Elsevier

Fuzzy Sets and Systems

Volume 297, 15 August 2016, Pages 141-144
Fuzzy Sets and Systems

Short Communication
Generalized uniform fuzzy partition: The solution to Holčapek's open problem

https://doi.org/10.1016/j.fss.2015.09.014Get rights and content

Abstract

In this note, we propose a solution to an open problem, which was presented by Mesiar and Stupňanová. We give two counterexamples to show that the hypothesis is not sufficient to get the result and we propose that the sufficient condition of the hypothesis is held by modifying the symmetry condition.

Introduction

Let R be the set of real numbers. A function K:R[0,1] is said to be a generating function, if K is an even function that is non-increasing on [0,) and K(x)>0 iff x(1,1) holds true. A generating function K is said to be normal if K(0)=1.

Triangular and raised cosine functions are typical examples of normal generating functions:KT(x)=max(1|x|,0) andKC(x)={12(1+cos(πx)),1x10,otherwise.

Let K be a generating function, h and r be positive real numbers and x0R. A system of fuzzy sets {Ai|iZ} on R defined byAi(x)=K(xx0irh)foriZ is said to be a generalized uniform fuzzy partition (GUFP) of the real line determined by the quadruplet (K,h,r,x0) if the Ruspini condition is satisfied:S(x)=iZAi(x)=1 holds for any xR. The parameters h,r and x0 are called the bandwidth, shift and the central node, respectively.

In [2], Holčapek et al. have proved a full characterization of generalized uniform fuzzy partitions by using the sum of suitable integrals.

Lemma 1

(See [2].) A quadruplet (K,h,r,x0) determines a generalized uniform fuzzy partition iff the quadruplet (K,h,r,0) determines it as well.

Corollary 1

(See [2].) Let β>0 be a real number. A triplet (K,h,r) determines a generalized uniform fuzzy partition iff (K,βh,βr) determines it as well.

Let K be a normal generating function. Define Kα(x)=αK(x) for α(0,1], where αK(x) is the common product of real numbers. The necessary and sufficient condition for GUFPs can be significantly simplified in the cases of triangular and raised cosine generating functions have proved in [2].

Theorem 1

(See [2].) Let K{KT,KC}. Then, (Krh,h,r,x0) determines a GUFP iff hrN.

Section snippets

Counterexamples

Below we present Problem 7.1 from [3], which was posed by Holčapek et al. during the conference FSTA 2014.

Problem 1

(See [2].) Let K be a normal generating function satisfying the symmetry condition.12y12+yK(x)dx=yfor ally[0,12] Then (Krh,h,r,x0) determines a GUFP iff hrN.

The following counterexamples show that the necessary and sufficient condition for GUFP do not hold under the symmetric condition given by (2).

Counterexample 1 The counterexample of necessary condition

Let K be a normal generating function defined byK(x)={1,x=034,0<|x|1214,12<|x|<10,

Modified result

In this section, we modify the symmetry condition and subsequently propose a sufficient condition. It is noted that if K is continuous and satisfies the symmetry condition (2), thenK(12+y)+K(12y)=1for ally[0,12] or equivalentlyK(y)+K(1y)=1for ally[0,1].

Triangular and raised cosine functions are good examples of satisfying the condition (3).

Note. If a normal generating function K satisfies the condition (3), then the graph of K on [0,1] has rotational symmetry with respect to the point (12,12

Acknowledgements

We would like to thank the reviewers for taking the time to carefully read the paper and for providing some very valuable feedback and suggestions. This work was supported Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology (2010-0021089).

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