Definition 1

Crisp set

It defines any set ξ based on a given universal set U such that an element belongs to ξ or not. One of the examples is a student who is either present or absent in the class. It does not define the exact membership of whether an element belongs to the set.

Definition 2

Fuzzy set (Zadeh, 1965)

Let us suppose E is a universe, then the FS(ξ) can be defined as mapping μ_X(k) : ξ → [0, 1] for each k ∈ ξ. In this case, each element is represented using the defined membership values μ_ξ within [0, 1]. It represents the degree of an element that belongs to the given set. In this method, it provides representation of any element in the given set via a soft boundary.

Definition 3

Intuitionistic fuzzy set (Atanassov, 1986)

The IFS is a generalization of FS. It represents the acceptation or rejection part of any attribute simultaneously. The IFS A can be defined by A = {x, μ_X(k), ν_X(k)/k ∈ ξ} where μ_A(k) : ξ → [0, 1],   ν_A(k) : ξ → [0, 1] for each k ∈ ξ such that 0 ≤ μ_A(k) + ν_A(k) ≤ 1. Here, μ_A(k) : ξ → [0, 1] denotes degrees of membership andν_A(k) : ξ → [0, 1] denotes nonmembership of k ∈ A, respectively.

Definition 4

Interval-valued fuzzy set

The interval-valued fuzzy set is nothing but an extension of FS. It provides a way to represent the membership for belonging of any attribute. The interval-valued fuzzy set A over a universe ξ is defined by

$A=\left\{[A^{-}\left(k\right),A^{+}\left(k\right)]/k\in \xi \right\},$

where A⁻(k), A⁺(k) represent the lower boundary and upper boundary for the given membership degrees within interval [0, 1].

Definition 5

Cubic set

The cubic set provides a way to represent the interval-valued fuzzy set with more predictive analytics. It can be defined with the help of an interval-valued fuzzy set A(x) as well as a single-valued fuzzy set μ(k) as Ξ = {< x, A(k), μ(k) > /k ∈ ξ}. It means one can also represent the cubic set as 〈A, μ〉 for precise representation of any event.

Example 1

The set of NS A of ξ defined by

$A=\left\{\begin{array}{l}〈\left[0.2,0.32\right],[0.32,0.42],[0.71,0.91]〉/x_1^{+}\\ 〈\left[0.12,0.22\right],[0.51,0.91],[0.12,0.91]〉/x_2^{+}\\ 〈\left[0.71,0.82\right],[0.11,0.11],[0.51,0.42]〉/x_3^{+}\end{array}\right\}$

and a NS λ is a set of ξ defined by

$\lambda =\left\{\begin{array}{l}〈0.02,0.21,0.41〉/x_1\\ 〈0.11,0.02,0.21〉/x_2\\ 〈0.31,0.11,0.71〉/x{}_3\end{array}\right\}$

then τ = 〈A, λ〉 n is a neutrosophic cubic set.

Definition 6

Interval cubic set

The interval cubic set is nothing but an extension of the cubic set where the single-valued fuzzy set is replaced by interval-valued set, that is, Ξ = 〈A, μ〉 where A(k) is the interval-valued fuzzy set and the μ(k) lies between them as:

$A^{-}\left(x\right)\le \mu \left(x\right)\le A^{+}\left(x\right),\forall k\in \xi \text{.}$

Example 2

Let τ = 〈A, λ〉 ∈ C_N^X where C_N^X is the set of cubic sets. A(x) = 〈[0.1, 0.3], [0.4, 0.6], [0.7, 0.8]〉 and λ(x) = 〈0.2, 0.5, 0.6〉 for every k in ξ. Then τ = 〈A, λ〉 is an interval cubic set.

Definition 7

External cubic set

The external cubic set Ξ = 〈A, μ〉 is a set in which the FS μ(k) membership values do not belong to the given interval set A(x) meaning

$\mu \left(x\right)\notin \left(A^{-},\left(k\right),A^{+}\left(k\right)\right)\forall k\in \xi \text{.}$

Example 3

Let τ = 〈A, λ〉 ∈ C_N^X where C_N^X is the set of cubic sets. A(k) = 〈[0.1, 0.3], [0.4, 0.6], [0.7, 0.8]〉 and λ(k) = 〈0.4, 0.2, 0.3〉 for every k in ξ. Then τ = 〈A, λ〉 is an external cubic set.

Definition 8

Neutrosophic set

The NS consists of reptile functions, namely truth, indeterminacy, and false, (T, I, F), independently. Each of these values lies between 0 and 1 and does not depend on them. The boundary conditions of the sum of these membership degrees are 0 ≤ T + I + F ≤ 3. In this, 0 is hold for the universal false cases and 3 are the universal truth cases three memberships, that is,

$\lambda =\left\{〈x:T,I,F〉:x\in \xi \right\}.$

Definition 9

Interval neutrosophic set

The interval-valued neutrosophic set consists of reptile functions, namely truth, indeterminacy, and false, (T, I, F). Each of these values is defined in the following form [T⁻, T⁺], [I⁻, I⁺], and [F⁻, F⁺]. All these values lie between 0 and 1, and we denote this as

$A=\left\{〈k:\left[A_T^{-},A_T^{+}\right],\left[A_I^{-},A_I^{+}\right],\left[A_F^{-},A_F^{+}\right]〉:k\in \xi \right\}.$

