This document summarizes Lotfi Zadeh's 1965 paper that introduced fuzzy set theory. It defines fuzzy sets as sets with imprecise or unclear boundaries, where elements can partially belong through degrees of membership between 0 and 1. It provides key definitions for fuzzy sets, including complement, subset, union, intersection and algebraic operations. Convex combinations and fuzzy relations are also introduced. The document concludes with a definition for convex fuzzy sets.

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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