$$\begin{aligned} u\_{L,n}^1(\alpha) &= \left(\mu\_L^1(\alpha)\,\mu\_R^1(\alpha)\right)^{\frac{\mu}{2}} u\_{L,0}^1(\alpha) \\ u\_{R,n}^1(\alpha) &= \left(\mu\_L^1(\alpha)\,\mu\_R^1(\alpha)\right)^{\frac{\mu}{2}} u\_{R,0}^1(\alpha) \\ u\_{L,n}^2(\beta) &= \left(\mu\_L^2(\beta)\,\mu\_R^2(\beta)\right)^{\frac{\mu}{2}} u\_{L,0}^2(\beta) \\ u\_{R,n}^2(\beta) &= \left(\mu\_L^2(\beta)\,\mu\_R^2(\beta)\right)^{\frac{\mu}{2}} u\_{R,0}^3(\beta) \\ u\_{L,n}^3(\gamma) &= \left(\mu\_L^3(\gamma)\,\mu\_R^3(\gamma)\right)^{\frac{\mu}{2}} u\_{L,0}^3(\gamma) \\ u\_{R,n}^3(\gamma) &= \left(\mu\_L^3(\gamma)\,\mu\_R^3(\gamma)\right)^{\frac{\mu}{2}} u\_{R,0}^3(\gamma) \end{aligned} \tag{34}$$

The above equations show that the solution for *n* is even only. When *n* is odd, the solutions, which follow from Equation (32), are as follows:

$$\begin{array}{l} u\_{L,n}^{1}(\alpha) = -\left(\mu\_{L}^{1}(\alpha)\right)^{\frac{n-1}{2}} \left(\mu\_{R}^{1}(\alpha)\right)^{\frac{n+1}{2}} u\_{L,0}^{1}(\alpha) \\\ u\_{R,n}^{1}(\alpha) = -\left(\mu\_{L}^{1}(\alpha)\right)^{\frac{n+1}{2}} \left(\mu\_{R}^{1}(\alpha)\right)^{\frac{n-1}{2}} u\_{R,0}^{1}(\alpha) \\\ u\_{L,n}^{2}(\beta) = -\left(\mu\_{L}^{2}(\beta)\right)^{\frac{n-1}{2}} \left(\mu\_{R}^{2}(\beta)\right)^{\frac{n+1}{2}} u\_{L,0}^{2}(\beta) \\\ u\_{R,n}^{2}(\beta) = -\left(\mu\_{L}^{2}(\beta)\right)^{\frac{n+1}{2}} \left(\mu\_{R}^{2}(\beta)\right)^{\frac{n-1}{2}} u\_{R,0}^{2}(\beta) \\\ u\_{L,n}^{3}(\gamma) = -\left(\mu\_{L}^{3}(\gamma)\right)^{\frac{n-1}{2}} \left(\mu\_{R}^{3}(\gamma)\right)^{\frac{n+1}{2}} u\_{L,0}^{3}(\gamma) \\\ u\_{R,n}^{3}(\gamma) = -\left(\mu\_{L}^{3}(\gamma)\right)^{\frac{n+1}{2}} \left(\mu\_{R}^{3}(\gamma)\right)^{\frac{n-1}{2}} u\_{R,0}^{3}(\gamma) \end{array} \tag{35}$$

*4.2. Solution of Homogeneous Di*ff*erence Equation of Type II*

#### 4.2.1. The Solution When *a* = 1 and the Initial Condition *u*0 Is a Neutrosophic Number

Taking the (α, β, γ)-cut of Equation (8), we have the following:

$$\begin{array}{l} u\_{L,n+1}^1(\boldsymbol{a}) = u\_{R,n}^1(\boldsymbol{a}) \\ u\_{R,n+1}^1(\boldsymbol{\alpha}) = u\_{L,n}^1(\boldsymbol{\alpha}) \\ u\_{L,n+1}^2(\boldsymbol{\beta}) = u\_{R,n}^2(\boldsymbol{\beta}) \\ u\_{R,n+1}^2(\boldsymbol{\beta}) = u\_{L,n}^2(\boldsymbol{\beta}) \\ u\_{L,n+1}^3(\boldsymbol{\alpha}) = u\_{R,n}^3(\boldsymbol{\alpha}) \\ u\_{R,n+1}^3(\boldsymbol{\alpha}) = u\_{L,n}^3(\boldsymbol{\alpha}) \end{array} \tag{36}$$

In the matrix form, the first pairs of Equation (36) can be written as:

$$
\begin{pmatrix} u\_{L,n+1}^1(\alpha) \\ u\_{R,n+1}^1(\alpha) \end{pmatrix} = \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix} \begin{pmatrix} u\_{L,n}^1(\alpha) \\ u\_{R,n}^1(\alpha) \end{pmatrix} \tag{37}
$$

