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## Original Article

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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.

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

### 1.1 Uncertainty

Classical mathematics is one of the most important requirements in real-life problems, some of which involve imprecise data. Thus, solving these problems involves using mathematical principles based on uncertainty (not crisp). Therefore, many scientists and engineers are interested in non-crisp modeling to describe and debrief useful information hidden in uncertain data. Many researchers have proposed a set of theories to deal with this uncertainty, such as fuzzy set theory [1], intuitionistic fuzzy set theory [2], vague set theory [3], theory of interval mathematics [4], theory of rough set [5], theory of shadow set [6], and neutrosophic set theory [7].

### 1.2 Soft Set

Soft set, which is a theory defined as a new mathematical tool for dealing with uncertainties that traditional mathematical tools cannot handle, was initiated in 1999 by Molodtsov [8]. Since then, researchers have raced to enrich this topic in all aspects, and the related research has rapidly. They have worked on a combination of soft set theory and other theories starting from Maji et al. [9], who introduced the concept of fuzzy soft set, a more general concept as a combination of fuzzy set and soft set, and studied its properties. Then, in 2003 [10], the theory of soft sets and their properties were studied. In 2001, they introduced an intuitionistic fuzzy soft set, a combination of intuitionistic fuzzy sets and soft sets. In addition, Xu et al. combined a vague set with a soft set to define a vague soft set in 2010.

Rough soft set theory is a combination concept defined by Feng et al. [11]. Yang et al. presented a combination of an interval-valued fuzzy set and soft set in 2009 [12]. A neutrosophic soft set is a new concept defined by Maji in 2013 [13]. Many researchers have studied soft set theory, attempted to develop it, and studied its applications in decision-making and medical diagnosis problems. For example, Alkhazaleh et al. [14] introduced the concept of soft multiset as a generalization of the soft set. They also generalized this fuzzy soft multiset theory concept in 2012 [15]. Majumdar and Samanta [16] introduced and studied a generalized fuzzy soft set, where the degree is attached to the parameterizations of fuzzy sets while defining a fuzzy soft set. Alkhazaleh et al. also defined fuzzy parameterized interval-valued fuzzy soft set [17] and possibility fuzzy soft set [18]. They provided their applications in decision-making and medical diagnosis problems. Alkhazaleh and Salleh [19] introduced the concept of a soft expert set, in which the user can know the opinions of all experts in one model without any operations. Even after any operation, the user can know the opinions of all experts. Then, they generalized this concept to a fuzzy soft expert set and established its application in decision-making. Adam and Hassan in 2014 [20] introduced the theory of Q–fuzzy soft set with its application in decision-making and medical diagnosis problems.

### 1.3 Shadow Set

In this section, we recall the definitions related to this work, starting from the fuzzy set, the definition of the shadow set, where we define the shadow set, its properties, and basic operations. Thereafter, we recall the concepts related to soft set theory and fuzzy soft sets based on their properties and operations. Since Zadeh published his new classic paper almost 60 years ago, fuzzy set theory has received increasing attention from researchers in various scientific areas, especially in the past few years. The difference between a binary set and a fuzzy set is that in a “normal” set, every element is either a member or a non-member. Here, it either must be A or notA. In a fuzzy set, an element can be a member of a set to some degree and, simultaneously, a non-member of the same set to some degree. In classical set theory, the membership of elements in a set is assessed in binary terms according to a bivalent condition: an element either belongs or does not belong to the set. By contrast, fuzzy set theory permits the gradual assessment of the membership of elements in a set, which is described with a membership function valued in the closed unit interval [0, 1]. Fuzzy sets generalize classical sets because the indicator functions of classical sets are special cases of the membership functions of fuzzy sets if the latter only takes values 0 or 1. Therefore, a fuzzy set A in a universe of discourse X is a function A : X → [0, 1], usually referred to as the membership function, denoted by μA(x). Some mathematicians use the notation A(x) to denote: membership function instead of μA(x). Fuzzy set A is written symbolically in various

A={x,μA(x):xX}.

### Definition 2.1 ([1])

Let X be a nonempty set. A fuzzy set A in X is characterized by its membership function

μA:U[0,1],

and μA(x) is interpreted as the degree of membership of element x in the fuzzy set A for each xX.

A is completely determined by the set of pairs

A={(x,μA(x))xX}.

The family of all the fuzzy sets in X is denoted by F(X). If X = {x1, …, xn} is a finite set and A is a fuzzy set in X, we often use the notation

A={μA(x1)/x1,,μA(xn)/xn},

where the term μA(xi), i = 1, …, n signifies that μi is the membership grade of xi in A. Here, we use the following notation for the fuzzy set.

A={xμA(x):xX}.

In what follows, we summarize the quintessence of shadowed sets, a concept presented by Pedrycz. [6]. Pedrycz introduced the concept of a shadow set using a different formula. In general, by considering a fuzzy set A, the main idea is to elevate all membership values that are high enough and reduce those that are viewed as substantially low, where the elevation and reduction components are very radical as we go for 1 and 0, respectively. Mathematically, we define the shadow set as follows:

### Definition 2.2 ([6, 21])

Let X be a set of objects called the universe, and μA : X → [0, 1] be a fuzzy set over X. A shadowed set on X is any mapping B : X → {0, [0, 1], 1}, B : X → {0, a, 1}. by

Bαf(x)={0,if μA(x)α,1,if μA(x)1-α,a,O.W.

The shadowed set B is a mapping defined in X into a three-element set, where the elements of X for which B(x) reaches 1 constitute its core. The zones of X, where B(x) = [0, 1] or B(x) = a form a shadow of this construct. Eventually, the elements with B(x) = 0 are excluded from B. Pedrycz presented a new way of understanding uncertainty. This definition has evolved and undergone some development and modernization, either through Pedrycz or other researchers. To easily demonstrate how to produce a shadow set, one way was to choose α to generate this set by selecting the excluded zone for all values less than α and shadowed set between α and α − 1 where the core zone of all values are greater than α − 1. In contrast, the other direction selects α and β for the same generation process. Let X = {x1, x2, …, x10} be a set of universes, FX={x10.3,x20.4,x30.8,x40.5,x50.7,x60.9,x10,x80.6,x91,x100.2} be a fuzzy set over X. Suppose α = 0.3, then we can write the shadow set Shdw(X) as

Shdw(X)={x10,x2a,x31,x4a,x50,x61,x10,x8a,x91,x100}.

