This document discusses computational intelligence and fuzzy set theory. It defines soft computing constituents like neural networks, fuzzy set theory, genetic algorithms, and conventional AI. It then discusses fuzzy set theory, including fuzzy sets, membership functions, α-cuts, and set theoretic operations like union, intersection, and complement. It also defines two-dimensional membership functions like cylindrical extensions and projections of fuzzy sets.

1. Computational Intelligence
Parveen Malik
Assistant Professor
School of Electronics Engineering
KIIT University

2. Outline
• Soft Computing Constituents
• Fuzzy Set Theory
• Set Theoretic Operations
• Two Dimensional MFs

3. Soft Computing Constituents
• Neural Network
• Fuzzy Set Theory
• Derivative Free Optimization( Genetic Algorithms)
• Conventional AI
Text Book : Chuen-Tsai Sun, Eiji Mizutani, and Jyh-Shing Roger Jang. “Neuro-Fuzzy and Soft Computing: A
Computational Approach to Learning and Machine Intelligence”

4. Hidden Layer
Output Layer
Input Layer
Neural Network

5. Fuzzy Set Theory
Degree
of
Belief
Fuzzy
Hot
Cold
Degree
of
Belief
Non-Fuzzy (Crisp)
1.0
0.0
Cold
Hot
1.0
0.0
𝟒𝟏𝟎𝑪
𝟑𝟗𝟎𝑪
𝟒𝟎𝟎
𝑪
𝟐𝟎𝟎𝑪
𝟏𝟐𝟎𝑪
𝟑𝟎𝟎𝑪
𝟐𝟓𝟎
𝑪
𝟒𝟏𝟎𝑪
𝟑𝟗𝟎𝑪
𝟒𝟎𝟎
𝑪
𝟐𝟎𝟎𝑪
𝟏𝟐𝟎𝑪
𝟑𝟎𝟎𝑪
𝟐𝟓𝟎
𝑪

6. History of Conventional AI approaches

7. Fuzzy Set Theory Vs Classical Set Theory
Classical Set Theory :
• Crisp, clear or unambiguous boundary
𝐴 = 𝑥 𝑥 > 6}
• Important tool for mathematic and computer science.
• Sharp Transition between “Inclusion” and Exclusion”.
• Incapable of addressing human concepts and thoughts
which tend to be abstract and imprecise.
A=“Tall Person” x =“Height”
B= Water is hot” C=“Temp. is High”

8. Fuzzy Set Theory Vs Classical Set Theory
Fuzzy Set Theory :
• Introduced by “L.A.Zadeh” in 1965 paper “Fuzzy Sets”
• Unclear or ambiguous boundary
• Capable of addressing human concepts and thoughts which
tend to be abstract and imprecise.
A=“Tall Person” x =“Height”
B= Water is hot” C=“Temp. is High”
• Introduction of membership function
• Gradual Transition from “Belong to a set” To “Not belong to
set”.

9. Fuzzy Set Theory
Definition 1.1 : Fuzzy Set and membership functions
If X is a collection of objects denoted generically by x, then a fuzzy set A in X
is defined as a set of ordered pairs:
A= 𝒙, 𝝁𝑨(𝒙) | 𝒙 ∈ 𝑿
Where 𝜇𝐴(𝑥) is the membership function for the fuzzy set A
Definition 1.2 : support
The support of A fuzzy set A is the set of all points x in X such that 𝜇𝐴(𝑥) >0
support(A)= 𝒙 | 𝝁𝑨 𝒙 > 𝟎
Definition 1.3 : core
The core of A fuzzy set A is the set of all points x in X such that 𝜇𝐴(𝑥) =1
core(A)= 𝒙|𝝁𝑨 𝒙 = 𝟎
Definition 1.4 : Normality
A fuzzy set A is normal if its core is non-empty.

10. Fuzzy Set Theory
Definition 1.5 : Cross Over Points
A crossover point of a fuzzy set A is a point 𝑥 ∈ 𝑋 at which 𝜇𝐴(𝑥) = 0.5.
crossover(A)= 𝒙|𝝁𝑨 𝒙 = 𝟎. 𝟓
Where 𝜇𝐴(𝑥) is the membership function for the fuzzy set A.
Definition 1.6 : Fuzzy singleton
A fuzzy set whose support is a single point in X with 𝜇𝐴(𝑥) = 1 is called a Fuzzy
Singleton.
Definition 1.7 : α-cut, strong α-cut
α-cut, α-level set of a fuzzy set A is crisp set defined by
𝑨𝜶= 𝒙|𝝁𝑨 𝒙 ≥ 𝟎
Strong α-cut, strong α-level set of a fuzzy set A is defined by
𝑨𝜶
′
= 𝒙|𝝁𝑨 𝒙 > 𝟎

