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Re: [ontolog-forum] Ontology, Information Models and the 'Real World': C

To: "[ontolog-forum]" <ontolog-forum@xxxxxxxxxxxxxxxx>
From: Waclaw Kusnierczyk <Waclaw.Marcin.Kusnierczyk@xxxxxxxxxxx>
Date: Mon, 28 May 2007 18:57:46 +0200
Message-id: <465B0A0A.1040604@xxxxxxxxxxx>
John,    (01)

That's a very useful summary.  I do not think there were any doubt how 
the usual definition of 'asymmetric' goes.  I had some doubt as to 
whether Pat has adopted another definition, but this issue has been 
succesfully solved.    (02)

Thanks anyway.    (03)

vQ    (04)

John F. Sowa wrote:
> Folks,
> 
> I have a web page that gives a brief review and summary
> of some basic topics and definitions in math and logic:
> 
>     http://www.jfsowa.com/logic/math.htm
>     Mathematical Background
> 
> Following is the table of contents (each section takes about
> 3 or 4 pages, if printed out).  After that is an excerpt from
> Section 5 on relations.
> 
> John Sowa
> _____________________________________________________________
> 
>     1. Sets, Bags, and Sequences
>     2. Functions
>     3. Lambda Calculus
>     4. Graphs
>     5. Relations
>     6. Representing Relations by Graphs
>     7. Lattices
>     8. Propositional Logic
>     9. Predicate Logic
>    10. Axioms and Proofs
>    11. Formal Grammars
>    12. Game Graphs
>    13. Model Theory
>    14. References
> 
> An excerpt from Section 5:
> 
> The following table lists some common types of relations, an axiom that 
> states the defining constraint for each type, and an example of the 
> type. The symbol  represents an arbitrary dyadic relation.
> 
>    Type               Axiom                      Example
>    ----          -----                      -------
>    Reflexive     (Ax)xx                    x is as old as y
>    Irreflexive        (Ax)not(xx)               x is the mother of y
>    Symmetric  (Ax,y)(xy -> yx)         x is the spouse of y
>    Asymmetric         (Ax,y)(xy -> not yx)     x is the husband of y
>    Antisymmetric (Ax,y)(xy & yx -> x=y)   x was present at y's birth
>    Transitive    (Ax,y)(xy & yz -> xz)   x is an ancestor of y
> 
> The symbol A, called the universal quantifier, may be read "for every" 
> or "for all".  It is discussed further in Section 9 on predicate logic. 
> Some important types of relations -- such as partial order, linear 
> order, and equivalence -- satisfy two or more of the above axioms:
> 
>  
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>      (05)

-- 
Wacek Kusnierczyk    (06)

------------------------------------------------------
Department of Information and Computer Science (IDI)
Norwegian University of Science and Technology (NTNU)
Sem Saelandsv. 7-9
7027 Trondheim
Norway    (07)

tel.   0047 73591875
fax    0047 73594466
------------------------------------------------------    (08)

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