Without loss of generality, we can indicate [0, 1] or a with a value 0.5.

Defining operations on shadowed sets as union, intersection, and complement; here, a = [0, 1].

### Definition 2.3 ([6, 21])

The complement of the shadowed set Shdw(X) = {0, [0, 1], 1} is shadowed set Shdw(X)c = {1, [0, 1], 0}.

### Definition 2.4

The union of the two shadowed sets ShdwA(X) and ShdwB(X) over a common universe X is the shadowed set ShdwC(X) presented in Table 1.

### Definition 2.5

The intersection of the two shadowed sets ShdwA(X) and ShdwB(X) over a common universe X is the shadowed set ShdwC(X) presented in Table 2.

Note that for any a in [0, 1] we have min([0, 1], a) = [0, a] and max([0, 1], a) = [a, 1].

Molodtsov defined a soft set as a mapping from a set of parameters to a crisp subset of the universe, where the approximate function of a soft set is defined from a crisp set of parameters to a crisp subset of a universal set. Let U be a universe and E be a set of parameters. Let P(U) denote the power set of U and AE.

### Definition 2.6 ([8])

A pair (F,A) is called a soft set over U, where F is a mapping

F:AP(U).

In other words, a soft set over U is a parameterized family of subsets of the universe U. For ɛA, F (ɛ) may be considered as the set of ɛ-approximate elements of the soft set (F,A).

In 2001, Maji et al. [9] generalized the soft set to a fuzzy soft set. In the following, we review some definitions and properties related to fuzzy soft-set theory, which we use in our work. The following definitions and propositions were provided by Maji et al. [9]:

### Definition 2.7

Let U be an initial universal set and E be a set of parameters. Let IU denote all fuzzy subsets of U. Let AE. A pair (F,E) be a fuzzy soft set over U where F is a mapping given by

F:AIU.

### Definition 2.8

The union of two fuzzy soft sets (F,A) and (G,B) over a common universe U is the fuzzy soft set (H,C) where C = AB, and ∀ɛC,

H(ɛ)={F(ɛ),if ɛA-B,G(ɛ),if ɛB-A,s(F(ɛ),G(ɛ)),if ɛAB,

where s denotes any s-norm.

### Definition 2.9

The intersection of the two fuzzy soft sets (F,A) and (G,B) over a common universe U is the fuzzy soft set (H,C) where C = AB, and ∀ɛC,

H(ɛ)={F(ɛ),if ɛA-B,G(ɛ),if ɛB-A,t(F(ɛ),G(ɛ)),if ɛAB,

where t denotes any t-norm.

In this section, we describe soft shadow sets. In our definition, we introduced different shadow sets for each fuzzy set related to each parameter. Some operations and properties are presented and studied with some examples.

### Definition 3.1

Let U = {x1, x2, …, xn} be the universal set of elements, (F,E) be a fuzzy soft set over U where F is a mapping given by F : EIU, E = {e1, e2, …, em} be the set of parameters, and let shdw = {(α1, β1), (α2, β2), …, (αm, βm)} be the shadow parameter set related to E. Let shdw (U) be the set of all shadow subsets on U. A pair (F,E)shdw is called a shadow soft set over U, where F is a mapping F(αi,βi) : Eshdw (U), ∀i = 1, 2, …, m. Thus, we can write F(αi,βi) as

F(αi,βi)(ei)={xjfi(xj)},

i = 1, 2, …, m, j = 1, 2, …, n, where

fi(xj)={0,if μi(xj)αi,1,if μi(xj)βi,a,αi<μi(xj)<βi.

### Example 3.1

Let U = {u1, u2, u3, u4, u5} be the universal set of elements and E = {e1, e2, e3, e4} be the set of parameters

F(e1)={u10.2,u20.4,u30.8,u40.5,u50},F(e2)={u10.2,u20.4,u30.8,u40.5,u50},F(e3)={u10.7,u20.7,u30.1,u40.8,u50.7},F(e4)={u10.9,u20.5,u30.5,u40.2,u50.7}

be the fuzzy soft set over U. Let

shdw={(0.2,0.7),(0.3,0.8),(0.3,0.7),(0.4,0.8)}

be the shadow parameters set that related to E. Then the shadow soft set (F,E)shdw can be defined as follows:

F(0.2,0.7)(e1)={u10,u2[0,1],u31,u4[0,1],u50},F(0.3,0.8)(e2)={u10,u2[0,1],u31,u41,u50},F(0.3,0.7)(e3)={u11,u21,u30,u41,u51},F(0.4,0.8)(e4)={u11,u2[0,1],u3[0,1],u40,u5[0,1]}.

Then, we can view the shadow soft set (F,E)shdw, comprising the following collection of approximations:

(F,E)shdw={(e1,{u10,u2[0,1],u31,u4[0,1],u50}),(e2,{u10,u2[0,1],u31,u41,u50}),(e3,{u11,u21,u30,u41,u51}),(e4,{u11,u2[0,1],u3[0,1],u40,u5[0,1]})}.

Here, a = [0, 1].

### Definition 3.2

Let (F,A)shdw and (G,B)shdw be two shadow soft sets over common universe U. (F,A)shdw is said to be a shadow-soft subset of (G,B)shdw if AB; and F(αi,βi) (ei) (x) is a shadow subset of G(αi,βi) (ei) (x); ∀eiAB; xU.. We denote this by (F,A)shdw ⊆ (F,A)shdw, where 0 < [0, 1] < 1,

### Definition 3.3

A null shadow soft set (φ,E)shdw over common universe U is a shadow soft set with φ(e)shdw(x) = 0, ∀eE; xU.

### Definition 3.4

An absolute shadow soft set (Ψ,E)shdw over common universe U is a shadow soft set with Ψ(e)shdw(x) = 1, ∀eE; xU.

### Definition 3.5

A completely shadowed soft set (Ω,E)shdw over common universe U is a shadow soft set with Ω(e)shdw(x) = [0, 1], ∀eE; xU.

### 3.1 Basic Operations

Here, we introduce some basic operations on the shadow soft set, namely complement, union, and intersection, and provide some properties related to these operations.

Definition 3.6

Let (F,E)shdw be a soft shadow set over U. Then, the complement of (F,E)shdw, denoted by (F,E)shdwc, is defined by (F,E)shdwc=c(F,E)shdw, where c is a shadow complement.