Definition 10

Neutrosophic set (Smarandache, 1995)

This set contains triplets having true, false, and indeterminacy membership values that can be characterized independently, T_N, I_N, FN, in [0,1]. It can be abbreviated as follows:

There is no restriction on the sum of T_N(k), I_N(k), and F_N(k). So

$0^{-}\le +T_N\left(k\right)+I_N\left(k\right)+F_N\left(k\right)\le 3^{+}.$

Definition 11

Nonstandard neutrosophic set (Smarandache, 1995)

Let ξ be a nonempty set and its element is k, the NS N in ξ is termed by

$A=\left\{〈k;T_A\left(k\right),I_A\left(k\right),F_A\left(k\right)〉\left|k\in \xi \right.\right\}$

which is characterized by a TMF T_N(k), an IMF I_N(k), and an FMF F_N(k), respectively, where

$T_N\left(k\right):\xi \to \left]{}^{-}01^{+}\right[$

$I_N\left(k\right):\xi \to \left]{}^{-}01^{+}\right[$

$F_N\left(k\right):\xi \to \left]{}^{-}01^{+}\right[$

The functions T_N(k), I_N(k), F_N(k) in ξ are real standard or nonstandard subsets of ]⁻ 0, 1⁺[. The sum of T_N(k), I_N(k), F_N(k) does not have any restrictions, that is

${}^{-}0\le sup{T}_N\left(x\right)+sup{I}_N\left(k\right)+sup{F}_N\left(k\right)\le 3^{+}.$

Here ]⁻ 0, 1⁺[ is named the nonstandard subset, which is the extension of real standard subsets [0,1] where the nonstandard number 1⁺ = 1 + ɛ, “1” is named the standard part, and “ɛ” is named the nonstandard part. ⁻ 0 = 0 − ɛ, “0” is the standard part and “ɛ” is named the nonstandard part, where ɛ is closed to positive real number zero.

Definition 12

Standard neutrosophic set (Wang, Smarandache, Zhang, et al., 2010)

It is well known that the NS (N) in use contains a TMF T_N(k), an IMF I_N(k), and a FMF F_N(k), respectively. Each of them can contain the membership values as given below in case of the standard format:

$T_N\left(k\right):\xi \to \left[0,1\right]$

$I_N\left(k\right):\xi \to \left[0,1\right]$

$F_N\left(k\right):\xi \to \left[0,1\right]$

Then,

$N=\left\{〈k;T_N\left(k\right),I_N\left(k\right),F_N\left(k\right)〉\left|k\in \xi \right.\right\}$

is termed an SVNS.

If the nonempty set ξ has only one element x, then we call the NS N the single-valued neutrosophic number (SVNN). We abbreviate it as N = 〈k; T_N, I_N, F_N〉.

Definition 13

(Wang et al., 2010)

Suppose N and M are two SVNSs, N is contained in M, if

$T_N\left(k\right)\le T_M\left(k\right),I_N\left(k\right)\ge I_M\left(k\right),F_N\left(k\right)\ge F_M\left(k\right)$

for each k in ξ.

Definition 14

(Wang et al., 2010)

Suppose N is an SVNS, and its complement is termed as below:

$N^C=\left\{〈k;F_N\left(k\right),1-I_N\left(k\right),T_N\left(k\right)〉\left|k\in \xi \right.\right\}.$

Definition 15

(Wang et al., 2010)

Suppose N = 〈T_N, I_N, F_N〉 and M = 〈T_M, I_M, F_M〉 are two SVNNs, and λ > 0, then

$N\oplus M=〈T_N+T_M-T_N{T}_M,I_N{I}_M,F_N{F}_M〉;$

$N\otimes M=〈T_N{T}_M,I_N+I_M-I_N{I}_M,F_N+F_M-F_N{F}_M〉;$

$\mathit{\lambda N}=〈1-{\left(1-T_N\right)}^{\lambda },{\left(I_N\right)}^{\lambda },{\left(F_N\right)}^{\lambda }〉;$

$N^{\lambda }=〈{\left(T_N\right)}^{\lambda },1-{\left(1-I_N\right)}^{\lambda },1-{\left(1-F_N\right)}^{\lambda }〉.$

Definition 16

Single-valued neutrosophic overset (Smarandache, 2007)

Let us suppose that ξ is a series of real number points presented by k, then the NS will be a subset of those points, that is, N ⊂ ξ having T_N(k), I_N(k), and F_N(k). It describes the TM degree, the IM degree, and the FM degree for the given element k ∈ ξ with respect to the NS N. The overset of NS can be defined as follows:

$N_{\text{SVNOV}}=\left\{\left(x,T_N\left(k\right),I_N\left(k\right),F_N\left(k\right)\right),k\in \xi \text{and}{T}_N\left(k\right),I_N\left(k\right),F_N\left(k\right)\in \left[0,\mathrm{\Omega }\right]\right\},$

where T_N(k), I_N(k), F_N(k): ξ → [0, Ω], 0 < 1 < Ω and Ω are named overlimit, then there exists at least one element in N such that it has at least one neutrosophic component > 1, and no element has a neutrosophic component < 0.

