A First Course in Fuzzy Logic

The Concept of Fuzziness

Examples. Mathematical modeling. Some operations on fuzzy sets. Fuzziness as uncertainty.

Some Algebra of Fuzzy Sets

Boolean algebras and lattices. Equivalence relations and partitions. Composing mappings. Isomorphisms and homomorphisms. Alpha-cuts. Images of alpha-level sets.

Fuzzy Quantities

Fuzzy quantities. Fuzzy numbers. Fuzzy intervals.

Logical Aspects of Fuzzy Sets

Classical two-valued logic. A three-valued logic. Fuzzy logic. Fuzzy and Lukasiewicz logics. Interval-valued fuzzy logic.

Basic Connectives

t-norms. Generators of t-norms. Isomorphisms of t-norms. Negations. Nilpotent t-norms and negations. T-conforms. De Morgan systems. Groups and t-norms. Interval-valued fuzzy sets. Type-2 fuzzy sets.

Additional Topics on Connectives

Fuzzy implications. Averaging operators. Powers of t-norms. Sensitivity of connectives. Copulas and t-norms.

Fuzzy Relations

Definitions and examples. Binary fuzzy relations. Operations on fuzzy relations. Fuzzy partitions. Fuzzy relations as Chu spaces. Approximate reasoning. Approximate reasoning in expert systems. A simple form of generalized modus ponens. The compositional rule of inference.

Universal Approximation

Fuzzy rule bases. Design methodologies. Some mathematical background. Approximation capability.

Possibility Theory

Probability and uncertainty. Random sets. Possibility measures.

Partial Knowledge

Motivations. Belief functions and incidence algebras. Monotonicity. Beliefs, densities, and allocations. Belief functions on infinite sets. Mobius transforms of set-functions. Reasoning with belief functions. Decision making using belief functions. Rough sets. Conditional events.

Fuzzy Measures

Motivation and definitions. Fuzzy measures and lower probabilities. Fuzzy measures in other areas. Conditional fuzzy measures.

The Choquet Integral

The Lebesgue integral. The Sugeno integral. The Choquet integral.

Fuzzy Modeling and Control

Motivation for fuzzy control. The methodology of fuzzy control. Optimal fuzzy control. An analysis of fuzzy control techniques.