Example 3.2

Consider a shadow soft set (F,E)shdw over U as in Example 3.1. Then, we find the complement of (F,E)shdw as follows:

(F,E)shdwc={(e1,{u11,u2[0,1],u30,u4[0,1],u51}),(e2,{u11,u2[0,1],u30,u00,u51}),(e3,{u10,u20,u31,u40,u50}),(e4,{u10,u2[0,1],u3[0,1],u41,u5[0,1]})}.
Proposition 3.1

Let (F,E)shdw be a soft shadow set over U. Then, the following holds:

((F,E)shdwc)c=(F,E)shdw.
Proof

The proof is straightforward based on Definitions 3.1 and 3.6. For each eE and xX we have Table 3.

### Definition 3.7

The union of two shadow soft sets (F,A)shdw and (G,B)shdw over a common universe U and a shadow parameter set shdw is a shadow soft set (H,C)shdw where C = AB, and ∀ɛC,

H(ɛ)shdw={Fshdw(ɛ),if ɛA-B,Gshdw(ɛ),if ɛB-A,S(Fshdw(ɛ),Gshdw(ɛ)),if ɛAB,

where S denotes shadow union.

### Example 3.3

Let U = {u1, u2, u3, u4, u5} be the universal set of elements and E = {e1, e2, e3, e4} be the set of parameters. Let the shadow soft set (F,E)shdw comprise the following collection of approximations.

(F,A)shdw={(e1,{u10,u2[0,1],u31,u4[0,1],u50}),(e2,{u10,u2[0,1],u31,u41,u50}),(e3,{u11,u21,u30,u41,u51}),(e4,{u11,u2[0,1],u3[0,1],u40,u5[0,1]})}.

Let (G,B)shdw be another shadow soft set over U defined as follows:

(G,B)shdw={(e1,{u1[0,1],u2[0,1],u30,u41,u5[0,1]}),(e2,{u10,u21,u3[0,1],u40,u5[0,1]}),(e4,{u11,u2[0,1],u31,u41,u51})}.

Here, a = [0, 1].

Using the shadow soft union and shadow union, we have (H, C)shdw as follows:

(H,C)shdw={(e1,{u1[0,1],u2[0,1],u31,u41,u5[0,1]}),(e2,{u10,u21,u31,u41,u5[0,1]}),(e3,{u11,u2[0,1],u30,u41,u51}),(e4,{u11,u21,u31,u41,u51})}.

### Proposition 3.2

Let (F,E)shdw, (G,E)shdw and (H,E)shdw be any three shadow soft sets. Then, the following results hold.

• FshdwGshdw = GshdwFshdw.

• Fshdw ∪ (GshdwHshdw) = (FshdwGshdw) ∪ Hshdw.

• Fshdwφshdw = Fshdw.

• Fshdw ∪ Ψshdw = Ψshdw.

• Fshdw ∪ Ωshdw ⊆ Ψshdw.

Proof

The proof is straightforward based on Definitions 3.2, 3.3, 3.4, 3.5 and 3.7.

### Definition 3.8

The intersection of two shadow soft sets (F, A)shdw and (G,B)shdw over a common universe U and a shadow parameter set shdw is a shadow soft set (H,C)shdw where C = AB, and ∀ɛC,

K(ɛ)shdw={Fshdw(ɛ),if ɛA-B,Gshdw(ɛ),if ɛB-A,S(Fshdw(ɛ),Gshdw(ɛ)),if ɛAB,

where T denotes shadow intersection.

### Example 3.4

Let U = {u1, u2, u3, u4, u5} be the universal set of elements and E = {e1, e2, e3, e4} be the set of parameters. Consider the shadow soft sets (F,A)shdw and (G,B)shdw as in Example 3.3. Using the shadow soft intersection and shadow intersection, we have (K,C)shdw as follows:

(K,C)shdw={(e1,{u10,u2[0,1],u30,u4[0,1],u50}),(e2,{u10,u2[0,1],u3[0,1],u40,u50}),(e3,{u11,u2[0,1],u30,u41,u51}),(e4,{u11,u2[0,1],u3[0,1],u40,u5[0,1]})}.

### Proposition 3.3

Let (F,E)shdw, (G,E)shdw and (H,E)shdw be any three shadow soft sets. Then, the following results hold.

• FshdwGshdw = GshdwFshdw.

• Fshdw ∩ (GshdwHshdw) = (FshdwGshdw) ∩ Hshdw.

• Fshdwφshdw = φshdw.

• Fshdw ∩ Ψshdw = Fshdw.

• Fshdw ∪ ΩshdwFshdw.

Proof

The proof is straightforward based on Definitions 3.2, 3.3, 3.4, 3.5 and 3.8.

### Proposition 3.4

Let (F,E)shdw and (G,E)shdw be any two shadowed soft sets. Then, DeMorgan’s laws hold.

• (FshdwGshdw)c = FshdwcGshdwc.

• (FshdwGshdw)c = FshdwcGshdwc.

Proof

The proof is straightforward, using the truth table method. See Tables 4 and 5.

### Proposition 3.5

Let (F,E)shdw, (G,E)shdw and (H,E)shdw be any of the three shadow soft sets. Then, the following results hold.

• Fshdw ∪ (GshdwHshdw) = (FshdwGshdw) ∩ (FshdwHshdw).

• Fshdw ∩ (GshdwHshdw) = (FshdwGshdw) ∪̃ (FshdwHshdw).

Proof

The proof is straightforward, using the truth table method.

### 3.2 AND and OR Operations on Shadow Soft Set

In this section, we provide definitions of AND and OR operations on the shadow soft set and study some of their properties.

### Definition 3.9

If (F,A)shdw and (G,B)shdw are two shadow soft sets then “(F,A)shdw AND (G,B)shdw” denoted by

(F,A)shdw(G,B)shdw

is defined by

(F,A)shdw(G,B)shdw=(H,A×B)shdw,

where Hshdw (α, β) = Fshdw(α) ∩ Gshdw(β) and ∀ (α, β) ∈ A × B.

