Fuzzy Set Theory

Fuzzy Set Theory

• Fuzzy logic was first proposed by Lotfi A. Zadeh in 1965.
• Crisp logic demands a binary output (0/1 or Yes/No), while fuzzy logic deals with degrees of truth.
• Classical set theory was proposed by George Cantor.
• Crisp sets involve operations and properties.

Crisp Sets

• Crisp sets involve definite boundaries.
• Elements either belong or do not belong to the set.

Fuzzy Sets

• Fuzzy sets allow partial membership.
• A membership function assigns a degree of membership between 0 and 1 to each element.

Fuzzy Set Operations

• Let X be the universe of discourse.
• Let A and B be fuzzy sets with membership functions µA(x) and µB(x) respectively.

Union

• The membership function for the union of two fuzzy sets A and B is defined as µA∪B(x) = max(µA(x), µB(x)).

Intersection

• The membership function for the intersection of two fuzzy sets A and B is defined as µA∩B(x) = min(µA(x), µB(x)).

Complement

• The membership function for the complement of fuzzy set A is defined as µAc(x) = 1 - µA(x).

Product

• The membership function for the product of two fuzzy sets A and B is defined as µA.B(x) = µA(x) * µB(x).

Equality

• Two fuzzy sets A and B are considered equal when µA(x) = µB(x).

Product with a Crisp Number

• The membership function is adjusted to be a multiple of the original.

Power of a Fuzzy Set

• Concentration (CON) is raising a fuzzy set to its second power – taking the square of the membership values.
• Dilation (DIL) is taking the square root of the membership values.

Difference

• Defined as A - B = A intersect B complement.

Disjunctive Sum

• Defined as (A intersect B complement) union (A complement intersect B).

Properties of Fuzzy Sets

• Inclusion (A ⊆ B): µA(x) ≤ µB(x) for all x.

Properties of Fuzzy Sets (Laws)

• Commutative: A ∪ B = B ∪ A; A ∩ B = B ∩ A
• Associative: A ∪ (B ∪ C) = (A ∪ B) ∪ C; A ∩ (B ∩ C) = (A ∩ B) ∩ C
• Distributive: A ∪ (B ∩ C) = (A ∪ B) ∩ (A ∪ C); A ∩ (B ∪ C) = (A ∩ B) ∪ (A ∩ C)
• Idempotence: A ∪ A = A; A ∩ A = A

Fuzzy Set Characteristics

• Normality: A fuzzy set F is normal if there exists an element x in the universe such that µF(x) = 1.
• Height: The height of a fuzzy set F is the maximal membership value obtained by its elements.
• Support: The support of a fuzzy set F is the set of all elements of the universe with non-zero membership in F.
• Core: The core of a fuzzy set F is the set of all elements of the universe with complete membership (membership value = 1) to F.
• Cardinality: The sum of all membership values of the fuzzy set.

Fuzziness vs Probability

• Fuzziness represents vagueness, while probability represents uncertainty.

Membership Functions

Crisp Membership Function

• An element x is either in the set (µs(x) = 1) or not (µs(x) = 0).

Partial Membership Function

• An element can belong to a set to a certain degree, within the range [0, 1].

Standard Fuzzy Membership Functions

Triangular Function

• Defined by three parameters, a, m, and b, with a linear increase from a to m and a linear decrease from m to b.

Trapezoidal Function

• Similar to a triangular function, but with a flat peak between two values.

Gaussian Function

• Characterized by differentiability everywhere, using the exponential function.

S-Function

• Differentiable everywhere, defining membership based on a curve.

Fuzzy Transformations

• Normalization: Converts a subnormal fuzzy set to a normal fuzzy set, by dividing by the height of the fuzzy set: Norm(F,x) = MF(x) / height(F)
• Dilation: DIL(F,x) = [MF(x)]^(1/2)
• Concentration: CON(F,x) = [MF(x)]^2
• Contrast Intensification: A transformation that increases membership values > 0.5 and decreases those < 0.5

Fuzzy Relations

• Crisp Relation: Based on Cartesian product, resulting in ordered pairs.
• Fuzzy Relation: A fuzzy set defined on the Cartesian product, where n-tuples can have varying degrees of membership.

Operations on Relations

Union

• Represented as R ∪ S; resulting in maximized membership function.

Intersection

• Represented as R ∩ S; resulting in minimized membership function.

Complement

• R'(x, y) = 1 - R(x, y)

Composition of Relations

• Max-min composition: Involves comparing the minimum of the relationship between x and y from R and the relationship between y and z from S, then taking the maximum of these minimums.

Operations on Fuzzy Relations

• Union: R ∪ S (x,y) = max(Ř (x,y), Š (x,y))

• Intersection: R ∩ S (x,y) = min(Ř (x,y), Š (x,y))

• Complement: R’ (x,y) = 1 - R(x,y)

• Inclusion: R ⊆ S if ∀x, y Ř (x,y) ≤ Š(x,y)

• Dominance: R ≥ S if ∀x, y Ř (x,y) ≥ Š(x,y)

• Equality: R = S if ∀x, y Ř (x,y) = Š (x,y)

• Max-min Composition of Fuzzy Relations

Logic

• Crisp Logic: Operates on binary values (0 or 1).
• Fuzzy Logic: Handles multiple states of membership.
• Propositional Logic: Deals with declarative statements which are either true or false.
• Predicate Logic: Deals with statements with variables.

Crisp Logic Principles

• Uses propositions that take either a true or false value.
• Employs logical operators like 'and', 'or', and 'not'.
• Has truth tables to define the outcomes of these operators.
• Tautology: Formula that is always true.
• Contradiction: Formula that is always false.

Laws of Propositional Logic

• Commutative: (P ∨ Q) = (Q ∨ P); (P ∧ Q) = (Q ∧ P)
• Associative: (P ∨ Q) ∨ R = P ∨ (Q ∨ R); (P ∧ Q) ∧ R = P ∧ (Q ∧ R)
• Distributive: (P ∨ Q) ∧ R = (P ∧ R) ∨ (Q ∧ R); (P ∧ Q) ∨ R = (P ∨ R) ∧ (Q ∨ R)
• Identity: P ∨ false = P; P ∧ true = P
• Negation: P ∧ ¬P = false; P ∨ ¬P = true

Inference in Propositional Logic

• It is a technique that derives a goal from facts or premises.
• Rules of Inference: Modus Ponens, Modus Tollens, Chain Rule

Modus Ponens

• If P → Q and P are true, then Q is true.

Predicate Logic Fundamentals

• It enables quantifications, and uses constants, variables, predicates, and functions.
• Constants: Represent objects that don't change values.
• Variables: Symbols that represent values acquired by the objects.
• Predicates: Show the relationship or properties of objects.
• Quantifiers: Used to express the extent to which a predicate is true over a range of elements.

Types of Quantifiers

• Universal Quantifier (∀): For all.
• Existential Quantifier (∃): There exists.