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equivalence relation

From: Henry Story <henry.story@bblfish.net>
Date: Mon, 7 Feb 2005 09:02:06 +0100
Message-Id: <e52f0598a254be02e176b916e9f22a81@bblfish.net>
To: semantic-web@w3.org

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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