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International Journal of Fuzzy Logic and Intelligent Systems 2022; 22(4): 422-432

Published online December 25, 2022

https://doi.org/10.5391/IJFIS.2022.22.4.422

© The Korean Institute of Intelligent Systems

Shadow Soft Set Theory

Shawkat Alkhazaleh

Department of Mathematics, Faculty of Science and Information Technology, Jadara University, Irbid, Jordan

Correspondence to :
Shawkat Alkhazaleh (shmk79@gmail.com)

Received: May 4, 2022; Revised: August 16, 2022; Accepted: December 12, 2022

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

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The shadow set is a new concept defined as a new tool with uncertainty, where the membership values are taken from 0, 1, and [0, 1]. Here, we introduce the concept of a shadow soft set as a combination of the shadow set and soft set. An example is presented that shows the existing significance of the shadow soft set. Subsequently, some properties of the shadow soft set are discussed. The operations on shadow soft sets such as complement, union, intersection, AND, and OR are given with their properties, and an application to decision-making is shown.

Keywords: Soft set, Fuzzy soft set, Shadow set, Shadow soft set

Acknowledgments

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The authors acknowledge the financial support received from Jadara University.

Conflict of Interest

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The authors declare no potential conflicts of interest relevant to this article.

Biography

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Shawkat Alkhazaleh. is a Professor of Mathematics at Jadara University in Jordan. He received his MA degree and PhD from the National University of Malaysia (UKM). He specializes in fuzzy sets, soft fuzzy sets, and topics related to uncertainty and has conducted extensive research in this field. He is currently working as a dean of student affairs at Jadara University, in addition to being a faculty member at the College of Science and Information Technology.

Article

Original Article

International Journal of Fuzzy Logic and Intelligent Systems 2022; 22(4): 422-432

Published online December 25, 2022 https://doi.org/10.5391/IJFIS.2022.22.4.422

Copyright © The Korean Institute of Intelligent Systems.

Shadow Soft Set Theory

Shawkat Alkhazaleh

Department of Mathematics, Faculty of Science and Information Technology, Jadara University, Irbid, Jordan

Correspondence to:Shawkat Alkhazaleh (shmk79@gmail.com)

Received: May 4, 2022; Revised: August 16, 2022; Accepted: December 12, 2022

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

The shadow set is a new concept defined as a new tool with uncertainty, where the membership values are taken from 0, 1, and [0, 1]. Here, we introduce the concept of a shadow soft set as a combination of the shadow set and soft set. An example is presented that shows the existing significance of the shadow soft set. Subsequently, some properties of the shadow soft set are discussed. The operations on shadow soft sets such as complement, union, intersection, AND, and OR are given with their properties, and an application to decision-making is shown.

Keywords: Soft set, Fuzzy soft set, Shadow set, Shadow soft set

Table 1 . ShdwC(X) = ShdwA(X) ∪ ShdwB(X).

A\B01[0, 1]
001[0, 1]
1111
[0, 1][0, 1]1[0, 1]

Table 2 . ShdwC(X) = ShdwA(X) ∩ ShdwB(X).

A\B01[0, 1]
0000
101[0, 1]
[0, 1]0[0, 1][0, 1]

Table 3 . ((F,E)shdwc)c=(F,E)shdw.

(F,E)shdw(F,E)shdwc((F,E)shdwc)c
010
101
[0, 1][0, 1][0, 1]

Table 4 . (Fshdw ∪ Gshdw)c = Fshdwc ∩ Gshdwc.

FshdwGshdw(Fshdw ∪ Gshdw)(Fshdw ∪ Gshdw)CFshdwCGshdwC(FshdwC ∩ GshdwC)
0001111
0110100
0[0, 1][0, 1][0, 1]1[0, 1][0, 1]
1010010
1110000
1[0, 1]100[0, 1]0
[0, 1]0[0, 1][0, 1][0, 1]1[0, 1]
[0, 1]110[0, 1]00
[0, 1][0, 1][0, 1][0, 1][0, 1][0, 1][0, 1]

Table 5 . (Fshdw ∩ Gshdw)c = Fshdwc ∪ Gshdwc..

FshdwGshdw(Fshdw ∩ Gshdw)(Fshdw ∩ Gshdw)CFshdwCGshdwC(FshdwC ∪ GshdwC)
0001111
0101101
0[0, 1]011[0, 1]1
1001011
1110000
1[0, 1][0, 1][0, 1]0[0, 1][0, 1]
[0, 1]001[0, 1]11
[0, 1]1[0, 1][0, 1][0, 1]0[0, 1]
[0, 1][0, 1][0, 1][0, 1][0, 1][0, 1][0, 1]

Table 6 . Representation of (Hshdw,A × B).

Uu1u2u3u4u5
E
(e1, e1)0[0, 1]0[0, 1]0
(e1, e2)0[0, 1]0[0, 1]0
(e1, e3)0[0, 1]0[0, 1]0
(e2, e1)0[0, 1][0, 1]00
(e2, e2)0[0, 1][0, 1]00
(e2, e3)0[0, 1]000
(e3, e1)[0, 1][0, 1]01[0, 1]
(e3, e2)0[0, 1]01[0, 1]
(e3, e3)1[0, 1]011

Table 7 . Score of ui.

ScoreS1S[0,1]S0
U
u1117
u2090
u3027
u4333
u5126

Table 8 . Representation of (Hshdw,A × B).

Uu1u2u3u4u5
E
(e1, e1)[0, 1][0, 1]11[0, 1]
(e1, e2)0111[0, 1]
(e1, e3)1[0, 1]011
(e2, e1)[0, 1][0, 1]11[0, 1]
(e2, e2)0111[0, 1]
(e2, e3)1[0, 1][0, 1]11
(e3, e1)[0, 1][0, 1]01[0, 1]
(e3, e2)1[0, 1]111
(e3, e3)11111

Table 9 . Score of ui.

ScoreS1S[0,1]S0
U
u1432
u2360
u3612
u4900
u5450

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