Some Important Theorem on Balanced Fuzzy Set

by Rahul Deo Awasthi*, DR. P. K. MISHRA, Dr. K. K. Jain, Dr. Yogesh Sharma,

- Published in Journal of Advances in Science and Technology, E-ISSN: 2230-9659

Volume 2, Issue No. 2, Nov 2011, Pages 0 - 0 (0)

Published by: Ignited Minds Journals

ABSTRACT

In this paperwe have obtained some important result on balanced fuzzy set and theirproperties.

KEYWORD

important theorem, balanced fuzzy set, properties, result

INTRODUCTION – 1.1

Let X is a non-empty set called universal set. Then, by a fuzzy set on X is meant a function A:X (0,1), ‘A’ is called a membership function, A(x) is called the membership grade of X and we write A = {(x, A(x)) : xX} A fuzzy set A in a linear space E is set to be convex if for every [0,1]

A+(1-)AA

A fuzzy set A in a vector space E is said to be balanced if AA for every scalar  with 1

THEOREM – 1.2

Let A and B are balanced fuzzy set in a vector space E over K then, A+B is also a balanced fuzzy set in a vector space E.

PROOF

Let us assume that A and B are balanced fuzzy set in a vector space E over the field K, then we have AA for every scalarwith 1 and also BB for all scalar  with 1. Now we have

(A+B)= A+B A+B

This shows that (A+B)  A+B. Hence (A+B) is a balanced fuzzy set in E.

THEOREM – 1.3

Let us consider {Ai}iIis a family of balanced fuzzy set in a vector space E, then A=iIAi is a balanced fuzzy set in E.

PROOF

Let {Ai}iIbe a family of balanced fuzzy set in a vector space E. Then, we have Ai Ai for every scalar with 1 That is Ai(x)Ai(x) for every scalar 1 ….(i) Again let A=iIAi thus A (y) = infiI Ai(y) for every yE Hence A(x )= infiI Ai(x) now from (i) we have A(x ) infiI Ai(x) =A(x) for all scalar with 1 and every xE Hence A=Ai is a balanced fuzzy set in E.

REFERENCES:

(1) Ganesh, M.(2006) Introduction Of Fuzzy Sets And Fuzzy Logic, P.H.I. New Delhi. (2) Katsaras, A.K. And Liu, D.B. (1977), ”Fuzzy Vector Spaces And Fuzzy Topological Vector Spaces”, J.Math. Anal. Appl., 58, Pp. 135-146. (3) Nanda, S. Fuzzy Fields And Fuzzy Spaces, Fuzzy Sets And System 19(1986) 89-94.