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)
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