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)x®x x is as old as y
> Irreflexive (Ax)not(x®x) x is the mother of y
> Symmetric (Ax,y)(x®y -> y®x) x is the spouse of y
> Asymmetric (Ax,y)(x®y -> not y®x) x is the husband of y
> Antisymmetric (Ax,y)(x®y & y®x -> x=y) x was present at y's birth
> Transitive (Ax,y)(x®y & y®z -> x®z) 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
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