Fuzzy Logic and Fuzzy Sets

Fuzzy Logic

• Fuzzy logic was developed by Lotfi Zadeh in 1965.
• Boolean logic uses binary truth values (true or false).
• Fuzzy logic allows for partial truth values.
• Fuzzy logic can represent uncertainty in events, unlike Boolean logic which cannot.
• Fuzzy logic derives statements from natural language statements.

Fuzzy Sets

• Fuzzy sets use linguistic identifiers (e.g., tall, young, bigger).
• They use a membership grade, μ(x), where 0 ≤ μ(x) ≤ 1.
• A fuzzy set is a set of ordered pairs, A = {(x, μ(x))}.

Membership Function

• Membership functions define the grade of membership for elements in a fuzzy set.
• An example is the "young" membership function for age.
• The function determines how much an element belongs to a set; it does not define sharp boundaries (crisp sets do) rather fuzzy boundaries.

Types of Membership Functions

• Piecewise linear functions: use straight lines, increasing or decreasing.
• Triangular functions: are defined by three points (x1, x2, and x3)
• Smooth curves: have continuously differentiable membership functions (e.g., s-curve, z-curve, π-curve).
• Different shapes for different functionalities.

Operations on Fuzzy Sets

• Union: A∪B = A or B, represented by maximum functions.
• Intersection: A∩B = A and B, represented by minimum functions.
• Complement: ¬A = not A, represented by (1 - μ(x)).

Algebraic Properties

• Fuzzy sets follow specific algebraic properties like commutativity, associativity, and distributivity.
• These properties facilitate calculations in fuzzy logic systems.
• DeMorgan's rules are also relevant for fuzzy sets.

Linguistic Modifiers

• Linguistic modifiers approximate or modify fuzzy sets.
• Examples include "very," "somewhat," "extremely," and more.
• These modifiers adjust and refine fuzzy sets based on their intents or degrees.
• Intensification and dilution alter the shape of fuzzy sets.