Definition 17

Single-valued neutrosophic underset (Smarandache, 2007)

Let us suppose that ξ is a series of points (objects) with basic elements in ξ presented by k and the NS N ⊂ ξ. Here T_N(k), I_N(k), F_N(k) ts are the TM degree, the IM degree, and the FM degree for the element x ∈ ξ with respect to the NS N. In this case, the underset of neutrosophic values can be defined as:

$N_{\text{SVNU}}=\left\{\left(k,T_{N}k,I_{N}k,F_N\left(k\right)\right),k\in \xi \text{and}{T}_{N}k,I_{N}k,F_N\left(k\right)\in \left[\Psi ,1\right]\right\}.$

Definition 18

A single-valued neutrosophic offset (Smarandache, 2007)

Let us suppose that ξ is a series of points (objects) with basic elements in ξ presented by k and the NS N ⊂ ξ. Let T_N(k), I_N(k), F_N(k) represent the TM degree, the IM degree, and the FM degree for the given element k ∈ ξ with respect to the NS N. The offset can be defined as follows:

$N_{\text{SVNOF}}=\left\{\left(k,T_N\left(k\right),I_N\left(k\right),F_N\left(k\right)\right),x\in \xi \text{and}{T}_{N}k,I_{N}k,F_N\left(k\right)\in \left[\Psi ,\Omega \right]\right\}.$

In this case, T_Nk, I_Nk, F_N(k): ξ → [Ψ, 1], Ψ < 0 < 1 < Ω and Ψ are named the underlimit while Ω is named the overlimit. It means there exists some elements in N such that at least one neutrosophic component > 1 and at least another neutrosophic component < 0.

Example 4

N = {(k₁, 〈1.2, 0.4, 0.1〉), (k₂, 〈0.2, 0.3, − 0.7〉)} because T(k₁) = 1.2 > 1, F(k₂) = − 0.7 < 0.

Definition 19

Complement of overset/underset/offset (Smarandache, 2016)

The complement of an SVN overset/underset/offset N is abbreviated as C(N) and is defined by

$C\left(N\right)=\left\{\left(k,⟨F_N\left(k\right),\Psi +\mathrm{\Omega }-I_N\left(x\right),T_N\left(k\right)>\right),k\in \xi \right\}\text{.}$

Definition 20

Union and intersection of overset/underset/offset (Smarandache, 2016)

The intersection of two SVN overset/underset/offset N and M is an SVNN overset/underset/offset (C) represented as follows: C = N ∩ M and is represented by

$C=M\cap M=\left\{\left(k,<,min\left(T_N\left(k\right),T_M\left(k\right)\right),max\left(I_N\left(k\right),I_M\left(k\right)\right),max\left(F_N\left(k\right),F_M\left(k\right)\right)\right),k\in \xi \right\}.$

Definition 21

Containment of interval neutrosophic set (Wang, Smarandache, Zhang, et al., 2005; Zhang, Pu, Wang, et al., 2014)

Suppose N and M are two INSs, N is contained in M,

$\begin{array}{ll}inf{T}_A\left(k\right)\le inf{T}_M\left(k\right),& sup{T}_N\left(x\right)\le sup{T}_M\left(x\right)\\ inf{I}_N\left(k\right)\ge inf{I}_M\left(k\right),& sup{I}_N\left(k\right)\ge sup{I}_M\left(k\right)\\ inf{F}_N\left(k\right)\ge inf{F}_M\left(k\right),& sup{F}_N\left(k\right)\ge sup{F}_M\left(k\right)\end{array}$

for every k in ξ.

Definition 22

(Wang et al., 2005; Zhang et al., 2014)

The complement of INS A is defined by

$N^C=\left\{〈k;T_{N^C}\left(k\right),I_{N^C}\left(k\right),F_{N^C}\left(k\right)〉\left|k\in \xi \right.\right\},$

where T_N^C = F_N(k) = [infF_N(k), supF_N(k)], I_N^C(k) = [1 − sup I_N (k), 1 − inf I_N(k)], and F_N^C(k) = T_N(k) = [infT_N(k), supT_N(k)].

Definition 23

(Wang et al., 2005; Zhang et al., 2014)

Suppose N = 〈T_N, I_N, F_N〉 and M = 〈T_M, I_M, F_M〉 are two INSs, and λ > 0. The operational laws will then be defined as below:

(1)

$\begin{array}{l}C=N\oplus M\\ inf{T}_C\left(k\right)=min\left(inf{T}_N\left(k\right)+inf{T}_M\left(k\right),1\right)\\ sup{T}_C\left(k\right)=min\left(sup{T}_N\left(k\right)+sup{T}_M\left(k\right),1\right)\\ inf{I}_C\left(k\right)=min\left(inf{T}_N\left(k\right)+inf{T}_M\left(k\right),1\right)\\ sup{I}_C\left(k\right)=min\left(sup{I}_N\left(k\right)+sup{I}_M\left(k\right),1\right)\\ inf{F}_C\left(k\right)=min\left(inf{F}_N\left(k\right)+inf{F}_M\left(k\right),1\right)\\ sup{F}_C\left(k\right)=min\left(sup{F}_N\left(k\right)+sup{F}_M\left(k\right),1\right)\end{array}$

(2)

