good for knowledge.

1. *Zadeh, L. A. (1965), Fuzzy sets. Information and control, 8(3), 338-353
FUZZY SETS*

2.  Introduction
 Differences between crisp sets & Fuzzy sets
 Some definitions W.R.T to Fuzzy sets (FSS)
 Graphical Representation of union & intersection
 Algebraic operations on FSs
 Convex combination
 Fuzzy Relation
 Convexity
TABLE OF CONTENTS

3.  Consider a collection/set of animals. Dogs, horses, etc. are part of it but
rocks, fluids, etc. are not part of it.
 What about bacteria, starfish?
 Consider a set that contains “all real numbers much greater than 1”.
Where does ‘2’ & ‘10’ fall?
 Such classification problems occur frequently in real life problems.
INTRODUCTION

4.  Consider few more examples: class of beautiful women, class of tall
men.
 Even though such sets have an inherent type of imprecision in
information that they convey, still they are very important in a number of
fields such as pattern recognition, etc. Such sets are called FUZZY
SETS.
INTRODUCTION….

5.  Fuzzy Sets: Those collection of objects where it is not possible to make
a sharp distinction between the belongingness or non-belongingness to
the collection.
 These are useful in cases where the source of imprecision is the absence
of sharply defined criteria of the class of membership rather that the
probability theory.
INTRODUCTION….

6.  Let the universal set be denoted by X and its elements by x i.e. X= {x}.
We define a set A on X such that 𝐴 ⊂ 𝑋. We define the term grade of
membership denoted by fA(x) which represents the information
regarding the extent of belongingness of x to set A.
 If 𝑥 ∈ 𝑋, if fA(x)= 0 or 1 only and no intermediate value, then the set A
is called the crisp set and if the value of fA x belongs to the closed
interval [0, 1], then A is called the Fuzzy set. Eg: For the set X= set of
real numbers close to 1, we have fA 0 = 0; fA 10 = 0.2; fA 500 = 1
DIFFERENCES BETWEEN CRISP
SETS & FUZZY SETS

7.  Empty FS: 𝑓𝐴 𝑥 = 0 ∀𝑥 ∈ 𝑋
 Equal FSs: Given two FSs A & B, then if 𝑓𝐴 𝑥 = 𝑓𝐵(𝑥)∀𝑥 ∈ 𝑋, then A=B
 Complement(𝑨′): It is defined as 𝑓 𝐴′ 𝑥 = 1 − 𝑓𝐴(𝑥)
 Subset: 𝐴 ⊂ 𝐵 ↔ 𝑓𝐴 ≤ 𝑓𝐵 𝑖. 𝑒. 𝑓𝐴 𝑥 ≤ 𝑓𝐵 𝑥 ∀𝑥 ∈ 𝑋
 Union: Let 𝐶 = 𝐴 ∪ 𝐵, 𝑡ℎ𝑒𝑛 𝑓𝑐 𝑥 = 𝑀𝑎𝑥 𝑓𝐴 𝑥 , 𝑓𝐵 𝑥 , ∀𝑥 ∈ 𝑋
Corollary: The union of A & B is the smallest fuzzy set containing both A & B.
 Intersection: Let 𝐶 = 𝐴 ∩ 𝐵, 𝑡ℎ𝑒𝑛 𝑓𝑐 𝑥 = 𝑀𝑖𝑛 𝑓𝐴 𝑥 , 𝑓𝐵 𝑥 , ∀𝑥 ∈ 𝑋
Corollary: The intersection of A & B is the largest fuzzy set containing both A & B.
SOME DEFINITIONS W.R.T TO FUZZY SETS (FSS)

8. Graphically, the union is shown by sections 1 & 2 of the graph. Intersection
is shown by sections 3 & 4:
GRAPHICAL REPRESENTATION OF UNION &
INTERSECTION

9.  Algebraic product: Denoted by AB, is defined for FSs A & B as:
𝑓𝐴𝐵 = 𝑓𝐴 𝑓𝐵
 Algebraic sum: Denoted by A+B, is defined as:
𝑓𝐴+𝐵 = 𝑓𝐴 + 𝑓𝐵
𝑝𝑟𝑜𝑣𝑖𝑑𝑒𝑑 𝑓𝐴 + 𝑓𝐵 ≤ 1 ∀𝑥 ∈ 𝑋
 Absolute difference: Denoted by |A-B|, is defined as:
𝑓|𝐴−𝐵| = |𝑓𝐴 − 𝑓𝐵|
ALGEBRAIC OPERATIONS ON FSS

10.  Let A, B & Λ be three FSs, then their convex combination is denoted by
𝐴, 𝐵; Λ = Λ𝐴 + Λ′ 𝐵
𝑤ℎ𝑒𝑟𝑒Λ′ is the complement of Λ
 Or in terms of the membership function as:
𝑓 𝐴,𝐵;Λ 𝑥 = 𝑓Λ 𝑥 𝑓𝐴 𝑥 + [1 − 𝑓Λ 𝑥 ]𝑓𝐵(𝑥)
 A basic property of convex combination of A, B and Λ is given by:
𝐴 ∩ 𝐵 ⊂ 𝐴, 𝐵; Λ ⊂ 𝐴 ∪ 𝐵 ∀Λ
CONVEX COMBINATION

11.  For FSs, the above expression is rewritten as:
𝑀𝑖𝑛 𝑓𝐴 𝑥 , 𝑓𝐵 𝑥 ≤ 𝜆𝑓𝐴 𝑥 + 1 − 𝜆 𝑓𝐵 𝑥 ≤ 𝑀𝑎𝑥 𝑓𝐴 𝑥 , 𝑓𝐵 𝑥
 Also, it is possible to find a FS ‘C’, s.t. C= 𝐴, 𝐵; Λ and the membership of this
set is given by:
𝑓Λ 𝑥 =
𝑓𝐶 𝑥 − 𝑓𝐵(𝑥)
𝑓𝐴 𝑥 − 𝑓𝐵(𝑥)
, 𝑥 ∈ 𝑋
CONVEX COMBINATION……

12.  A fuzzy relation in X is a fuzzy set in the product space 𝑋 × 𝑋.
 For example, the relation 𝑥 ≫ 𝑦, 𝑥, 𝑦 ∈ 𝑅, is expressed as a fuzzy set A
in 𝑅2, with the membership function value given by: 𝑓𝐴(𝑥, 𝑦). Lets say
𝑓𝐴 10,5 = 0; 𝑓𝐴 100,10 = 0.7 and 𝑓𝐴 100,1 = 1.
 However, the realtion values are subjective interpretations.
FUZZY RELATION

13.  A fuzzy set A is called the convex iff
𝑓𝐴 𝜆𝑥1 + 1 − 𝜆 𝑥2 ≥ 𝑀𝑖𝑛[𝑓𝐴 𝑥1 , 𝑓𝐴 𝑥2 ]
∀𝑥1, 𝑥2 ∈ 𝑋 𝑎𝑛𝑑𝜆 ∈ [0,1]
 However the above definition does not imply that 𝑓𝐴(𝑥) must be a
convex function of 𝑥.
CONVEXITY

14. THANK
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