The solution of Equation (37) is, when *n* is even:

$$\begin{cases} \boldsymbol{u}^1\_{L,n}(\alpha) = \boldsymbol{u}^1\_{L,0}(\alpha) \\ \boldsymbol{u}^1\_{R,n}(\alpha) = \boldsymbol{u}^1\_{R,0}(\alpha) \end{cases} \tag{38}$$

When *n* is odd, the solutions are:

$$\begin{cases} \boldsymbol{u}^1\_{L,n}(\boldsymbol{\alpha}) = \boldsymbol{u}^1\_{\mathcal{R},0}(\boldsymbol{\alpha}) \\ \boldsymbol{u}^1\_{\mathcal{R},n}(\boldsymbol{\alpha}) = \boldsymbol{u}^1\_{L,0}(\boldsymbol{\alpha}) \end{cases} \tag{39}$$

For both cases, when either *n* is even or odd, *u* 1 *L*,*n* (α) and *u* 1 *R*,*n* (α) leads to a convergent solution. In a similar way, solutions of remaining equations are as follows: when *n* is even:

$$\begin{cases} \begin{array}{l} \mathfrak{u}^{1}\_{\mathcal{L},\mathfrak{u}}(\beta) = \mathfrak{u}^{1}\_{\mathcal{L},0}(\beta) \\ \mathfrak{u}^{1}\_{\mathcal{R},\mathfrak{u}}(\beta) = \mathfrak{u}^{1}\_{\mathcal{R},0}(\beta) \\ \mathfrak{u}^{1}\_{\mathcal{L},\mathfrak{u}}(\mathcal{\gamma}) = \mathfrak{u}^{1}\_{\mathcal{L},0}(\mathcal{\gamma}) \\ \mathfrak{u}^{1}\_{\mathcal{R},\mathfrak{u}}(\mathcal{\gamma}) = \mathfrak{u}^{1}\_{\mathcal{R},0}(\mathcal{\gamma}) \end{array} \tag{40}$$

When *n* is odd:

$$\begin{cases} \begin{array}{l} \boldsymbol{u}^{1}\_{\mathrm{L},\boldsymbol{n}}(\boldsymbol{\beta}) = \boldsymbol{u}^{1}\_{\mathrm{R},0}(\boldsymbol{\beta})\\ \boldsymbol{u}^{1}\_{\mathrm{R},\boldsymbol{n}}(\boldsymbol{\beta}) = \boldsymbol{u}^{1}\_{\mathrm{L},0}(\boldsymbol{\beta})\\ \boldsymbol{u}^{1}\_{\mathrm{L},\boldsymbol{n}}(\boldsymbol{\gamma}) = \boldsymbol{u}^{1}\_{\mathrm{R},0}(\boldsymbol{\gamma})\\ \boldsymbol{u}^{1}\_{\mathrm{R},\boldsymbol{n}}(\boldsymbol{\gamma}) = \boldsymbol{u}^{1}\_{\mathrm{L},0}(\boldsymbol{\gamma}) \end{array} \tag{41}$$

4.2.2. The Solution When *a* > 0, a Real Valued Number, and the Initial Condition *u*0 Is a Neutrosophic Number

Taking the (α, β, γ)-cut of (8), we get the following equations:

$$\begin{array}{l} u\_{L,n+1}^1(\alpha) - a u\_{R,n}^1(\alpha) = 0 \\ u\_{R,n+1}^1(\alpha) - a u\_{L,n}^1(\alpha) = 0 \\ u\_{L,n+1}^1(\beta) - a u\_{R,n}^1(\beta) = 0 \\ u\_{R,n+1}^1(\beta) - a u\_{L,n}^1(\beta) = 0 \\ u\_{L,n+1}^1(\gamma) - a u\_{R,n}^1(\gamma) = 0 \\ u\_{R,n+1}^1(\gamma) - a u\_{L,n}^1(\gamma) = 0 \end{array} \tag{42}$$

In the matrix form, the first pair of Equation (42) can be written as:

$$
\begin{pmatrix} u\_{L,n+1}^1(\alpha) \\ u\_{R,n+1}^1(\alpha) \end{pmatrix} = \begin{pmatrix} 0 & a \\ a & 0 \end{pmatrix} \begin{pmatrix} u\_{L,n}^1(\alpha) \\ u\_{R,n}^1(\alpha) \end{pmatrix} \tag{43}
$$

The solutions of (43) are, when *n* is even:

$$\begin{cases} u\_{L,n}^1(\alpha) = a^n u\_{L,0}^1(\alpha) \\ u\_{R,n}^1(\alpha) = a^n u\_{R,0}^1(\alpha) \end{cases} \tag{44}$$