$\begin{array}{c}N\otimes M=\{\left[inf{T}_{N}inf{T}_M,sup{T}_{N}sup{T}_M\right],\\ \left[inf{I}_N+inf{I}_M-inf{I}_{N}inf{I}_M,sup{I}_N+sup{I}_M-sup{I}_{N}sup{I}_M\right],\\ \left[inf{F}_N+inf{F}_M-inf{F}_{N}inf{F}_M,sup{F}_N+sup{F}_M-sup{F}_{N}sup{F}_M\right]\}\end{array}$

(3)

$\mathit{\lambda N}=\left\{\left[\lambda inf{T}_N,\lambda sup{T}_N\right],\left[\lambda inf{I}_N,\lambda sup{I}_N\right],\left[\lambda inf{F}_N,\lambda sup{F}_N\right]\right\}$

(4)

$N^{\lambda }=\left\{\left[{\left(inf{T}_N\right)}^{\lambda },{\left(sup{T}_N\right)}^{\lambda }\right],,,\left[1-{\left(1-inf{I}_N\right)}^{\lambda },1-{\left(1-sup{I}_N\right)}^{\lambda }\right],,,\left[1-{\left(1-inf{F}_N\right)}^{\lambda },1-{\left(1-sup{F}_N\right)}^{\lambda }\right]\right\}$

$F_{N^C}\left(k\right)=T_N\left(k\right)=\left[inf{T}_N\left(k\right),sup{T}_N\left(k\right)\right],$

where T_N = T_N(k)，I_N = I_N(k), F_N = F_N(k).

Ye (2015) has developed the concepts of interval neutrosophic linguistic sets (INLS) and interval neutrosophic linguistic variables by combining a linguistic variable with an interval neutrosophic set (INS).

Definition 24

An interval-valued neutrosophic linguistic set (Ye, 2015)

Let ξ be a series of points with basic elements in ξ presented by k, then an interval neutrosophic linguistic set N (IVNLS) in ξ is defined as

$N_{\text{IVNLS}}=\left\{〈k,\left[s_{\theta \left(k\right)},\left(T_N\left(k\right),I_N\left(k\right),F_N\left(k\right)\right)\right]〉\left|k\in \xi \right.\right\},$

where $s_{\theta \left(x\right)}\in \stackrel{⌢}{s}$_,

$\begin{array}{l}{T}_N\left(k\right)=\left[inf{T}_N\left(k\right),sup{T}_N\left(k\right)\right]\subseteq \left[0,1\right],\\ I_N\left(k\right)=\left[inf{I}_N\left(k\right),sup{I}_N\left(k\right)\right]\subseteq \left[0,1\right],\\ F_N\left(k\right)=\left[inf{F}_N\left(k\right),sup{F}_N\left(k\right)\right]\subseteq \left[0,1\right],\end{array}$

with the condition 0 ≤ T_N(k) + I_N(k) + F_N(k) ≤ 3, for any k ∈ ξ. s_θ(x) is an uncertain linguistic term. The functions T_N(k), I_N(k), and F_N(k) express, respectively, the TM degree, the IM degree, and the FM degree of the element k in ξ belonging to the linguistic term s_θ(x), which is another continuous form of the linguistic set S.s_θ, s_ρ, s_μ, s_ν are four linguistic terms, and s₀ ≤ s_θ ≤ s_ρ ≤ s_μ ≤ s_ν ≤ s_l − 1 if 0 ≤ θ ≤ ρ ≤ μ ≤ ν ≤ l − 1, then the trapezoid linguistic variable (TLV) is termed as $\stackrel{⌢}{s}=\left[s_{\theta },s_{\rho },s_{\mu },s_{\nu }\right]$, and $\stackrel{⌢}{s}$ represents a set of the TLVs.

Definition 25

Linguistics variable

Let $\overline{S}=\left\{s_{\theta }\left|s_0\le s_{\theta }\le s_{l-1},\theta \in \left[0,l-1\right]\right.\right\}$ be the linguistic set in its continuous form S.s_θ, s_ρ, s_μ, s_ν are four linguistic terms, and s₀ ≤ s_θ ≤ s_ρ ≤ s_μ ≤ s_ν ≤ s_l − 1 if0 ≤ θ ≤ ρ ≤ μ ≤ ν ≤ l − 1, then the TLV is defined as $\stackrel{⌢}{s}=\left[s_{\theta },s_{\rho },s_{\mu },s_{\nu }\right]$, and $\stackrel{⌢}{s}$ represents a set of the TLVs.

Definition 26

Linguistic neutrosophic set (Ye, 2015)

The concept of SVNLS N in ξ can be defined in the following form

$N_{\text{SVNLS}}=\left\{〈k,\left[s_{\theta \left(x\right)},s_{\rho \left(x\right)}\right],\left(T_N\left(k\right),I_N\left(k\right),F_N\left(k\right)\right)〉\left|k\in \xi \right.\right\},$

where $s_{\theta \left(x\right)},s_{\rho \left(x\right)}\in \stackrel{⌢}{s}$, T_N(k) ⊆ [0, 1]，I_N(k) ⊆ [0, 1], F_N(k) ⊆ [0, 1] with the condition 0 ≤ T_N(k) + I_N(k) + F_N(k) ≤ 3, for any x ∈ ξ. [s_θ(x), s_ρ(x)] is an uncertain linguistic term. The functions T_N(k), I_N(k), and F_N(k) express, respectively, the TM degree, the IM degree, and the FM degree of the element x in ξ belonging to the linguistic term[s_θ(x), s_ρ(x)].

