Interval -Valued Fuzzy Menger Space

Interval -Valued Fuzzy Menger Space

Durgi Kol, Dr. Ruchi Singh, Manish Tiwari

(Department of Mathematics)

Pandit shambhunath Shukla University Shahdol Madhya Pradesh

Abstract : – The Interval -Valued Fuzzy Menger Space is a Mathematics concept that, combines fuzzy set theory and Menger Space. Its used to handle understand imprecies data. We define Interval -Valued Fuzzy Menger Space, discussing its characteristics and relation to Fuzzy Menger Space

Key words – Fuzzy Menger Space , Fuzzy metric space , interval -valued fuzzy set, interval -Valued Fuzzy Menger Space,

1. INTRODUCTION

   The concept of fuzzy set was introduced of Lotfi A. Zadeh in 1965 to model uncertainty using membership values between 0 and 1 Later, interval -valued fuzzy set were developed to represent uncertainty more effectively by assigning an interval (instead of a single number)as the membership degree. This allows a better representation of vagueness in real word problem. The development of Interval -Valued Fuzzy Menger Space involved contribution from several mathematics.An interval valued fuzzy Menger space set is characterized by an interval valued membership function, and it is taken as another generalization of the fuzzy set. In 2009, Li [20] introduced three kinds of distance between two interval valued fuzzy set ( or number s) defined on real line R moreover, he noted that each kind of distance is a metric on the corresponding set and in interval -valued fuzzy numbers metric space is complete. A.George and P. Veeramani Modified fuzzy metric spaces in 1994 and studies their properties [1][3][4]. S. Nadaban introduced fuzzy b- metric spaces and studies their properties. Khedmati yaghoobi and Morteza Saheli researched interval Valued Fuzzy Menger Space, other mathematics also contributed in this field, their work established interval -Valued Fuzzy Menger Space as a significant mathematical structure for studying uncertainty.

2. PRELIMINARIES :

   Diffinition 2.1 A triplet (X, F t) where X is a set, t is a continuous t- norm (like min or product) , and F: X × X(0,) [0,1] satisfies condition for all x, y, z X ,t ,s > 0

   1. F(x,y, t) > 0

   2. F( x, y, t) = 1 if and only if x= y ;

   3. F( x,y, t) = F(y,x,t);

   4. F( x,z,t+s) F( x,y, t) F(y,z, s) ( tringle inequality/ Menger property).

      Diffinition 2.2 The triple ( X,F, t) is said to be an Interval -Valued Fuzzy Menger Space if X is an arbitrary set, t is a continuous interval valued t- norm on [1] and F is an interval valued fuzzy set on X × (0,) satisfying the following condition: –

      1. F( x,y, t) > 0 ;

         –

      2. F( x, y, t) = 1 if and only if x= y

      3. F( x, y, t) = F( y,x, t) ;

      4. F( x,y,t) F( y, z, s) F (x, z, t+s) ;

      5. F(x, y,): (0,) [1] is continuous;

         –

      6. limn F(x,y, t)= [1] where x, y, z X and t, s > 0 ;

   Diffinition 2.3 An Interval -Valued Fuzzy Menger Space ( X, F, t) in which for every sequence is Convergent is called a complete Interval -Valued Fuzzy Menger Space.

   Diffinition 2.4 Let X be an ordinary non- empty set. The mapping A : X [I] is called an interval valued fuzzy set on X. All interval -valued fuzzy sets on X denoted by IVF (X).

   Diffinition 2.5 Commutative associative and non decreasing mapping t : [0,1] × [0,1] [0,1] is a t- norm if and only if t(a,1) = a for all a [0,1] , t(0,0) = 0 and t(c,d) for c a , d b.

   Diffinition 2.6 A mapping : [0,1]×[0,1][0,1] is called a t norm if for all a,b,c [0,1]

   1. (a,1) = a, (0,0) = 0

   2. (a,b) = (b,a) ;

   3. (c,d) (a,b) for c a ,d b;

   4. ( ( a,b,c) ) = (a,(b,c) );

   Diffinition 2.7 A fuzzy Menger Space ( X,F,t) with the continuous t- norm is said to be complete if every couchy sequence in X converges point in X.

3. RESULT:

Theorem : Let ( X,F,t) be an Interval -Valued Fuzzy Menger Space, for all x, y X ,iff s,t > 0 , then F(x, y, t) = F(x,y, t) . Proof According to diffinition 2.2 if s > t > 0,

then we know that F(x,y, t) F(y,y, s-t) F( x,y,s)

Since

F( y,y, s-t) = 1 we have

Therefore,

F( x,y, t) F ( y,y, s-t) = F ( x, y, t) 1 = F (x,y,t) F( x,y, t) F ( x,y,s)

Remark : In an Interval -Valued Fuzzy Menger Space ( X, F, t), whenever

– – – –

F ((x, y, t) 1- r for all x,y X , t> 0, r (I) there exists a t0 with 0< t0<t such

– –

that F ( x, y, t) > 1 – r .

CONCLUSION

An Interval -Valued Fuzzy Menger Space as a genrealization of Fuzzy Menger Space . Various properties and characterizetion of Interval -Valued Fuzzy Menger Space introduced by a continuous t- norm have been discussed. We have established several important results related to Convergence, Cauchy sequence and completeness in Interval- Valued Fuzzy Menger Space.The obtaind results extend and improve many known results of fuzzy metric spaces and Menger Space. This study shows that Interval -Valued Fuzzy Menger Space provide a more flexible and realistic framework for dealing uncertainty in mathematical modeling .

Interval -Valued Fuzzy Menger Space is an advanced tooling fuzzy mathematics, offering greater flexibility and accuracy then traditional fuzzy space.

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