11. Fuzzy Set Theory
Definition 1.8 : Convexity
A fuzzy set A is convex if and only if for any 𝑥1, 𝑥2 ∈ 𝑋 𝑎𝑛𝑑 for any λ ∈ 0,1
𝝁𝑨(𝝀𝒙𝟏 + (𝟏 − 𝝀)𝒙𝟐) ≥ Min 𝝁𝑨 𝒙𝟏 , 𝝁𝑨 𝒙𝟐
Alternatively, A is convex if all its 𝛼 − 𝑠𝑒𝑡𝑠 are convex.
Definition 1.9 : Fuzzy numbers
A fuzzy number A is a fuzzy set in the real line that satisfies the conditions for
Normality and convexity.
Definition 1.10 : Bandwidths of normal and convex fuzzy sets
For a normal and convex fuzzy set, the bandwidth or width is defined as the distance
between the two unique crossover points:
𝒘𝒊𝒅𝒕𝒉 𝑨 = 𝒙𝟐 − 𝒙𝟏
Where 𝝁𝑨 𝒙𝟏 = 𝝁𝑨(𝒙𝟐)=0.5

12. Fuzzy Set Theory
Definition 1.11 : Symmetry
A fuzzy set A is symmetric if its MF is symmetric around a certain
point x=c ,
𝝁𝑨 𝒄 + 𝒙 = 𝝁𝑨 𝒄 − 𝒙 for all 𝒙 ∈ 𝑿
Definition 1.12 : Open left, Open right, closed
A fuzzy set A is open left
if 𝒍𝒊𝒎𝒙→−∞𝝁𝑨 𝒙 =1 and 𝒍𝒊𝒎𝒙→+∞𝝁𝑨 𝒙 = 0
open right
if 𝒍𝒊𝒎𝒙→−∞𝝁𝑨 𝒙 =0 and 𝒍𝒊𝒎𝒙→+∞𝝁𝑨 𝒙 = 1
Closed
if 𝒍𝒊𝒎𝒙→−∞𝝁𝑨 𝒙 =𝒍𝒊𝒎𝒙→+∞𝝁𝑨 𝒙 =0

13. Set Theoretic Operations
Definition 2.1 : Containment or Subset
Fuzzy set A is contained in Fuzzy set B if and only if
𝑨 ⊆ 𝑩 ⇔ 𝝁𝑨 𝒙 ≤ 𝝁𝑩 𝒙
Definition 2.2 : Union (Disjunction)
The union of two sets A and B is a set C (𝐴 ∪ B) whose MF is
related by
𝝁 𝒙 = max {𝝁𝑨 𝒙 , 𝝁𝑩 𝒙 } = 𝝁𝑨 𝒙 ˅ 𝝁𝑩 𝒙
Definition 2.3 : Intersection (Conjunction)
The intersection of two sets A and B is a set C (𝐴 ∩ B) whose MF
is related by
𝝁 𝒙 = min {𝝁𝑨 𝒙 , 𝝁𝑩 𝒙 } = 𝝁𝑨 𝒙 ˄ 𝝁𝑩 𝒙

14. Set Theoretic Operations
Definition 2.4 : Complement (Negation)
The complement of Fuzzy set A is ഥ
𝐀 ¬𝐀 𝐨𝐫 𝐍𝐎𝐓 𝐀
𝝁ഥ
𝑨 𝒙 = 𝟏 − 𝝁𝑨 𝒙
Definition 2.5 : Cartesian product and co-product
Let A and B be fuzzy sets in X and Y respectively. The Cartesian
product of A and B denoted by 𝑨 × 𝑩 ,
𝝁𝑨×𝑩 𝒙, 𝒚 = min {𝝁𝑨 𝒙 , 𝝁𝑩 𝒚 }
The Co- product of A and B denoted by A + B,
𝝁𝑨+𝑩 𝒙, 𝒚 = max {𝝁𝑨 𝒙 , 𝝁𝑩 𝒚 }

15. Two Dimensional MFs
Definition 3.1 : Cylindrical extensions of 1D fuzzy sets
If A is a fuzzy set in X, then its cylindrical extension in 𝑿 × 𝒀 is a
fuzzy set c(A) defined by
𝒄(𝑨) = න
𝑿×𝒀
𝝁𝑨(𝒙)/(𝒙, 𝒚)

16. Two Dimensional MFs
Definition 3.1 : Projections of fuzzy sets
If R be a two-dimensional fuzzy set on 𝑿 × 𝒀. The projections of R onto X and Y are
defined as
𝑹𝑿 = න
𝑋
𝒎𝒂𝒙 {𝝁𝑹(𝒙, 𝒚)} /𝒙
𝑹𝒀 = න
𝑌
𝒎𝒂𝒙 {𝝁𝑹(𝒙, 𝒚)} /𝒚