Definition 27

(Ye, 2015): Operations on linguistics neutrosophic set

For any given two SVNULVNs ${\tilde{\alpha }}_1=〈\left[s_{\theta \left({\tilde{\alpha }}_1\right)},s_{\rho \left({\tilde{\alpha }}_1\right)}\right],\left(T\left({\tilde{\alpha }}_1\right),I\left({\tilde{\alpha }}_1\right),F\left({\tilde{\alpha }}_1\right)\right)〉$, ${\tilde{\alpha }}_2=〈\left[s_{\theta \left({\tilde{\alpha }}_2\right)},s_{\rho \left({\tilde{\alpha }}_2\right)}\right],\left(T\left({\tilde{\alpha }}_2\right),I\left({\tilde{\alpha }}_2\right),F\left({\tilde{\alpha }}_2\right)\right)〉$, λ > 0 is a constant, and their operational rules are defined as follows:

${\tilde{\alpha }}_1\oplus {\tilde{\alpha }}_2=〈\begin{array}{l}\left[s_{\theta \left({\tilde{\alpha }}_1\right)+\theta \left({\tilde{\alpha }}_2\right)},s_{\rho \left({\tilde{\alpha }}_1\right)+\rho \left({\tilde{\alpha }}_2\right)}\right],\left(T\left({\tilde{\alpha }}_1\right)+T\left({\tilde{\alpha }}_2\right)-T\left({\tilde{\alpha }}_1\right)T\left({\tilde{\alpha }}_2\right)\right),\\ I\left({\tilde{\alpha }}_1\right)I\left({\tilde{\alpha }}_2\right),F\left({\tilde{\alpha }}_1\right)F\left({\tilde{\alpha }}_2\right)\end{array}〉,$

${\tilde{\alpha }}_1\otimes {\tilde{\alpha }}_2=〈\begin{array}{l}\left[s_{\theta \left({\tilde{\alpha }}_1\right)\times \theta \left({\tilde{\alpha }}_2\right)},s_{\rho \left({\tilde{\alpha }}_1\right)\times \rho \left({\tilde{\alpha }}_2\right)}\right],\left(T\left({\tilde{\alpha }}_1\right)T\left({\tilde{\alpha }}_2\right)\right),\\ \left(I\left({\tilde{\alpha }}_1\right)+I\left({\tilde{\alpha }}_2\right)-I\left({\tilde{\alpha }}_1\right)I\left({\tilde{\alpha }}_2\right)\right),\left(F\left({\tilde{\alpha }}_1\right)+F\left({\tilde{\alpha }}_2\right)-F\left({\tilde{\alpha }}_1\right)F\left({\tilde{\alpha }}_2\right)\right)\end{array}〉,$

$\lambda {\tilde{\alpha }}_1=〈\left[s_{\mathit{\lambda \theta }\left({\tilde{\alpha }}_1\right)},s_{\mathit{\lambda \rho }\left({\tilde{\alpha }}_1\right)}\right],\left(1-{\left(1-T\left({\tilde{\alpha }}_1\right)\right)}^{\lambda },{\left(I\left({\tilde{\alpha }}_1\right)\right)}^{\lambda },{\left(F\left({\tilde{\alpha }}_1\right)\right)}^{\lambda }\right)〉,$

${{\tilde{\alpha }}_1}^{\lambda }=〈\left[s_{{\theta }^{\lambda }\left({\tilde{\alpha }}_1\right)},s_{{\rho }^{\lambda }\left({\tilde{\alpha }}_1\right)}\right],\left({\left(T\left({\tilde{\alpha }}_1\right)\right)}^{\lambda },\left(1-{\left(1-I\left({\tilde{\alpha }}_1\right)\right)}^{\lambda }\right),\left(1-{\left(1-F\left({\tilde{\alpha }}_1\right)\right)}^{\lambda }\right)\right)〉.$

Definition 28

Bipolar neutrosophic set (Deli et al., 2015)

Suppose ξ is a series of points (objects) with basic elements in ξ presented by x. A bipolar neutrosophic set N (BNS) in ξ is defined as in the following form

$N=\left\{〈k,\left(T_N^{+}\left(k\right),I_N^{+}\left(k\right),F_N^{+}\left(k\right),T_N^{-}\left(k\right),I_N^{-}\left(k\right),F_N^{-}\left(k\right)\right)〉:k\in \xi \right\},$

where T^P, I_N⁺, F^P : ξ → [1, 0] and T^P, I_N⁻, F^P : ξ → [− 1, 0] are the PMFs NMFs. We call T_N⁺(k), I_N⁺(k), F_N⁺(k) the TMF, IMF, and FMF of an element k ∈ s. The NMF degree T_N⁻(k), I_N⁻(k), F_N⁻(k) represents the TMF, IMF, and FMF of an element x ∈ ξ. If the set has only one element, then we call N a BNN and define it by

$N_{\mathit{BNS}}=〈T_A^{+},I_A^{+},F_A^{+},T_A^{-},I_A^{-},F_A^{-}〉.$

In case the uncertainty in the dataset fluctuates at given intervals of time, then the complex NS is defined.

