Crisp Sets: Fuzzy Logic and Neural Networks

• Course covers Fuzzy Logic and Neural Networks
• Lectures by Prof. Dilip Kumar Pratihar, Mechanical Engineering department

Classical/Crisp Sets

• These have well-defined boundaries
• Universal Set/Universe of Discourse (X): comprises all possible elements
• Example: All technical universities worldwide
• Classical or Crisp Sets have fixed and well-defined boundaries
• Example: Technical universities with at least five departments

Representing Crisp Sets

• Can be done by listing elements: A={a1, a2, ..., an}
• Or by defining a property: A={x|P(x)}, where P is a property
• Characteristic functions are use to represent a crisp set

Characteristic Functions

• μα(x) = 1, if x belongs to A
• μα(x) = 0, if x does not belong to A

Set Theory Notation

• Φ: Represents an Empty/Null set
• x ∈ A: Element x of the Universal set X belongs to set A
• x ∉ A: Element x does not belong to set A
• A ⊆ B: Set A is a subset of set B
• A ⊇ B: Set A is a superset of set B
• A = B: Sets A and B are equal
• A ≠ B: Sets A and B are not equal
• A ⊂ B: A is a proper subset of B
• A ⊃ B: A is a proper superset of B
• |A|: Cardinality refers to the total number of elements in set A
• p(A): Power set counts the maximum subsets from set A, including the null set
• |p(A)| = 2^|A|

Crisp Set Operations

• Difference: A − B = {x | x ∈ A and x ∉ B}, also known as relative complement of set B
• Absolute complement: Ā = AC = X − A = {x | x ∈ X and x ∉ A}
• Intersection: A ∩ B = {x | x ∈ A and x ∈ B}
• Union: A U B = {x | x ∈ A or x ∈ B}

Properties of Crisp Sets

• Law of involution: Ā = A
• Law of Commutativity: A ∪ B = B ∪ A; A ∩ B = B ∩ A
• Associativity: (A ∪ B) ∪ C = A ∪ (B ∪ C); (A ∩ B) ∩ C = A ∩ (B ∩ C)
• Distributivity: A ∩ (B ∪ C) = (A ∩ B) ∪ (A ∩ C); A ∪ (B ∩ C) = (A ∪ B) ∩ (A ∪ C)
• Laws of Tautology: A ∪ A = A; A ∩ A = A
• Laws of Absorption: A ∪ (A ∩ B) = A; A ∩ (A ∪ B) = A
• Laws of Identity: A ∪ X = X; A ∩ X = A; A ∪ Φ = A; A ∩ Φ = Φ
• De Morgan's Laws: (A ∩ B) = Ā ∪ B; (A ∪ B) = Ā ∩ B
• Law of contradiction: A ∩ Ā = Φ
• Law of excluded middle: A ∪ Ā = X

Fuzzy Sets

• Fuzzy sets handle imprecise/vague boundaries
• Introduced in 1965 by Prof. L.A. Zadeh, University of California, USA
• Fuzzy sets are useful for handling imprecision and uncertainties
• They are a more general concept than classical sets

Fuzzy Set Representation

• A(x) = {(x, µA(x)), x ∈ X}, where µA(x) is the membership function
• Probability: Frequency of likelihood that an element is in a class.
• Membership: Similarity of an element to a class

Types of Fuzzy Sets

• Discrete: A(x) = Σ µA(xi) / xi from i=1 to n, where n is the number of elements in the set
• Continuous: A(x) = ∫ µA(x) / x over X

Convex Fuzzy Sets

• A fuzzy set A(x) is convex if μA{λx1 + (1 - λ)x2} ≥ min{μA(x1), μA(x2)}, where 0.0 ≤ λ ≤ 1.0

Membership Functions

• Triangular membership: µtriangle = max(min((x-a)/(b-a), (c-x)/(c-b)), 0)
• Trapezoidal membership: µtrapezoidal = max(min((x-a)/(b-a), 1, (d-x)/(d-c)), 0)
• Gaussian membership: µGaussian = e^(-1/2)((x-m)/σ)^2
• Bell-shaped membership: µBell-shaped = 1 / (1 + |(x-c)/a|^(2b))
• Sigmoid membership: µSigmoid = 1 / (1 + e^(-a(x-b)))