### Example 3.5

Suppose the universe comprises five machines u1, u2, u3, u4, u5, that is, U = {u1, u2, u3, u4, u5} and consider the set of parameters E = {e1, e2, e3} that describe their performance according to certain specific tasks. Suppose a firm wants to buy one such machine depending on any two parameters. Let there be two observations Fshdw and Gshdw by experts A and B, respectively, defined as follows:

(F,A)shdw={(e1,{u10,u2[0,1],u30,u4[0,1],u50}),(e2,{u10,u2[0,1],u3[0,1],u40,u50}),(e3,{u11,u2[0,1],u30,u41,u51}).}.

Let Gshdw be another soft shadow set over (U,E) defined as follows:

(G,B)shdw={(e1,{u1[0,1],u2[0,1],u31,u41,u5[0,1]}),(e2,{u10,u21,u31,u41,u5[0,1]}),(e3,{u11,u2[0,1],u30,u41,u51})}.

To find the AND between the two shadow soft sets, we have (Fshdw,A) AND (Gshdw,B) = (Hshdw,A × B) where

Hshdw(e1,e1)={u10,u2[0,1],u30,u4[0,1],u50};Hshdw(e1,e2)={u10,u2[0,1],u30,u4[0,1],u50},Hshdw(e1,e3)={u10,u2[0,1],u30,u4[0,1],u50},Hshdw(e2,e1)={u10,u2[0,1],u3[0,1],u40,u50},Hshdw(e2,e2)={u10,u2[0,1],u3[0,1],u40,u50},Hshdw(e2,e3)={u10,u2[0,1],u30,u40,u50},Hshdw(e3,e1)={u1[0,1],u2[0,1],u30,u41,u5[0,1]},Hshdw(e3,e2)={u10,u2[0,1],u30,u41,u5[0,1]},Hshdw(e3,e3)={u11,u2[0,1],u30,u41,u51}.

In Table 6, we can see the table representation of (Hshdw,A× B). Now, we find the score of each element in U as in Table 7, where X0 is the number of repetitions of (0), X1 is the number of repetitions of (1) and X([0,1]) is the number of repetitions of ([0, 1])

According to Table 7, machines u1, u3 and u5 have the highest rejection scores, whereas machine u2 has the highest waiting scores and u4 has the highest acceptance score.

### Definition 3.10

If (F,A)shdw and (G,B)shdw are two shadow soft sets then “(F,A)shdw OR (G,B)shdw” denoted by

(F,A)shdw(G,B)shdw

is defined by

(F,A)shdw(G,B)shdw=(H,A×B)shdw,

where Hshdw (α, β) = Fshdw(α) ∪ Gshdw(β) and ∀ (α, β) ∈ A × B.

### Example 3.6

Consider Example 3.5. Suppose the firm wants to buy one such machine depending on any of the parameters. Then, we find the OR between the two shadow soft sets as follows: (Fshdw,A) OR (Gshdw,B) = (Hshdw,A×B) where

Hshdw(e1,e1)={u1[0,1],u2[0,1],u31,u41,u5[0,1]};Hshdw(e1,e2)={u10,u21,u31,u41,u5[0,1]},Hshdw(e1,e3)={u11,u2[0,1],u30,u41,u51},Hshdw(e2,e1)={u1[0,1],u2[0,1],u31,u41,u5[0,1]},Hshdw(e2,e2)={u10,u21,u31,u41,u5[0,1]},Hshdw(e2,e3)={u11,u2[0,1],u3[0,1],u41,u51},Hshdw(e3,e1)={u11,u2[0,1],u31,u41,u51},Hshdw(e3,e2)={u11,u21,u31,u41,u51},Hshdw(e3,e3)={u11,u2[0,1],u30,u41,u51}.

In Table 8, we can see the table representation of (Hshdw,A× B). Now, we find the score of each element in U as shown in Table 9. According to Table 9, machines u1, u3 and u4 have the highest acceptance scores, whereas machines u2 and u5 have the highest waiting scores, and there is no rejecting machine.

### Proposition 3.6

Let (Fshdw,A) and (Gshdw,B) be any two shadow soft sets. Then, the following results hold.

• ((Fshdw,A) ∧ (Gshdw,B))c = (Fshdw,A)c ∨(Gshdw,B)c,

• ((Fshdw,A) ∨ (Gshdw,B))c = (Fshdw,A)c ∧(Gshdw,B)c.

Proof

Straightforward from Definitions 3.6, 3.9 and 3.10.

### Proposition 3.7

Let (μ,A), (δ,B) and (λ, C) be any three shadow soft sets. Then, the following results hold.

• (Fshdw,A) ∧ ((Gshdw,B) ∧ (Hshdw, C)) = ((Fshdw,A) ∧ (Gshdw,B)) ∧ (Hshdw, C),

• (Fshdw,A) ∨ ((Gshdw,B) ∨ (Hshdw, C)) = ((Fshdw,A) ∨ (Gshdw,B)) ∨ (Hshdw, C),

• (Fshdw,A) ∨ ((Gshdw,B) ∧ (Hshdw, C)) = ((Fshdw,A) ∨ (Gshdw,B)) ∧ ((Fshdw,A) ∨ (Hshdw, C)),

• (Fshdw,A) ∧ ((Gshdw,B) ∨ (Hshdw, C)) = ((Fshdw,A) ∧ (Gshdw,B)) ∨ ((Fshdw,A) ∧ (Hshdw, C)).

Proof

Straightforward from Definitions 3.9 and 3.10.

### Remark 3.1

The commutativity property does not hold for the AND and OR operations because A × BB × A.

We introduced the concept of a shadow soft set as a generalization of a fuzzy soft set and studied some of its properties. Basic operations, such as complements, unions, and intersections, were defined on the shadow soft set. Subsequently, “AND” and “OR” operations were presented with their properties

The authors acknowledge the financial support received from Jadara University.

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

Table. 1.

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.

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.

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.

Table 4. (FshdwGshdw)c = FshdwcGshdwc.

FshdwGshdw(FshdwGshdw)(FshdwGshdw)CFshdwCGshdwC(FshdwCGshdwC)
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.

Table 5. (FshdwGshdw)c = FshdwcGshdwc..

FshdwGshdw(FshdwGshdw)(FshdwGshdw)CFshdwCGshdwC(FshdwCGshdwC)
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.

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.

Table 7. Score of ui.

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

Table. 8.

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.

Table 9. Score of ui.