Definition 29

Complex neutrosophic set (Ali & Smarandache, 2015)

Suppose that ξ is a set of some points denoted as k. A complex neutrosophic set (CNS) is defined on the set ξ through the definition of three membership functions, T_N(k), I_N(k), F_N(k), and we call them the TMF, the IMF, and the FMF, respectively. Here, we allocate a complex-valued grade of T_N(k), I_N(k), and F_N(k) in N for any k ∈ ξ. All the values T_N(k), I_N(k), and F_N(k) and the sum values of these three functions are all in the complex plane unit circle, and their forms are as follows.

$\begin{array}{l}{T}_N\left(k\right)={\rho }_N\left(k\right)\cdot e^{j{\mu }_N\left(k\right)},\\ I_N\left(k\right)=q_N\left(k\right)\cdot e^{{\mathit{jv}}_N\left(k\right)},\text{and}\\ F_N\left(k\right)=r_N\left(k\right)\cdot e^{j{\omega }_N\left(k\right)}\end{array}$

Here ρ_N(k), q_N(k), r_N(k) and μ_N(k), v_N(k), ω_N(k) are real values, respectively, and ρ_N(k),q_N(k),r_N(k) ∈ F_A(x)∈[0, 1] such that

${}^{-}0\le T_A\left(x\right)+I_A\left(x\right)+F_A\left(x\right)\le 3^{+}.$

The CNSN can be rewritten in a set form as

$N_{\mathrm{CNS}}=\left\{\left(k,T_N\left(k\right)=a_T,I_N\left(k\right)=a_I,F_N\left(k\right)=a_F\right):k\in \xi \right\},$

where

$\begin{array}{l}{T}_N:X\to \left\{a_T:a_T\in C,\left|a_T\right|\le 1\right\}\\ I_N:\xi \to \left\{a_I:a_I\in C,\left|a_I\right|\le 1\right\}\\ F_N:\xi \to \left\{a_F:a_F\in C,\left|a_F\right|\le 1\right\}\\ \left|T_N\left(k\right)+I_N\left(k\right)+F_N\left(k\right)\right|\le 3.\end{array}$

The complex grade of TMF is characterized by a truth amplitude term ρ_N(k) and a truth phase term μ_N(k).

In addition, the complex degree of IMF is defined as an indeterminate amplitude term, and the complex grade of IMF is defined as an indeterminate term q_N(k) and an indterminate phase term v_N(x).

The complex grade of the FMF is defined by the false amplitude term r_N(k) and a false phase term ω_N(x), respectively. It should be noted that the truth amplitude term ρ_N(x) is equal to | T_N(k)|, the amplitude of T_N(k). The indeterminate amplitude term q_N(x) is equal to | I_N(k)| and the false amplitude term r_N(k) is equal to | F_N(k)|.

Definition 30

Union and intersection of CNSs

Suppose N and M are two CNSs in ξ, where N = {(k, T_N(k), I_N(k), F_N(k)) : k ∈ ξ} andM = {(k, T_M(k), I_M(k), F_M(k)) : k ∈ ξ}.

Then the union of N and M is termed as N ∪_CNSM and we denote it as

$N{\cup }_{\mathrm{CNS}}M=\left\{\left(k,T_{N\cup M}\left(k\right),I_{N\cup M}\left(k\right),F_{N\cup M}\left(k\right)\right):k\in \xi \right\},$

where the TMF nT_N ∪ M(k), the IMF I_N ∪ M(k), and the FMF F_N ∪ M(k) are denoted by

$\begin{array}{l}{T}_{N\cup M}\left(k\right)=\left[\left(p_N\left(k\right)\vee p_M\left(k\right)\right)\right]\cdot e^{j\cdot {\mu }_{T_{N\cup M}}\left(k\right)},\\ I_{N\cup \mathit{MB}}\left(k\right)=\left[\left(q_N\left(k\right)\wedge q_M\left(k\right)\right)\right]\cdot e^{j\cdot {\nu }_{I_{N\cup M}}\left(k\right)},\\ F_{N\cup M}\left(k\right)=\left[\left(r_N\left(k\right)\wedge r_M\left(k\right)\right)\right]\cdot e^{j\cdot {\omega }_{F_{N\cup M}}\left(k\right)},\end{array}$

where ∨ and ∧ represent the operators max and min, respectively.

Definition 31

Bipolar complex neutrosophic set (Broumi, Bakali, et al., 2019; Broumi, Talea, et al., 2019)

Suppose ξ is a set of some points with basic elements in ξ presented by x. A bipolar complex neutrosophic set A (BCNS) in ξ is defined in the following form

$N=\left\{〈k,T_1^{+}{e}^{{\mathit{iT}}_2^{+}},I_1^{+}{e}^{{\mathit{iI}}_2^{+}},F_1^{+}{e}^{{\mathit{iF}}_2^{+}},T_1^{-}{e}^{{\mathit{iT}}_2^{-}},I_1^{-}{e}^{{\mathit{iI}}_2^{-}},F_1^{-}{e}^{{\mathit{iF}}_2^{-}}〉:k\in \xi \right\},$

where T^P, I₁⁺, F^P : X → [1, 0] and T^P, I₁⁻, F^P : X → [− 1, 0]. The positive membership degree T₁⁺(k), I₁⁺(k), F₁⁺(k) represents the TM, the IM, and the MF of an element x ∈ ξ corresponding to the given property, whereas the negative membership degree T₁⁻(k), I₁⁻(k), F₁⁻(k) represents the TM, the IM, and the FM of an element k ∈ ξ to some implicit counterproperty. A BCNN can be abbreviated as follows:

