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

FUZZY SETS*

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

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

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….

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….

Let the universal set be denoted by X and its elements by i.e. X= {x}. We define a set A on X such that . We define the term grade of membership denoted by which represents the information regarding the extent of belongingness of x to set A.

If , if = 0 or 1 only and no intermediate value, then the set A is called the crisp set and if the value of 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

DIFFERENCES BETWEEN CRISP SETS & FUZZY SETS

Empty FS: Equal FSs: Given two FSs A & B, then if , then A=B Complement): It is defined as Subset: Union: Let

Corollary: The union of A & B is the smallest fuzzy set containing both A & B.

Intersection: Let Corollary: The intersection of A & B is the largest fuzzy set containing both A & B.

SOME DEFINITIONS W.R.T TO FUZZY SETS (FSS)

Graphically, the union is shown by sections 1 & 2 of the graph. Intersection is shown by sections 3 & 4:

GRAPHICAL REPRESENTATION OF UNION & INTERSECTION

Algebraic product: Denoted by AB, is defined for FSs A & B as:

Algebraic sum: Denoted by A+B, is defined as:

Absolute difference: Denoted by |A-B|, is defined as:

ALGEBRAIC OPERATIONS ON FSS

Let A, B & be three FSs, then their convex combination is denoted by

is the complement of Or in terms of the membership function as:

A basic property of convex combination of A, B and is given by:

CONVEX COMBINATION

For FSs, the above expression is rewritten as:

Also, it is possible to find a FS ‘C’, s.t. C= and the membership of this set is given by:

CONVEX COMBINATION……

A fuzzy relation in X is a fuzzy set in the product space .

For example, the relation , is expressed as a fuzzy set A in , with the membership function value given by: . Lets say and .

However, the realtion values are subjective interpretations.

FUZZY RELATION

A fuzzy set A is called the convex iff

However the above definition does not imply that must be a convex function of .

CONVEXITY

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