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

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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

### 1.1 Uncertainty

Classical mathematics is one of the most important requirements in real-life problems, some of which involve imprecise data. Thus, solving these problems involves using mathematical principles based on uncertainty (not crisp). Therefore, many scientists and engineers are interested in non-crisp modeling to describe and debrief useful information hidden in uncertain data. Many researchers have proposed a set of theories to deal with this uncertainty, such as fuzzy set theory [1], intuitionistic fuzzy set theory [2], vague set theory [3], theory of interval mathematics [4], theory of rough set [5], theory of shadow set [6], and neutrosophic set theory [7].

### 1.2 Soft Set

Soft set, which is a theory defined as a new mathematical tool for dealing with uncertainties that traditional mathematical tools cannot handle, was initiated in 1999 by Molodtsov [8]. Since then, researchers have raced to enrich this topic in all aspects, and the related research has rapidly. They have worked on a combination of soft set theory and other theories starting from Maji et al. [9], who introduced the concept of fuzzy soft set, a more general concept as a combination of fuzzy set and soft set, and studied its properties. Then, in 2003 [10], the theory of soft sets and their properties were studied. In 2001, they introduced an intuitionistic fuzzy soft set, a combination of intuitionistic fuzzy sets and soft sets. In addition, Xu et al. combined a vague set with a soft set to define a vague soft set in 2010.

Rough soft set theory is a combination concept defined by Feng et al. [11]. Yang et al. presented a combination of an interval-valued fuzzy set and soft set in 2009 [12]. A neutrosophic soft set is a new concept defined by Maji in 2013 [13]. Many researchers have studied soft set theory, attempted to develop it, and studied its applications in decision-making and medical diagnosis problems. For example, Alkhazaleh et al. [14] introduced the concept of soft multiset as a generalization of the soft set. They also generalized this fuzzy soft multiset theory concept in 2012 [15]. Majumdar and Samanta [16] introduced and studied a generalized fuzzy soft set, where the degree is attached to the parameterizations of fuzzy sets while defining a fuzzy soft set. Alkhazaleh et al. also defined fuzzy parameterized interval-valued fuzzy soft set [17] and possibility fuzzy soft set [18]. They provided their applications in decision-making and medical diagnosis problems. Alkhazaleh and Salleh [19] introduced the concept of a soft expert set, in which the user can know the opinions of all experts in one model without any operations. Even after any operation, the user can know the opinions of all experts. Then, they generalized this concept to a fuzzy soft expert set and established its application in decision-making. Adam and Hassan in 2014 [20] introduced the theory of Q–fuzzy soft set with its application in decision-making and medical diagnosis problems.

### 2. Preliminary

In this section, we recall the definitions related to this work, starting from the fuzzy set, the definition of the shadow set, where we define the shadow set, its properties, and basic operations. Thereafter, we recall the concepts related to soft set theory and fuzzy soft sets based on their properties and operations. Since Zadeh published his new classic paper almost 60 years ago, fuzzy set theory has received increasing attention from researchers in various scientific areas, especially in the past few years. The difference between a binary set and a fuzzy set is that in a “normal” set, every element is either a member or a non-member. Here, it either must be A or notA. In a fuzzy set, an element can be a member of a set to some degree and, simultaneously, a non-member of the same set to some degree. In classical set theory, the membership of elements in a set is assessed in binary terms according to a bivalent condition: an element either belongs or does not belong to the set. By contrast, fuzzy set theory permits the gradual assessment of the membership of elements in a set, which is described with a membership function valued in the closed unit interval [0, 1]. Fuzzy sets generalize classical sets because the indicator functions of classical sets are special cases of the membership functions of fuzzy sets if the latter only takes values 0 or 1. Therefore, a fuzzy set A in a universe of discourse X is a function A : X → [0, 1], usually referred to as the membership function, denoted by μA(x). Some mathematicians use the notation A(x) to denote: membership function instead of μA(x). Fuzzy set A is written symbolically in various

$A={〈x,μA(x)〉:x∈X}.$

### Definition 2.1 ([1])

Let X be a nonempty set. A fuzzy set A in X is characterized by its membership function

$μA:U→[0,1],$

and μA(x) is interpreted as the degree of membership of element x in the fuzzy set A for each xX.

A is completely determined by the set of pairs

$A={(x,μA (x))∣x∈X}.$

The family of all the fuzzy sets in X is denoted by F(X). If X = {x1, …, xn} is a finite set and A is a fuzzy set in X, we often use the notation

$A={μA (x1)/x1,…,μA (xn)/xn},$

where the term μA(xi), i = 1, …, n signifies that μi is the membership grade of xi in A. Here, we use the following notation for the fuzzy set.

$A={xμA(x):x∈X}.$

In what follows, we summarize the quintessence of shadowed sets, a concept presented by Pedrycz. [6]. Pedrycz introduced the concept of a shadow set using a different formula. In general, by considering a fuzzy set A, the main idea is to elevate all membership values that are high enough and reduce those that are viewed as substantially low, where the elevation and reduction components are very radical as we go for 1 and 0, respectively. Mathematically, we define the shadow set as follows:

### Definition 2.2 ([6, 21])

Let X be a set of objects called the universe, and μA : X → [0, 1] be a fuzzy set over X. A shadowed set on X is any mapping B : X → {0, [0, 1], 1}, B : X → {0, a, 1}. by

$Bαf (x)={0,if μA(x)≤α,1,if μA(x)≥1-α,a,O.W.$

The shadowed set B is a mapping defined in X into a three-element set, where the elements of X for which B(x) reaches 1 constitute its core. The zones of X, where B(x) = [0, 1] or B(x) = a form a shadow of this construct. Eventually, the elements with B(x) = 0 are excluded from B. Pedrycz presented a new way of understanding uncertainty. This definition has evolved and undergone some development and modernization, either through Pedrycz or other researchers. To easily demonstrate how to produce a shadow set, one way was to choose α to generate this set by selecting the excluded zone for all values less than α and shadowed set between α and α − 1 where the core zone of all values are greater than α − 1. In contrast, the other direction selects α and β for the same generation process. Let X = {x1, x2, …, x10} be a set of universes, $FX={x10.3,x20.4,x30.8,x40.5,x50.7,x60.9,x10,x80.6,x91,x100.2}$ be a fuzzy set over X. Suppose α = 0.3, then we can write the shadow set Shdw(X) as

$Shdw(X)={x10,x2a,x31,x4a,x50,x61,x10,x8a,x91,x100}.$

Without loss of generality, we can indicate [0, 1] or a with a value 0.5.

