fuzzy logic and its application to switching systems

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FUZZY LOGIC AND ITS APPLICATION TO

SWITCHING SYSTEMS

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Introduction

The first step in this direction was made by Zadeh who propsed a so called “Fuzzy Set” theory dealing with events and situations having subjectively ascribed attributes

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Illustrations of Situations

• The set of all real numbers much greater than 100 (Is 125 a member of this set?)

• The set of all intelligent humans.(Do I belong to this set?)

• The set of all poor people.(Am I a member of this set?)

Obviously, all the above sets are very fuzzily defined and their members do not possess sharply defined attributes

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Fuzzy SetsLet X and Y be two fuzzy sets, and x and y be the membership grades of an object with respect to sets X and Y, respectively

Definition 1:

Two fuzzy sets X and Y are equal (X=Y) if, and only if, for every object -i one has that its membership grade Xi in X is equal to its membership grade Yi in Y

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Definition 2:A fuzzy set is the complement of another fuzzy set X and is denoted by X’ if and only if, for every object-i one has that its member-ship grade Xi’ in X’ is equal to 1-Xi, where Xi is the member ship grade of the object -i in X

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Definition 3:

A fuzzy set X is contained in another fuzzy set Y if and only if for every object i one has that

Xi <= Yi

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Definition 4:Two fuzzy sets X and Y form a union, denoted by Z = X + Y if and only if, for every object i one has that Zi = max(Xi,Yi)

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Definition 5:

Two fuzzy sets X and Y form an intersection, denoted by Z = X.Y, if and only if for every object i one has that Zi = min(Xi,Yi)

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CLASSIFICATION OF FUZZY FUNCTIONS

where, 1>a1>a2>……>an-1>0

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Relationship (9) is satisfied if and only if the variables(x,y,z)satisfy the following relations:

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2

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……(17)

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……(18)

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Example:

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