The solutions of (44) and (45) are, when *n* is odd:

$$\begin{cases} u\_{L,\boldsymbol{n}}^1(\boldsymbol{a}) = a^\boldsymbol{\boldsymbol{n}} u\_{R,0}^1(\boldsymbol{a}) \\ u\_{R,\boldsymbol{n}}^1(\boldsymbol{a}) = a^\boldsymbol{\boldsymbol{n}} u\_{L,0}^1(\boldsymbol{a}) \end{cases} \tag{45}$$

In a similar way, the solutions of the remaining Equation (42) are as follows: When *n* is even:

$$\begin{array}{l} u\_{L,\boldsymbol{n}}^{1}(\boldsymbol{\beta}) = a^{\boldsymbol{n}} u\_{L,0}^{1}(\boldsymbol{\beta}) \\ u\_{R,\boldsymbol{n}}^{1}(\boldsymbol{\beta}) = a^{\boldsymbol{n}} u\_{R,0}^{1}(\boldsymbol{\beta}) \\ u\_{L,\boldsymbol{n}}^{1}(\boldsymbol{\gamma}) = a^{\boldsymbol{n}} u\_{L,0}^{1}(\boldsymbol{\gamma}) \\ u\_{R,\boldsymbol{n}}^{1}(\boldsymbol{\gamma}) = a^{\boldsymbol{n}} u\_{R,0}^{1}(\boldsymbol{\gamma}) \end{array} \tag{46}$$

When *n* is odd:

$$\begin{array}{l} u\_{L,n}^{1}(\beta) = a^{n} u\_{R,0}^{1}(\beta) \\ u\_{R,n}^{1}(\beta) = a^{n} u\_{L,0}^{1}(\beta) \\ u\_{L,n}^{1}(\gamma) = a^{n} u\_{R,0}^{1}(\gamma) \\ u\_{R,n}^{1}(\gamma) = a^{n} u\_{L,0}^{1}(\gamma) \end{array} \tag{47}$$

4.2.3. The Solution When *a* < 0 and When the Initial Condition *u*0 Is a Neutrosophic Number

Let *a* = −*m* , *m* > 0, a real valued number.

From Equation (8), after taking the (α, β, γ)-cut, we have the following sets of equations:

$$\begin{array}{l} u\_{L,n+1}^1(\alpha) + mu\_{L,n}^1(\alpha) = 0\\ u\_{R,n+1}^1(\alpha) + mu\_{R,n}^1(\alpha) = 0\\ u\_{L,n+1}^1(\beta) + mu\_{L,n}^1(\beta) = 0\\ u\_{R,n+1}^1(\beta) + mu\_{R,n}^1(\beta) = 0\\ u\_{L,n+1}^1(\gamma) + mu\_{L,n}^1(\gamma) = 0\\ u\_{R,n+1}^1(\gamma) + mu\_{R,n}^1(\gamma) = 0 \end{array} \tag{48}$$

Solving the above equations, we get:

$$\begin{cases} \begin{array}{l} u\_{L,n}^{1}(\alpha) = (-m)^{n} u\_{L,0}^{1}(\alpha) \\ u\_{R,n}^{1}(\alpha) = (-m)^{n} u\_{R,0}^{1}(\alpha) \\ u\_{L,n}^{2}(\beta) = (-m)^{n} u\_{L,0}^{2}(\beta) \\ u\_{R,n}^{2}(\beta) = (-m)^{n} u\_{R,0}^{2}(\beta) \\ u\_{L,n}^{3}(\gamma) = (-m)^{n} u\_{L,0}^{3}(\gamma) \\ u\_{R,n}^{3}(\gamma) = (-m)^{n} u\_{R,0}^{3}(\gamma) \end{array} \tag{49}$$

4.2.4. The Solution When *a* > 0 Is a Positive Neutrosophic Number and the Initial Condition *u*0 Is Nota Neutrosophic Number

Let [e*a*] (α, β, γ) = h *a* 1 *L* (α), *a* 1 *R* (α); *a* 2 *L* (β), *a* 2 *R* (β); *a* 3 *L* (γ), *a* 3 *R* (γ) i **,**∀ α , β, γ ∈ [0, 1], and 0 ≤ α + β + γ ≤ 3. Taking the (α, β, γ)-cut of Equation (8), we have the following equation:

$$\begin{array}{l} u\_{L,n+1}^1(\alpha) = a\_L^1(\alpha) u\_{R,n}^1(\alpha) \\ u\_{R,n+1}^1(\alpha) = a\_R^1(\alpha) u\_{L,n}^1(\alpha) \\ u\_{L,n+1}^2(\beta) = a\_L^2(\beta) u\_{R,n}^2(\beta) \\ u\_{R,n+1}^2(\beta) = a\_R^2(\beta) u\_{L,n}^2(\beta) \\ u\_{L,n+1}^3(\gamma) = a\_L^3(\gamma) u\_{R,n}^3(\gamma) \\ u\_{R,n+1}^3(\gamma) = a\_R^3(\gamma) u\_{L,n}^3(\gamma) \end{array} \tag{50}$$