$N_{\mathit{BCNS}}=〈T_1^{+}{e}^{{\mathit{iT}}_2^{+}},I_1^{+}{e}^{{\mathit{iI}}_2^{+}},F_1^{+}{e}^{{\mathit{iF}}_2^{+}},T_1^{-}{e}^{{\mathit{iT}}_2^{-}},I_1^{-}{e}^{{\mathit{iI}}_2^{-}},F_1^{-}{e}^{{\mathit{iF}}_2^{-}}〉.$

Definition 32

Containment operations on bipolar neutrosophic sets (Broumi, Bakali, et al., 2019; Broumi, Talea, et al., 2019)

Suppose there are two BCNSs BCN₁ =(T₁⁺ e^iT₂+,I₁⁺ e^iI₂+,F₁⁺ e^iF₂+,T₁⁻ e^iT₂−,I₁⁻ e^iI₂−,F₁⁻ e^iF₂−) and BCN₂ =(T₃⁺ e^iT₄+,I₃⁺ e^iI₄+,F₃⁺ e^iF₄+,T₃⁻ e^iT₄−,I₃⁻ e^iI₄−, F₃⁻ e^iF₄−). BCN₁ is contained in the other BCN₂ detailed as BCN₁ ⊆ BCN₂ if

$\begin{array}{c}{T_1}_{{\mathrm{BCN}}_1}^{+}\left(x\right)\le {T_1}_{{\mathrm{BCN}}_2}^{+}\left(x\right),{I_1}_{{\mathrm{BCN}}_1}^{+}\left(x\right)\ge {I_1}_{{\mathrm{BCN}}_2}^{+}\left(x\right),{F_1}_{{\mathrm{BCN}}_1}^{+}\left(x\right)\ge {F_1}_{{\mathrm{BCN}}_2}^{+}\left(x\right)\text{and}\\ {T_2}_{{\mathrm{BCN}}_1}^{+}\left(x\right)\le {T_2}_{{\mathrm{BCN}}_2}^{+}\left(x\right),{I_2}_{{\mathrm{BCN}}_1}^{+}\left(x\right)\ge {I_2}_{{\mathrm{BCN}}_2}^{+}\left(x\right),{F_2}_{{\mathrm{BCN}}_1}^{+}\left(x\right)\ge {F_2}_{{\mathrm{BCN}}_2}^{+}\left(x\right)\end{array}$

and

${T_3}_{{\mathrm{BCN}}_1}^{-}\left(x\right)\ge {T_3}_{{\mathrm{BCN}}_2}^{-}\left(x\right),{I_3}_{{\mathrm{BCN}}_1}^{-}\left(x\right)\le {I_3}_{{\mathrm{BCN}}_2}^{-}\left(x\right),{F_3}_{{\mathrm{BCN}}_1}^{-}\left(x\right)\le {F_3}_{{\mathrm{BCN}}_2}^{-}\left(x\right)$

and

${T_4}_{{\mathrm{BCN}}_1}^{-}\left(x\right)\ge {T_4}_{{\mathrm{BCN}}_2}^{-}\left(x\right),{I_4}_{{\mathrm{BCN}}_1}^{-}\left(x\right)\le {I_4}_{{\mathrm{BCN}}_2}^{-}\left(x\right),{F_4}_{{\mathrm{BCN}}_1}^{-}\left(x\right)\le {F_4}_{{\mathrm{BCN}}_2}^{-}\left(x\right)\forall x\in \xi .$

Definition 33

Union and intersection of BCNSs

Suppose N and M are two BCNSs in ξ, where

$N=\left(T_1^{+}{e}^{{\mathit{iT}}_2^{+}},I_1^{+}{e}^{{\mathit{iI}}_2^{+}},F_1^{+}{e}^{{\mathit{iF}}_2^{+}},T_1^{-}{e}^{{\mathit{iT}}_2^{-}},I_1^{-}{e}^{{\mathit{iI}}_2^{-}},F_1^{-}{e}^{{\mathit{iF}}_2^{-}}\right)\text{and}M=\left(T_3^{+}{e}^{{\mathit{iT}}_4^{+}},I_3^{+}{e}^{{\mathit{iI}}_4^{+}},F_3^{+}{e}^{{\mathit{iF}}_4^{+}},T_3^{-}{e}^{{\mathit{iT}}_4^{-}},I_3^{-}{e}^{{\mathit{iI}}_4^{-}},F_3^{-}{e}^{{\mathit{iF}}_4^{-}}\right)$

Then, the union of N and M is termed as N ∪_BCNM

$\begin{array}{c}N{\cup }_{\mathit{BCN}}M=\left\{\right(k,T_{+}^{N\cup M}\left(x\right),I_{+}^{N\cup M}\left(x\right),F_{+}^{N\cup M}\left(k\right),T_{-}^{N\cup M}\left(k\right)I_{-}^{N\cup M}\left(k\right)\\ F_{-}^{N\cup M}\left(k\right)):k\in \xi \}.\end{array}$

where PTMF ${T^{+}}_{N\cup M}\left(k\right)$, PIMF ${I^{+}}_{N\cup M}\left(k\right)$ and PFMF ${F^{+}}_{N\cup M}\left(k\right)$, NTMF ${T^{-}}_{N\cup M}\left(k\right)$, NIMF ${I^{-}}_{N\cup M}\left(k\right)$ and NFMF ${F^{-}}_{N\cup M}\left(k\right)$ is termed by

