner----> "Henry Story"
>
>
> My thought was that a relation that is functional, symmetric and
> transitive
>  is just such a relation. Here is how I come to this conclusion.
>
> 1) Functional and symmetric
>
>  If a relation is functional and symmetric, then it is also
>  inverse functional. It is a 1 to 1 mapping.
>
> 2) If it is functional, inverse functional and symmetric
>
>    then for all aRb we also have bRa
>
>    this still allows a and b to be different
>
> 3) if it is transitive then for any a, b and c, where
>
>    [1] aRb
>    [2] bRc
>
>    then
>
>    [3] aRc
>
>    but since R is symmetric
>
>    from [2] bRc we deduce that
>
>    [4] cRb
>
>     and since R is inverse functional
>
>    from  [1] aRb and [4] cRb we deduce that a==c
>
>    similarly from [3] aRb, [1] aRc and the functional nature of R
>    we deduce that c == b.
>
>    So a == c and c == b and so a == b.
>
> Is this reasoning ok?
> I was hoping it would be, cause then I can just specify in OWL that
> properties
> are functional, symmetric and transitive if I want them to be
> equivalence relations (or is there a shorthand for this)
>
> Henry Story
>
> [1] http://en.wikipedia.org/wiki/Equivalence_relation
>
```
Received on Monday, 7 February 2005 13:57:46 UTC

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