# equivalence relation

From: Henry Story <henry.story@bblfish.net>
Date: Mon, 7 Feb 2005 09:02:06 +0100
Message-Id: <e52f0598a254be02e176b916e9f22a81@bblfish.net>

```
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 08:02:33 UTC

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