$T_{A\cup B}^{+}\left(k\right)=\left(T_1^{+}\vee T_3^{+}\right)e^{i\left(T_2^{+}\cup T_4^{+}\right)},T_{A\cup B}^{-}\left(k\right)=\left(T_1^{-}\wedge T_3^{-}\right)e^{i\left(T_2^{-}\cup T_4^{-}\right)}{I}_{A\cup B}^{+}\left(k\right)=\left(I_1^{+}\wedge I_3^{+}\right)e^{i\left(I_2^{+}\cup I_4^{+}\right)},T_{A\cup B}^{-}\left(k\right)=\left(I_1^{-}\vee I_3^{-}\right)e^{i\left(I_2^{-}\cup I_4^{-}\right)}$

$F_{A\cup B}^{+}\left(k\right)=\left(F_1^{+}\wedge F_3^{+}\right)e^{i\left(F_2^{+}\cup F_4^{+}\right)},F_{A\cup B}^{-}\left(k\right)=\left(F_1^{-}\vee F_3^{-}\right)e^{i\left(F_2^{-}\cup F_4^{-}\right)}$

Definition 34

(Ali et al., 2018)

Suppose ξ is a series of points (objects) basic elements in ξ presented by k. An ICNS is defined on ξ, which is characterized through a function ITMF ${\tilde{T}}_N\left(k\right)=\left[T_N^L\left(k\right),T_N^U\left(k\right)\right],$ an interval IMF ${\tilde{I}}_N\left(k\right),$ and an interval FMF ${\tilde{F}}_N\left(k\right)$ that assigns a complex-valued membership grade to ${\tilde{T}}_N\left(k\right),{\tilde{I}}_N\left(k\right),{\tilde{F}}_N\left(k\right)$ for any k ∈ ξ. The values of ${\tilde{T}}_N\left(k\right),{\tilde{I}}_N\left(k\right),{\tilde{F}}_N\left(k\right)$ and their sum take some values within a complex plane unit circle. The forms of the functions are below:

${\tilde{T}}_N\left(k\right)=\left[p_N^L\left(k\right),p_N^U\left(k\right)\right]\cdot e^{i\left[{\mu }_N^L\left(k\right),{\mu }_N^U\left(k\right)\right]},$

${\tilde{I}}_N\left(k\right)=\left[q_N^L\left(k\right),q_N^U\left(k\right)\right]\cdot e^{i\left[v_N^L\left(k\right),v_N^U\left(k\right)\right]},\text{and}$

${\tilde{F}}_N\left(k\right)=\left[r_N^L\left(x\right),r_N^U\left(k\right)\right]\cdot e^{i\left[{\omega }_N^L\left(k\right),{\omega }_N^U\left(k\right)\right]}.$

All the amplitude and phase terms are real valued. And p_N^L(k), p_N^U(k), q_N^L(k), q_N^U(k), r_N^L(k) and r_N^U(k) ∈ [0, 1], whereas μ_N(k), ν_N(k), ω_N(k) ∈ (0, 2π], such that the condition

$0\le p_N^U\left(k\right)+q_N^U\left(k\right)+r_N^U\left(k\right)\le 3$

is satisfied. T_A(x) an ICNS $\tilde{A}$ can thus be termed in the defined form as:

$\tilde{N}=\left\{〈k,T_N\left(k\right)=a_T,I_N\left(k\right)=a_I,F_N\left(k\right)=a_F〉:k\in \xi \right\},$

where T_N : ξ. → {a_T : a_T ∈ C, | a_T | ≤ 1}, I_N : ξ. → {a_I : a_I ∈ C, | a_I | ≤ 1}, F_N : ξ. → {a_F : a_F ∈ C, | a_F | ≤ 1}, and also | T_N^U(k) + I_N^U(k) + F_N^U(k)| ≤ 3.

Definition 35

(Ali et al., 2018)

For two ICNS N and M in ξ, the union of N and M detailed as N ∪_ICNSM is abbreviated as:

$N{\cup }_{\text{ICNS}}M=\left\{\left(k,{\tilde{T}}_{N\cup M}\left(k\right),{\tilde{I}}_{N\cup M}\left(k\right),{\tilde{F}}_{N\cup M}\left(k\right)\right):k\in \xi \right\},$

where ${\tilde{T}}_{N\cup M}\left(k\right),{\tilde{I}}_{N\cup M}\left(k\right),{\tilde{F}}_{N\cup M}\left(k\right)$ are given by

$T_{N\cup M}^L\left(k\right)=\left[\left(p_N^L\left(k\right)\vee p_M^L\left(k\right)\right)\right]\cdot e^{j.{\mu }_{T_{A\cup B}}^L\left(k\right)},T_{N\cup M}^U\left(k\right)=\left[\left(p_N^U\left(k\right)\vee p_M^U\left(k\right)\right)\right]\cdot e^{j.{\mu }_{T_{A\cup B}}^U\left(k\right)}$