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

 To: "[ontolog-forum]" "John F. Sowa" Mon, 28 May 2007 11:28:46 -0400 <465AF52E.1000102@xxxxxxxxxxx>
 ```Folks,    (01) I have a web page that gives a brief review and summary of some basic topics and definitions in math and logic:    (02) http://www.jfsowa.com/logic/math.htm Mathematical Background    (03) 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.    (04) John Sowa _____________________________________________________________    (05) 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    (06) An excerpt from Section 5:    (07) 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.    (08) 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    (09) 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:    (010) _________________________________________________________________ Message Archives: http://ontolog.cim3.net/forum/ontolog-forum/ Subscribe/Config: http://ontolog.cim3.net/mailman/listinfo/ontolog-forum/ Unsubscribe: mailto:ontolog-forum-leave@xxxxxxxxxxxxxxxx Shared Files: http://ontolog.cim3.net/file/ Community Wiki: http://ontolog.cim3.net/wiki/ To Post: mailto:ontolog-forum@xxxxxxxxxxxxxxxx    (011) ```
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