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## Original Article

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International Journal of Fuzzy Logic and Intelligent Systems 2022; 22(1): 59-68

Published online March 25, 2022

https://doi.org/10.5391/IJFIS.2022.22.1.59

© The Korean Institute of Intelligent Systems

## Transitive Closure of Vague Soft Set Relations and its Operators

Yousef Al-Qudah1 , Khaleed Alhazaymeh2, Nasruddin Hassan3, Hamza Qoqazeh1, Mohammad Almousa1, and Mohammad Alaroud1

1Department of Mathematics, Faculty of Arts and Science, Amman Arab University, Amman, Jordan
2Department of Basic Sciences and Mathematics, Faculty of Science, Philadelphia University, Amman, Jordan
3School of Mathematical Sciences, Faculty of Science and Technology, Universiti Kebangsaan Malaysia, Bangi, Malaysia

Correspondence to :
Yousef Al-Qudah (alquyousef82@gmail.com)

Received: July 17, 2021; Revised: September 28, 2021; Accepted: October 12, 2021

This is an Open Access article distributed under the terms of the Creative Commons Attribution Non-Commercial License (http://creativecommons.org/licenses/by-nc/3.0/) which permits unrestricted noncommercial use, distribution, and reproduction in any medium, provided the original work is properly cited.

### Abstract

A vague soft set is a mapping from a parameter set to the collection of vague subsets of the universal set. In this study, a vague soft relation is presented based on the Cartesian product of vague soft sets. The basic properties of these relations are studied to explain the concept of transitive closure of a vague soft relation. The symmetric, reflexive, and transitive closures of a vague soft set are introduced followed by examples to illustrate these relations. The concepts are further extended by proposing some of their properties. The existence and uniqueness of the transitive closure of a vague soft relation are established, and an algorithm to compute the transitive closure of a vague soft relation is also provided.

Keywords: Vague soft set, Transitive closure, Symmetric closure, Fuzzy set

### Biographies

Yousef Al-Qudah received the M.Sc. and Ph.D. degrees in mathematics from Universiti Kebangsaan Malaysia, Malaysia. He is currently an Assistant Professor in the Department of Mathematics, Amman Arab University, Jordan. His research interests include decision-making, fuzzy sets, fuzzy topology, fuzzy algebra, and complex fuzzy sets.

E-mail: alquyousef82@gmail.com

Khaleed Alhazaymeh is an Associate Professor in the Department of Mathematics at Philadelphia University in Jordan. He received his M.Sc. and Ph.D. from the National University of Malaysia (UKM). He specializes in fuzzy sets, vague soft sets, and topics related to uncertainty, and has conducted extensive research in this field.

Nasruddin Hassan received the B.Sc. degree in mathematics from Western Illinois University, USA, the M.Sc. degree in applied mathematics from Western Michigan University, USA, and the Ph.D. degree in applied mathematics from Universiti Putra Malaysia, Malaysia. His research interests include decision making, operations research, fuzzy sets, and numerical convergence.

Hamza Qoqazeh received his M.Sc. from Al Al-Bayt University and Ph.D. from the University of Jordan. Since 2019, he has been at Amman Arab University. His research interests include topology and fuzzy topologies.

Mohammad Almousa received his M.Sc. from Al Al-bayt University, and a Ph.D. degree in applied mathematics from Universiti Sains Malaysia, Malaysia. His research interests include numerical optimization and partial differential equations.

Mohammad Alaroud received the M.Sc. and Ph.D. degrees in mathematics from Universiti Kebangsaan Malaysia, Malaysia. Since 2020, he is working as an Assistant Professor in the Faculty of Science and Arts, Amman Arab University. His research interests include numerical topology and partial differential equations.

### Article

#### Original Article

International Journal of Fuzzy Logic and Intelligent Systems 2022; 22(1): 59-68

Published online March 25, 2022 https://doi.org/10.5391/IJFIS.2022.22.1.59

## Transitive Closure of Vague Soft Set Relations and its Operators

Yousef Al-Qudah1 , Khaleed Alhazaymeh2, Nasruddin Hassan3, Hamza Qoqazeh1, Mohammad Almousa1, and Mohammad Alaroud1

1Department of Mathematics, Faculty of Arts and Science, Amman Arab University, Amman, Jordan
2Department of Basic Sciences and Mathematics, Faculty of Science, Philadelphia University, Amman, Jordan
3School of Mathematical Sciences, Faculty of Science and Technology, Universiti Kebangsaan Malaysia, Bangi, Malaysia

Correspondence to:Yousef Al-Qudah (alquyousef82@gmail.com)

Received: July 17, 2021; Revised: September 28, 2021; Accepted: October 12, 2021

This is an Open Access article distributed under the terms of the Creative Commons Attribution Non-Commercial License (http://creativecommons.org/licenses/by-nc/3.0/) which permits unrestricted noncommercial use, distribution, and reproduction in any medium, provided the original work is properly cited.

### Abstract

A vague soft set is a mapping from a parameter set to the collection of vague subsets of the universal set. In this study, a vague soft relation is presented based on the Cartesian product of vague soft sets. The basic properties of these relations are studied to explain the concept of transitive closure of a vague soft relation. The symmetric, reflexive, and transitive closures of a vague soft set are introduced followed by examples to illustrate these relations. The concepts are further extended by proposing some of their properties. The existence and uniqueness of the transitive closure of a vague soft relation are established, and an algorithm to compute the transitive closure of a vague soft relation is also provided.

Keywords: Vague soft set, Transitive closure, Symmetric closure, Fuzzy set