Fuzzy Measure: A Different View

Vol. 11, No. 2, 2016, 133-137

ISSN: 2279-087X (P), 2279-0888(online) Published on 22 June 2016

www.researchmathsci.org

133

Annals of

Fuzzy Measure: A Different View

Department of Mathematics, Government Arts College, Coimbatore, Tamilnadu, India E.mail: [email protected]

Department of Mathematics, Sri Vasavi College, Erode, Tamilnadu, India E.mail: [email protected]

Received 10 May 2016; accepted 3 June 2016

Abstract. In this paper the existing fuzzy measure “” is defined in a different manner so as to ensure and enhance the effectiveness of fuzzy measure. Some examples relating to various contexts where “” describes a fuzzy measure and where “” fails to be a fuzzy measure are stated.

Keywords: Fuzzy measure, fuzzy relation

AMS Mathematics Subject Classification (2010): 05C12, 03E72, 05C72

1. Introduction

Fuzzy sets are defined as a model, fit for vague concepts and subjective judgment. Such a model is used where the deterministic or probabilistic models fail to describe the system. Fuzzy set theory is used in various domains as decision making under uncertainty, information retrieval, machine fault diagnosis in large scale industries and image processing etc.

Various definitions of fuzzy measure exist. For the definition the reader is directed to refer [1, 4, 5, 6].

We use fuzzy measure theory for decision making. Our new fuzzy measure increases the efficiency so as to make decision easy. This measure is monotonic.

2. A new definition of fuzzy measure

A relation R:A–›B is a subset of A ×B where A and B are any two sets. Let A, B be any two sets. A relation ∶ → B is said to be a fuzzy relation if

1) D() = A where is the domain of and 2) there exists a membership function : –› 0,1

Let us consider an example

Let = , , = , Let = ,,,,_".!,,_".$!, % Then D() = A

The membership function : –›0,1 is

, = , = 1

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Remarks:

1. For the sake of simplicity ₍, is simply denoted by ₍, . 2. If (a,b) ∈ we write it as = .

3. By₍, = * we mean that x is the membership value (in [0,1]) of the case that = . That is = = * is simply denoted by ₍, = *.

The new definition of fuzzy measure

Suppose X is any set, + be a class of sub sets of X and ,, + be a measurable space. A fuzzy relation ∶ + → - is said to be a fuzzy measure if the following conditions are satisfied

1) µ_m( ,0) 1φ =

2) (i)

( ) ( ) A B Sup x Sup y

m A x m B y

⊆ ⇒ _≤

= =

(ii)A B⊆ ;A≠φ⇒ µ_m( ,A Sup x) ≤µ_m( ,B Sup y)

Making use of this definition, various instances where ‘m’ forms a measure are given below.

Example 2.1. Let X={1,2,3}and+ = P X( )=

{

{ } { } { } { } { } { } {

}

}

= {, , , _., _/, ₀, _$, ₁} (say) Define a measure m A: →Rasm A( i)= Ai

= = = 1; _. = _/ = ₀ = 2 ;

$ = 3, 1 = 0

Let the membership function be defined as

0

( , ) 1

1 0

x when x x

Ai x m

when x µ



≠ 

=  +

 ₌



Then the corresponding membership values are

μ4, 1 = μ4, 1 = μ4, 1 =_{2 ; μ}1 4., 2 = μ40, 2 = μ4/, 2 =2₃

μ4$, 3 =3 _{4 ; μ}41, 0 = 1

Then the following cases arise

⊆ . = 1 < . = 2 and μ₄, 1 =< μ_4., 2 =

⊆ / = 1 < / = 2 and μ₄, 1 =≤ μ₄:_{/ ,}2; =

⊆ $ = 1 < $ = 3 and μ₄, 1 =≤ μ_4$, 3 =_.

⊂ . = 1 < . = 2 and μ₄, 1 =< μ_4., 2 =

⊂ 0 = 1 < 0 = 2 and μ4, 1 =< μ40 ,2 =

⊂ $ = 1 < $ = 3 and μ₄, 1 =< μ_4$, 3 =_.

⊂ / = 1 < / = 2 and μ₄, 1 =< μ_4/, 2 =

135

⊂ $ = 1 < $ = 3 and μ₄, 1 =< μ_4$, 3 =_.

.⊂ $ .=2<_$=3 and μ₄( _., 2) = <μ₄( _$, 3) =_.

/ ⊂ $ / = 2<_$)=3 and μ_{4 / ,} 2 ) = <μ₄(_$, 3) = _.

0 ⊂ $ 0=2<_$ =3 and μ₄(₀ ,2)=<μ₄(_$, 3) = _. and m( ₁) = 0 but μ₄( ₁,0) = 1.

In all the above cases “m” satisfies the conditions laid down in the definition 2.1 “m ” is a fuzzy measure.