Defining operations on shadowed sets as union, intersection, and complement; here, a = [0, 1].

### Definition 2.3 ([6, 21])

The complement of the shadowed set Shdw(X) = {0, [0, 1], 1} is shadowed set Shdw(X)c = {1, [0, 1], 0}.

### Definition 2.4

The union of the two shadowed sets ShdwA(X) and ShdwB(X) over a common universe X is the shadowed set ShdwC(X) presented in Table 1.

### Definition 2.5

The intersection of the two shadowed sets ShdwA(X) and ShdwB(X) over a common universe X is the shadowed set ShdwC(X) presented in Table 2.

Note that for any a in [0, 1] we have min([0, 1], a) = [0, a] and max([0, 1], a) = [a, 1].

Molodtsov defined a soft set as a mapping from a set of parameters to a crisp subset of the universe, where the approximate function of a soft set is defined from a crisp set of parameters to a crisp subset of a universal set. Let U be a universe and E be a set of parameters. Let P(U) denote the power set of U and AE.

### Definition 2.6 ([8])

A pair (F,A) is called a soft set over U, where F is a mapping

$F:A→P(U).$

In other words, a soft set over U is a parameterized family of subsets of the universe U. For ɛA, F (ɛ) may be considered as the set of ɛ-approximate elements of the soft set (F,A).

In 2001, Maji et al. [9] generalized the soft set to a fuzzy soft set. In the following, we review some definitions and properties related to fuzzy soft-set theory, which we use in our work. The following definitions and propositions were provided by Maji et al. [9]:

### Definition 2.7

Let U be an initial universal set and E be a set of parameters. Let IU denote all fuzzy subsets of U. Let AE. A pair (F,E) be a fuzzy soft set over U where F is a mapping given by

$F:A→IU.$

### Definition 2.8

The union of two fuzzy soft sets (F,A) and (G,B) over a common universe U is the fuzzy soft set (H,C) where C = AB, and ∀ɛC,

$H (ɛ)={F (ɛ),if ɛ∈A-B,G (ɛ),if ɛ∈B-A,s (F (ɛ),G (ɛ)),if ɛ∈A∩B,$

where s denotes any s-norm.

### Definition 2.9

The intersection of the two fuzzy soft sets (F,A) and (G,B) over a common universe U is the fuzzy soft set (H,C) where C = AB, and ∀ɛC,

$H (ɛ)={F (ɛ),if ɛ∈A-B,G (ɛ),if ɛ∈B-A,t (F (ɛ),G (ɛ)),if ɛ∈A∩B,$

where t denotes any t-norm.

### 3. Shadow Soft Set

In this section, we describe soft shadow sets. In our definition, we introduced different shadow sets for each fuzzy set related to each parameter. Some operations and properties are presented and studied with some examples.

### Definition 3.1

Let U = {x1, x2, …, xn} be the universal set of elements, (F,E) be a fuzzy soft set over U where F is a mapping given by F : EIU, E = {e1, e2, …, em} be the set of parameters, and let shdw = {(α1, β1), (α2, β2), …, (αm, βm)} be the shadow parameter set related to E. Let shdw (U) be the set of all shadow subsets on U. A pair (F,E)shdw is called a shadow soft set over U, where F is a mapping F(αi,βi) : Eshdw (U), ∀i = 1, 2, …, m. Thus, we can write F(αi,βi) as

$F(αi,βi) (ei)={xjfi (xj)},$

i = 1, 2, …, m, j = 1, 2, …, n, where

$fi (xj)={0,if μi (xj)≤αi,1,if μi (xj)≥βi,a,αi<μi (xj)<βi.$

### Example 3.1

Let U = {u1, u2, u3, u4, u5} be the universal set of elements and E = {e1, e2, e3, e4} be the set of parameters

$F (e1)={u10.2,u20.4,u30.8,u40.5,u50},F (e2)={u10.2,u20.4,u30.8,u40.5,u50},F (e3)={u10.7,u20.7,u30.1,u40.8,u50.7},F (e4)={u10.9,u20.5,u30.5,u40.2,u50.7}$

be the fuzzy soft set over U. Let

$shdw={(0.2,0.7),(0.3,0.8),(0.3,0.7),(0.4,0.8)}$

be the shadow parameters set that related to E. Then the shadow soft set (F,E)shdw can be defined as follows:

$F(0.2,0.7) (e1)={u10,u2[0,1],u31,u4[0,1],u50},F(0.3,0.8) (e2)={u10,u2[0,1],u31,u41,u50},F(0.3,0.7) (e3)={u11,u21,u30,u41,u51},F(0.4,0.8) (e4)={u11,u2[0,1],u3[0,1],u40,u5[0,1]}.$

Then, we can view the shadow soft set (F,E)shdw, comprising the following collection of approximations:

$(F,E)shdw={(e1,{u10,u2[0,1],u31,u4[0,1],u50}), (e2,{u10,u2[0,1],u31,u41,u50}), (e3,{u11,u21,u30,u41,u51}), (e4,{u11,u2[0,1],u3[0,1],u40,u5[0,1]})}.$

Here, a = [0, 1].

### Definition 3.2

Let (F,A)shdw and (G,B)shdw be two shadow soft sets over common universe U. (F,A)shdw is said to be a shadow-soft subset of (G,B)shdw if AB; and F(αi,βi) (ei) (x) is a shadow subset of G(αi,βi) (ei) (x); ∀eiAB; xU.. We denote this by (F,A)shdw ⊆ (F,A)shdw, where 0 < [0, 1] < 1,

### Definition 3.3

A null shadow soft set (φ,E)shdw over common universe U is a shadow soft set with φ(e)shdw(x) = 0, ∀eE; xU.

### Definition 3.4

An absolute shadow soft set (Ψ,E)shdw over common universe U is a shadow soft set with Ψ(e)shdw(x) = 1, ∀eE; xU.

### Definition 3.5

A completely shadowed soft set (Ω,E)shdw over common universe U is a shadow soft set with Ω(e)shdw(x) = [0, 1], ∀eE; xU.

### 3.1 Basic Operations

Here, we introduce some basic operations on the shadow soft set, namely complement, union, and intersection, and provide some properties related to these operations.

Definition 3.6

Let (F,E)shdw be a soft shadow set over U. Then, the complement of (F,E)shdw, denoted by $(F,E)shdwc$, is defined by $(F,E)shdwc=c(F,E)shdw$, where c is a shadow complement.

