2007 fuzzy book review

"Fuzzy Logic for Business, Finance, and Management" Bojadziev, 2007, worldscibooks.com

Four decades have passed since Lotfi Zadeh delivered fuzzy set theory in 1965.
Hundreds of fuzzy set theory books and papers have been written since then.
Here is the latest ... the publisher gives away Chapter One for free download.

Here is the outline of Chapter One.

(1.1) - classical set theory review: relations and functions

bi-variate membership rule, universal set, empty set, interval, subset, complement, intersection, union, disjoint, convex, complementary

cartesian product, relation, function

indicator (or characteristic) function

membership function

tall example

(1.2) - fuzzy set theory introduction

fuzzy set is a generalization of Cantor crisp set

fuzzy set members have a degree of membership - is formally equal to it's membership function

fuzzy set normalization

fuzzy singleton set

alpha-level interval, or a-cut defined- is a crisp set, is a confidence level indicator

fuzzy set convexity - iff all alpha-level intervals are convex

tall example, revisited

(1.3) - basic operations on fuzzy sets

most classical set operations hold, except: the principle of the excluded middle does *NOT*

(1.4) - fuzzy numbers

fuzzy number is a fuzzy set that is convex and normalized (concave != convex)

fuzzy number have an interval [a1; a2] called the supporting interval, as well as a maximum

piecewise-quadratic fuzzy number - bell-shape with 2 parameters: peak point and bandwidth

(1.5/6) - triangular / trapezoidal fuzzy numbers

(1.7) - fuzzy relations

2 notions of generalization .... (? intentional or accidental)

fuzzy relations generalize fuzzy sets from 2-space to 3-space

fuzzy relations generalize crisp relations (more expressive domain:-> range mapping )

... enter 'linguistic relations'

(1.8) fuzzy relation operations

R1 = { (x,y), ur1(x,y) } , (x,y) in A x B

previous operations confirmed, plus

direct min prouduct, and

direct max prouduct

Concluding Notes

examples: Heap, Tall

fuzzy sets are not a new probability type, there are substantial differences.

For instance, grade or degree of membership is not a probablistic concept.