Numerical Examples of Determining Membership

• Examples are provided for finding the membership value (μ) for Triangular, Trapezoidal, Guassian and Bell Shaped and Sigmoid Membership Functions

Crisp vs Fuzzy Sets

• Difference between the crisp and fuzzy sets

Definitions in Fuzzy Sets

• α-cut of a fuzzy set αµA(x): Set of elements x from the Universal set X
• Membership values ≥ α: αµA(x) = {x | μA(x) ≥ α}
• Strong α-cut of a fuzzy set: Similar to α-cut but uses strict inequality: α+µA(x) = {x | μA(x) > α}

Support of Fuzzy Set A(x)

• A set with all x ∈ X where µA(x) > 0
• Denoted as supp(A) = {x ∈ X | µA(x) > 0}, similar to its strong 0-cut

Scalar Cardinality

• The scalar cardinality of a Fuzzy SetA(x): sum of all the member ships
• |A(x)| = Σ µA(x) over all x ∈ X

Core and Height in Fuzzy Sets

• Core of Fuzzy Set A(x): Equivalent to its 1-cut
• Height of Fuzzy Set A(x): The largest of the memberships of the elements contained in that set
• Normal Fuzzy Set: A fuzzy set is considered normal if h(A) = 1.0
• Sub-normal Fuzzy Set: h(A) < 1.0

Standard Operations in Fuzzy Sets

• Proper Subset of a Fuzzy Set if A(x) ⊂ B(x), then µA(x) < µB(x)
• Equal fuzzy sets: Denoted if A(x) = B(x) ,Then if µA(x) = µB(x)
• Complement: Is defined as Ã(x) = 1 − A(x)

Fuzzy Set Operations

• Intersection: µ(A∩B)(x) = min{µA(x), µB(x)}
• Union: µ(A∪B)(x) = max{µA(x), µB(x)}
• Algebraic product of Fuzzy Sets: A(x).B(x) = {(x, µA(x) . µB(x)), x ∈ X}
• Multiplication of a Fuzzy Set by a crisp Number: d .A(x) = {(x,d ×µA(x)), x ∈ X}
• Power of a Fuzzy set:Defined as μAp(x) = {μA(x)}p, x ∈ X
  • Concentration: p=2
  • Dilation: p=½
• The algebraic Sum of Two Fuzzy sets is: A(x) + B(x) = {(x,μA+B(x)), x ∈ X} where μa+B(x) = μA(x) + μB(x) – μA(x). μB(x)
• Bounded sum of two fuzzy sets: A(x) ⊕ B(x) = {(x, µA⊕B(x)), x ∈ X} where μA⊕B(x) = min{1, μA(x) + μB(x)}
• The difference bewtween algerbraic and bounded is that the bounded function contains an upper bound of 1

ALgerbraic Difference of Two Fuzzy Sets

• Difference A(x)-B(x) = {(x, μA-B(x)), x ∈ X}
  • μA-B(x) = μA ∩ B(x)
• Bounded difference of two fuzzy sets: A(x)ΘB(x) = {(x, µAΘB(x)), x ∈ X where μΑΘΒ(x) = max{0, μΑ(x) + μΒ(x) – 1}

Cartesian Products

• Two Fuzzy sets are defined as A(x) and B(y), whereCartesian products of the two fuzzy sets is defined by: μA×B(x, y) = min{μA(x), μB(y)}

Fuzzy Relations

• Composition of fuzzy relations
  • Let A = [aij] and B = [bjk] be two fuzzy relations expressed in the matrix form.
  • Composition of these two fuzzy relations, that is, C is represented as follows: C=A o B
  • In matrix form [cik] = [aij] o [bjk]
  • Where cik =max[min(aij, bjk)]

Properties

• Fuzzy sets follow the properties of crisp sets except the following two:
  • Law of excluded middle
  • Law of contradiction

Fuzziness Measurements

• Fuzziness for Fuzzy Sets is used
  • Let X - {x1,x2,...,Xn} be the discrete unrest of scores
  • H(A) = -∑ 〖〖UM(x1)| 〖og(UM(x1)} +{1 - UN (x1)} ||og{1 - UM (X1)}}}

Sets and Their Inaccuaracy

• I(A; B) -∑ [4 (x) x log[µg (x1)] + {+ (x1)}log 1 ~ (x1)}})