Example 3.2

Consider a shadow soft set (F,E)shdw over U as in Example 3.1. Then, we find the complement of (F,E)shdw as follows:

$(F,E)shdwc={(e1,{u11,u2[0,1],u30,u4[0,1],u51}), (e2,{u11,u2[0,1],u30,u00,u51}), (e3,{u10,u20,u31,u40,u50}), (e4,{u10,u2[0,1],u3[0,1],u41,u5[0,1]})}.$
Proposition 3.1

Let (F,E)shdw be a soft shadow set over U. Then, the following holds:

$((F,E)shdwc)c=(F,E)shdw.$
Proof

The proof is straightforward based on Definitions 3.1 and 3.6. For each eE and xX we have Table 3.

### Definition 3.7

The union of two shadow soft sets (F,A)shdw and (G,B)shdw over a common universe U and a shadow parameter set shdw is a shadow soft set (H,C)shdw where C = AB, and ∀ɛC,

$H (ɛ)shdw={Fshdw (ɛ),if ɛ∈A-B,Gshdw (ɛ),if ɛ∈B-A,S (Fshdw (ɛ),Gshdw (ɛ)),if ɛ∈A∩B,$

where S denotes shadow union.

### Example 3.3

Let U = {u1, u2, u3, u4, u5} be the universal set of elements and E = {e1, e2, e3, e4} be the set of parameters. Let the shadow soft set (F,E)shdw comprise the following collection of approximations.

$(F,A)shdw={(e1,{u10,u2[0,1],u31,u4[0,1],u50}), (e2,{u10,u2[0,1],u31,u41,u50}), (e3,{u11,u21,u30,u41,u51}), (e4,{u11,u2[0,1],u3[0,1],u40,u5[0,1]})}.$

Let (G,B)shdw be another shadow soft set over U defined as follows:

$(G,B)shdw={(e1,{u1[0,1],u2[0,1],u30,u41,u5[0,1]}), (e2,{u10,u21,u3[0,1],u40,u5[0,1]}), (e4,{u11,u2[0,1],u31,u41,u51})}.$

Here, a = [0, 1].

Using the shadow soft union and shadow union, we have (H, C)shdw as follows:

$(H,C)shdw={(e1,{u1[0,1],u2[0,1],u31,u41,u5[0,1]}), (e2,{u10,u21,u31,u41,u5[0,1]}), (e3,{u11,u2[0,1],u30,u41,u51}), (e4,{u11,u21,u31,u41,u51})}.$

### Proposition 3.2

Let (F,E)shdw, (G,E)shdw and (H,E)shdw be any three shadow soft sets. Then, the following results hold.

• FshdwGshdw = GshdwFshdw.

• Fshdw ∪ (GshdwHshdw) = (FshdwGshdw) ∪ Hshdw.

• Fshdwφshdw = Fshdw.

• Fshdw ∪ Ψshdw = Ψshdw.

• Fshdw ∪ Ωshdw ⊆ Ψshdw.

Proof

The proof is straightforward based on Definitions 3.2, 3.3, 3.4, 3.5 and 3.7.

### Definition 3.8

The intersection of two shadow soft sets (F, A)shdw and (G,B)shdw over a common universe U and a shadow parameter set shdw is a shadow soft set (H,C)shdw where C = AB, and ∀ɛC,

$K (ɛ)shdw={Fshdw (ɛ),if ɛ∈A-B,Gshdw (ɛ),if ɛ∈B-A,S (Fshdw (ɛ),Gshdw (ɛ)),if ɛ∈A∩B,$

where T denotes shadow intersection.

### Example 3.4

Let U = {u1, u2, u3, u4, u5} be the universal set of elements and E = {e1, e2, e3, e4} be the set of parameters. Consider the shadow soft sets (F,A)shdw and (G,B)shdw as in Example 3.3. Using the shadow soft intersection and shadow intersection, we have (K,C)shdw as follows:

$(K,C)shdw={(e1,{u10,u2[0,1],u30,u4[0,1],u50}), (e2,{u10,u2[0,1],u3[0,1],u40,u50}), (e3,{u11,u2[0,1],u30,u41,u51}), (e4,{u11,u2[0,1],u3[0,1],u40,u5[0,1]})}.$

### Proposition 3.3

Let (F,E)shdw, (G,E)shdw and (H,E)shdw be any three shadow soft sets. Then, the following results hold.

• FshdwGshdw = GshdwFshdw.

• Fshdw ∩ (GshdwHshdw) = (FshdwGshdw) ∩ Hshdw.

• Fshdwφshdw = φshdw.

• Fshdw ∩ Ψshdw = Fshdw.

• Fshdw ∪ ΩshdwFshdw.

Proof

The proof is straightforward based on Definitions 3.2, 3.3, 3.4, 3.5 and 3.8.

### Proposition 3.4

Let (F,E)shdw and (G,E)shdw be any two shadowed soft sets. Then, DeMorgan’s laws hold.

• (FshdwGshdw)c = FshdwcGshdwc.

• (FshdwGshdw)c = FshdwcGshdwc.

Proof

The proof is straightforward, using the truth table method. See Tables 4 and 5.

### Proposition 3.5

Let (F,E)shdw, (G,E)shdw and (H,E)shdw be any of the three shadow soft sets. Then, the following results hold.

• Fshdw ∪ (GshdwHshdw) = (FshdwGshdw) ∩ (FshdwHshdw).

• Fshdw ∩ (GshdwHshdw) = (FshdwGshdw) ∪̃ (FshdwHshdw).

Proof

The proof is straightforward, using the truth table method.

### 3.2 AND and OR Operations on Shadow Soft Set

In this section, we provide definitions of AND and OR operations on the shadow soft set and study some of their properties.

### Definition 3.9

If (F,A)shdw and (G,B)shdw are two shadow soft sets then “(F,A)shdw AND (G,B)shdw” denoted by

$(F,A)shdw∧(G,B)shdw$

is defined by

$(F,A)shdw∧(G,B)shdw=(H,A×B)shdw,$

where Hshdw (α, β) = Fshdw(α) ∩ Gshdw(β) and ∀ (α, β) ∈ A × B.