Example 2.2.(Definition of atom) If + is any non empty class of subsets of sets of , and for any point * ∈ , , the set ∩ >|* ∈ > ∈ + is called atom of + ` at * and is denoted by A*|+. X = x, x, x

{ } { }{ } { }{ } { } { }

{

}

( ) _{1 2 3}, , , _{1 2}, , _{2 3}, , _{1 3}, , ₁, ₂, ₃, P X = x x x x x x x x x x x x φ

={ C, C, C, C_., C_/, C₀, C_$, C₁} (say)

:*/E,; denotes the atom of E, F *

:*/E,; = C ∩ C ∩ C. ∩ C/= C/ ; :*/E,; = C ∩ C ∩ C_. ∩ C₀= C₀

:*/E,; = C$

Let the measure be defined by ∶ E, → - where

CG = H |*G /E,| IJ∈KJ

C = 3 ; C = C = C. = 2; C/ = C0 = C$ = 1;C₁ = 0. Letμ ∶ → 0,1be defined as

1

0 2

( , ) 1

1 0

x when x x

Si x

m

when x µ

− 

≠ 

=₊

 ₌



μ4C, 3 = 0.7 , μ4C, 2 = μ4C, 2 = μ4C., 2 = 0.6

μ4C/, 1 = μ4C0, 1 = μ4C$, 1 = 0.5 ; μ4C1, 0 = 1.

Clearly ‘m’ satisfies the conditions of a fuzzy measure.

Example 2. 3. Let X={a,b,c}

{ } { } {

}

{

}

A= a b b a b c φ . Here A is a subclass of P(X).

= {, , , _.} (say)

∶ → - is defined as _G = |N_|O|J|

Then = ; = ; = 1 ; _. = 0

μ ∶ → 0,1 be defined as

(

)

(_Ai, ) 1x 1 (1 x) m

µ = − + ;μ₄P,Q = _/ 0= 0.64;μ₄P,Q = ₀ $ = 0.437 ,

μ4, 1 = 0.75.

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Example 2.4. Let A be a R-algebra given by X={a,b,c} ; A=

{

{ } { } {

}

}

={, , , _.} (say) Define measurem A: →Ras m(_G)=|_G |

Then = 1; = 2 ; = 3 ; _. = 0

μ ∶ → 0,1 be defined as

2 2

(_Ai, )x (x x 1) 1 x m

µ = − + + ; , 1 = , , 2 = _/, , 3 = _"$ Then is a fuzzy measure.

Example 2.5. Let X={1,2,3,4,5},A=

{

{ } { } { }

}

The R-algebra generated by A is denoted by R and is given by

{ } {

} { } {

} { } {

} {

}

{

}

( )A 1, 2 , 3, 4, 5 , 3, 4 , 1, 2, 5 , 5 , 1, 2, 3, 4 , 1, 2, 3, 4, 5 ,

σ = φ

={A A A A A A A A₁, ₂, ₃, ₄, ₅, ₆, ₇, ₈}(say)

Letm: ( )σ A →Rbe defined as _G = |_G|

Then = 2 ; = 3 ; = 2 ; _. = 3; _/ = 1 ;

0 = 4 and _$ = 5 and ₁ = 0 The membership function µ:m→

[ ]

( , ) 1 2 1 x x Ai m x µ = − + 3 (A₁,2) ₅ m

µ = ; _{( ,3) ( ,3)} 7

2 4 10

A A

m m

µ =µ = ; ( ₃,2) 3

5 A m

µ = ; _{( ,1)} 1

5 2

A m

µ = ; _{( ,4)} 13

6 17

A m

µ = ;

21 (A₇,5) ₂₆ m

µ = ; µ_m(A₈,0)=1

Then is a fuzzy measure.

Example 2.6. Let X ={1,2,3}

P(X) is ordered in descending order of cardinalities.

{

} { } { } { } { } { } { }

{

}

( ) 1, 2, 3 , 1, 2 , 1, 3 , 2, 3 , 1 , 2 , 3 ,

P X = φ

{A A A A A A A A_{1 2}, , ₃, ₄, ₅, ₆, ₇, ₈}

= (say)

Letm P X: ( )→Rbe defined as _G =|N_GJ| . Then

= 3 ; = 1 ; = ; _. =_.

/ =_/ ; 0 =₀ ; $ =_$ ; 1 = 0

μ ∶ → 0,1 is defined as

2 2 ( , ) 1

1 x Ai x x Ai m Ai φ µ φ  ≠  = +  = 

μ4, 3 =_"S ;μ₄, 1 =;μ₄P,Q =.;μ₄P_.,_.Q =_/;μ₄P_/,_/Q =₀

137

Example 2.7. Consider X, P(X) and the measure ‘m’ be as in Example 2.6. If the

membership function be defined asµm(Ai, )x = −1 Ai (1+x2) . We see that

2 5

A ⊂A andm A( ₂) = 1₅ <m A( ₂) = 1but 1 1

( ₅, ) 5 26 A m

µ = >0 =µ_m(A₂,1)

Therefore is not a fuzzy measure.

3. Conclusion

We have given different contexts where ‘m’has proved to be a fuzzy measure by means of examples. The domain of the measure ‘m’ has been the ordinary power set, the subclass of the power set, the sigma-algebra, and the sigma algebra generated by the subclass of the power set. Measure ‘m’ is also defined in a variety of ways and is proved to be a fuzzy measure according to the new definition.

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