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Limit properties in the metric semi-linear space of picture fuzzy numbers

The picture fuzzy set (PFS) just appeared in 2014 and was introduced by Cuong, which is a generalization of intuitionistic fuzzy sets (Atanassov in Fuzzy Sets Syst 20(1):87–96, 1986) and fuzzy sets (Zadeh Inf Control 8(3):338–353, 1965). The picture fuzzy number (PFN) is an ordered value triple, inc...

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Autores principales: Phu, Nguyen Dinh, Hung, Nguyen Nhut, Ahmadian, Ali, Salahshour, Soheil
Formato: Online Artículo Texto
Lenguaje:English
Publicado: Springer Berlin Heidelberg 2022
Materias:
Acceso en línea:https://www.ncbi.nlm.nih.gov/pmc/articles/PMC9017969/
https://www.ncbi.nlm.nih.gov/pubmed/35465468
http://dx.doi.org/10.1007/s00500-022-07017-8
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author Phu, Nguyen Dinh
Hung, Nguyen Nhut
Ahmadian, Ali
Salahshour, Soheil
author_facet Phu, Nguyen Dinh
Hung, Nguyen Nhut
Ahmadian, Ali
Salahshour, Soheil
author_sort Phu, Nguyen Dinh
collection PubMed
description The picture fuzzy set (PFS) just appeared in 2014 and was introduced by Cuong, which is a generalization of intuitionistic fuzzy sets (Atanassov in Fuzzy Sets Syst 20(1):87–96, 1986) and fuzzy sets (Zadeh Inf Control 8(3):338–353, 1965). The picture fuzzy number (PFN) is an ordered value triple, including a membership degree, a neutral-membership degree, a non-membership degree, of a PFS. The PFN is a useful tool to study the problems that have uncertain information in real life. In this paper, the main aim is to develop basic foundations that can become tools for future research related to PFN and picture fuzzy calculus. We first establish a semi-linear space for PFNs by providing two new definitions of two basic operations, addition and scalar multiplication, such that the set of PFNs together with these two operations can form a semi-linear space. Moreover, we also provide some important properties and concepts such as metrics, order relations between two PFNs, geometric difference, multiplication of two PFNs. Next, we introduce picture fuzzy functions with a real domain that is also known as picture fuzzy functions with time-varying values, called geometric picture fuzzy function (GPFFs). In this framework, we give definitions about the limit of GPFFs and sequences of PFN. The important limit properties are also presented in detail. Finally, we prove that the metric semi-linear space of PFNs is complete, which is an important property in the classical mathematical analysis.
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spelling pubmed-90179692022-04-20 Limit properties in the metric semi-linear space of picture fuzzy numbers Phu, Nguyen Dinh Hung, Nguyen Nhut Ahmadian, Ali Salahshour, Soheil Soft comput Foundations The picture fuzzy set (PFS) just appeared in 2014 and was introduced by Cuong, which is a generalization of intuitionistic fuzzy sets (Atanassov in Fuzzy Sets Syst 20(1):87–96, 1986) and fuzzy sets (Zadeh Inf Control 8(3):338–353, 1965). The picture fuzzy number (PFN) is an ordered value triple, including a membership degree, a neutral-membership degree, a non-membership degree, of a PFS. The PFN is a useful tool to study the problems that have uncertain information in real life. In this paper, the main aim is to develop basic foundations that can become tools for future research related to PFN and picture fuzzy calculus. We first establish a semi-linear space for PFNs by providing two new definitions of two basic operations, addition and scalar multiplication, such that the set of PFNs together with these two operations can form a semi-linear space. Moreover, we also provide some important properties and concepts such as metrics, order relations between two PFNs, geometric difference, multiplication of two PFNs. Next, we introduce picture fuzzy functions with a real domain that is also known as picture fuzzy functions with time-varying values, called geometric picture fuzzy function (GPFFs). In this framework, we give definitions about the limit of GPFFs and sequences of PFN. The important limit properties are also presented in detail. Finally, we prove that the metric semi-linear space of PFNs is complete, which is an important property in the classical mathematical analysis. Springer Berlin Heidelberg 2022-04-19 2022 /pmc/articles/PMC9017969/ /pubmed/35465468 http://dx.doi.org/10.1007/s00500-022-07017-8 Text en © The Author(s), under exclusive licence to Springer-Verlag GmbH Germany, part of Springer Nature 2022 This article is made available via the PMC Open Access Subset for unrestricted research re-use and secondary analysis in any form or by any means with acknowledgement of the original source. These permissions are granted for the duration of the World Health Organization (WHO) declaration of COVID-19 as a global pandemic.
spellingShingle Foundations
Phu, Nguyen Dinh
Hung, Nguyen Nhut
Ahmadian, Ali
Salahshour, Soheil
Limit properties in the metric semi-linear space of picture fuzzy numbers
title Limit properties in the metric semi-linear space of picture fuzzy numbers
title_full Limit properties in the metric semi-linear space of picture fuzzy numbers
title_fullStr Limit properties in the metric semi-linear space of picture fuzzy numbers
title_full_unstemmed Limit properties in the metric semi-linear space of picture fuzzy numbers
title_short Limit properties in the metric semi-linear space of picture fuzzy numbers
title_sort limit properties in the metric semi-linear space of picture fuzzy numbers
topic Foundations
url https://www.ncbi.nlm.nih.gov/pmc/articles/PMC9017969/
https://www.ncbi.nlm.nih.gov/pubmed/35465468
http://dx.doi.org/10.1007/s00500-022-07017-8
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