### Example 3.5

Suppose the universe comprises five machines u1, u2, u3, u4, u5, that is, U = {u1, u2, u3, u4, u5} and consider the set of parameters E = {e1, e2, e3} that describe their performance according to certain specific tasks. Suppose a firm wants to buy one such machine depending on any two parameters. Let there be two observations Fshdw and Gshdw by experts A and B, respectively, defined as follows:

$(F,A)shdw={(e1,{u10,u2[0,1],u30,u4[0,1],u50}), (e2,{u10,u2[0,1],u3[0,1],u40,u50}), (e3,{u11,u2[0,1],u30,u41,u51}).}.$

Let Gshdw be another soft shadow set over (U,E) defined as follows:

$(G,B)shdw={(e1,{u1[0,1],u2[0,1],u31,u41,u5[0,1]}), (e2,{u10,u21,u31,u41,u5[0,1]}), (e3,{u11,u2[0,1],u30,u41,u51})}.$

To find the AND between the two shadow soft sets, we have (Fshdw,A) AND (Gshdw,B) = (Hshdw,A × B) where

$Hshdw(e1,e1)={u10,u2[0,1],u30,u4[0,1],u50};Hshdw(e1,e2)={u10,u2[0,1],u30,u4[0,1],u50},Hshdw(e1,e3)={u10,u2[0,1],u30,u4[0,1],u50},Hshdw(e2,e1)={u10,u2[0,1],u3[0,1],u40,u50},Hshdw(e2,e2)={u10,u2[0,1],u3[0,1],u40,u50},Hshdw(e2,e3)={u10,u2[0,1],u30,u40,u50},Hshdw(e3,e1)={u1[0,1],u2[0,1],u30,u41,u5[0,1]},Hshdw(e3,e2)={u10,u2[0,1],u30,u41,u5[0,1]},Hshdw(e3,e3)={u11,u2[0,1],u30,u41,u51}.$

In Table 6, we can see the table representation of (Hshdw,A× B). Now, we find the score of each element in U as in Table 7, where X0 is the number of repetitions of (0), X1 is the number of repetitions of (1) and X([0,1]) is the number of repetitions of ([0, 1])

According to Table 7, machines u1, u3 and u5 have the highest rejection scores, whereas machine u2 has the highest waiting scores and u4 has the highest acceptance score.

### Definition 3.10

If (F,A)shdw and (G,B)shdw are two shadow soft sets then “(F,A)shdw OR (G,B)shdw” denoted by

$(F,A)shdw∨(G,B)shdw$

is defined by

$(F,A)shdw∨(G,B)shdw=(H,A×B)shdw,$

where Hshdw (α, β) = Fshdw(α) ∪ Gshdw(β) and ∀ (α, β) ∈ A × B.

### Example 3.6

Consider Example 3.5. Suppose the firm wants to buy one such machine depending on any of the parameters. Then, we find the OR between the two shadow soft sets as follows: (Fshdw,A) OR (Gshdw,B) = (Hshdw,A×B) where

$Hshdw(e1,e1)={u1[0,1],u2[0,1],u31,u41,u5[0,1]};Hshdw(e1,e2)={u10,u21,u31,u41,u5[0,1]},Hshdw(e1,e3)={u11,u2[0,1],u30,u41,u51},Hshdw(e2,e1)={u1[0,1],u2[0,1],u31,u41,u5[0,1]},Hshdw(e2,e2)={u10,u21,u31,u41,u5[0,1]},Hshdw(e2,e3)={u11,u2[0,1],u3[0,1],u41,u51},Hshdw(e3,e1)={u11,u2[0,1],u31,u41,u51},Hshdw(e3,e2)={u11,u21,u31,u41,u51},Hshdw(e3,e3)={u11,u2[0,1],u30,u41,u51}.$

In Table 8, we can see the table representation of (Hshdw,A× B). Now, we find the score of each element in U as shown in Table 9. According to Table 9, machines u1, u3 and u4 have the highest acceptance scores, whereas machines u2 and u5 have the highest waiting scores, and there is no rejecting machine.

### Proposition 3.6

Let (Fshdw,A) and (Gshdw,B) be any two shadow soft sets. Then, the following results hold.

• ((Fshdw,A) ∧ (Gshdw,B))c = (Fshdw,A)c ∨(Gshdw,B)c,

• ((Fshdw,A) ∨ (Gshdw,B))c = (Fshdw,A)c ∧(Gshdw,B)c.

Proof

Straightforward from Definitions 3.6, 3.9 and 3.10.

### Proposition 3.7

Let (μ,A), (δ,B) and (λ, C) be any three shadow soft sets. Then, the following results hold.

• (Fshdw,A) ∧ ((Gshdw,B) ∧ (Hshdw, C)) = ((Fshdw,A) ∧ (Gshdw,B)) ∧ (Hshdw, C),

• (Fshdw,A) ∨ ((Gshdw,B) ∨ (Hshdw, C)) = ((Fshdw,A) ∨ (Gshdw,B)) ∨ (Hshdw, C),

• (Fshdw,A) ∨ ((Gshdw,B) ∧ (Hshdw, C)) = ((Fshdw,A) ∨ (Gshdw,B)) ∧ ((Fshdw,A) ∨ (Hshdw, C)),

• (Fshdw,A) ∧ ((Gshdw,B) ∨ (Hshdw, C)) = ((Fshdw,A) ∧ (Gshdw,B)) ∨ ((Fshdw,A) ∧ (Hshdw, C)).

Proof

Straightforward from Definitions 3.9 and 3.10.

### Remark 3.1

The commutativity property does not hold for the AND and OR operations because A × BB × A.

### 4. Conclusion

We introduced the concept of a shadow soft set as a generalization of a fuzzy soft set and studied some of its properties. Basic operations, such as complements, unions, and intersections, were defined on the shadow soft set. Subsequently, “AND” and “OR” operations were presented with their properties

ShdwC(X) = ShdwA(X) ∪ ShdwB(X).

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

ShdwC(X) = ShdwA(X) ∩ ShdwB(X).

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

((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]

(FshdwGshdw)c = FshdwcGshdwc.

FshdwGshdw(FshdwGshdw)(FshdwGshdw)CFshdwCGshdwC(FshdwCGshdwC)
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]

(FshdwGshdw)c = FshdwcGshdwc..

FshdwGshdw(FshdwGshdw)(FshdwGshdw)CFshdwCGshdwC(FshdwCGshdwC)
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]

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

Score of ui.

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

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

Score of ui.

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

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