Introduction to Fuzzy Logic Fuzzy Sets Shadi T

Introduction to Fuzzy Logic Fuzzy Sets Shadi T. Kalat Session number 03/18/2016 1

Fuzzy Sets • Fuzzy sets • In mathematics, fuzzy sets are sets whose elements have degrees of membership • Crisp sets: • In a crisp set, an element is either a member of the set or not.

Fuzzy Sets •

Fuzzy Sets Continuous Universe Membership Grade Discrete Universe X= Number of children X=Age

Fuzzy Sets • A fuzzy set (A) could be written in two forms: Discrete Universe Continuous Universe

Fuzzy Sets Support and Core • Support • Height • Core Membership Grade

Fuzzy Sets Crossover points • Height Core α-cut

Fuzzy Sets • Convexity Convex Non-convex

Fuzzy Sets • Scalar Cardinality

Membership Functions • Triangular membership function Membership Grade

Membership Functions • Trapezoidal membership function Membership Grade

Membership functions • Gaussian membership function Membership Grade

Membership functions • Generalized bell membership function Membership Grade

• Fuzzy subset • Fuzzy negation Membership Grade • Equivalence Membership Grade Operations on fuzzy sets

Operations on fuzzy sets • Sugeno’s complement • Yager’s complement

Operations on fuzzy sets Fuzzy sets A and B Fuzzy sets A OR B Fuzzy set NOT A Fuzzy sets A AND B

Membership Grade 2 D fuzzy set

T-Norm (Triangle norm)

T-norms • Min • Algebraic product • Bounded product • Drastic product

T-norms Min A&B Algebraic product

T-norms Bounded product Drastic product

T-norms

S-norms

S-norms • Max • Algebraic sum • Bounded sum • Drastic sum

S-norms Max A&B Algebraic sum

S-norms Bounded sum Drastic sum

Fuzzy Rules • Extension Principle F, y, x belong to a crisp set

Fuzzy Rules • Assume A is a fuzzy member of X

Example

Cylindrical extension of A Membership Grade Base fuzzy set A

Projections of fuzzy sets 2 -D Membership Function Projection onto X Projection onto Y

Fuzzy relations

Fuzzy Relations • Max-Min Composition • Max-Dot Product

Fuzzy Relations

Fuzzy Relations • Max-Min • Max-Dot 35

Linguistic Variables • Primary terms: Young, old, … • Negation: Not young, Not old • Hedge: Very old, Extremely young, more or less old 36

Linguistic Variables • Concentration (very) Contrast intensifier effect • Not • Contrast Intensification Membership Grade • Dilation (more or less) 37