In the matrix form, among the above equations, the first pair of Equation (50) can be written as:

$$
\begin{pmatrix} u\_{L,n+1}^1(\alpha) \\ u\_{R,n+1}^1(\alpha) \end{pmatrix} = \begin{pmatrix} 0 & a\_R^1(\alpha) \\ a\_L^1(\alpha) & 0 \end{pmatrix} \begin{pmatrix} u\_{L,n}^1(\alpha) \\ u\_{R,n}^1(\alpha) \end{pmatrix} \tag{51}
$$

The solution of Equation (51), when *n* is even:

$$\begin{array}{l} u\_{L,n}^1(\alpha) = \left( a\_L^1(\alpha) \, a\_R^1(\alpha) \right)^{\frac{\mu}{2}} u\_0\\ u\_{R,n}^1(\alpha) = \left( a\_L^1(\alpha) \, a\_R^1(\alpha) \right)^{\frac{\mu}{2}} u\_0\\ u\_{L,n}^2(\beta) = \left( a\_L^2(\beta) \, a\_R^2(\beta) \right)^{\frac{\mu}{2}} u\_0\\ u\_{R,n}^2(\beta) = \left( a\_L^2(\beta) \, a\_R^2(\beta) \right)^{\frac{\mu}{2}} u\_0\\ u\_{L,n}^3(\gamma) = \left( a\_L^3(\gamma) \, a\_R^3(\gamma) \right)^{\frac{\mu}{2}} u\_0\\ u\_{R,n}^3(\gamma) = \left( a\_L^3(\gamma) \, a\_R^3(\gamma) \right)^{\frac{\mu}{2}} u\_0 \end{array} \tag{52}$$

When *n* is odd:

$$\begin{array}{l} u\_{\mathcal{L},n}^{1}(\alpha) = \left(a\_{\mathcal{L}}^{1}(\alpha)\right)^{\frac{n-1}{2}} \left(a\_{\mathcal{R}}^{1}(\alpha)\right)^{\frac{n+1}{2}} u\_{0} \\\ u\_{\mathcal{R},n}^{1}(\alpha) = \left(a\_{\mathcal{L}}^{1}(\alpha)\right)^{\frac{n+1}{2}} \left(a\_{\mathcal{R}}^{1}(\alpha)\right)^{\frac{n-1}{2}} u\_{0} \\\ u\_{\mathcal{L},n}^{2}(\beta) = \left(a\_{\mathcal{L}}^{2}(\beta)\right)^{\frac{n-1}{2}} \left(a\_{\mathcal{R}}^{2}(\beta)\right)^{\frac{n+1}{2}} u\_{0} \\\ u\_{\mathcal{R},n}^{2}(\beta) = \left(a\_{\mathcal{L}}^{2}(\beta)\right)^{\frac{n+1}{2}} \left(a\_{\mathcal{R}}^{2}(\beta)\right)^{\frac{n-1}{2}} u\_{0} \\\ u\_{\mathcal{L},n}^{3}(\gamma) = \left(a\_{\mathcal{L}}^{3}(\gamma)\right)^{\frac{n-1}{2}} \left(a\_{\mathcal{R}}^{3}(\gamma)\right)^{\frac{n+1}{2}} u\_{0} \\\ u\_{\mathcal{R},n}^{3}(\gamma) = \left(a\_{\mathcal{L}}^{3}(\gamma)\right)^{\frac{n+1}{2}} \left(a\_{\mathcal{R}}^{3}(\gamma)\right)^{\frac{n-1}{2}} u\_{0} \\\end{array} \tag{53}$$

4.2.5. The Solution When a < 0 Is a Neutrosophic Number and when the Initial Condition *u*0 Is a Crisp Number

$$\begin{array}{l} \text{Let } a = -m \text{, } m > 0. \text{ Let } [\widetilde{m}]\_{(a,\beta,\gamma)} = \left[ m\_{\mathcal{L}}^{1}(\alpha), m\_{\mathcal{R}}^{1}(\alpha); m\_{\mathcal{L}}^{2}(\beta), m\_{\mathcal{R}}^{2}(\beta); m\_{\mathcal{R}}^{3}(\gamma), m\_{\mathcal{R}}^{3}(\gamma) \right] \text{and} \\\ [\widetilde{u}\_{0}]\_{(a,\beta,\gamma)} = \left[ u\_{\mathcal{L},0}^{1}(\alpha), u\_{\mathcal{R},0}^{1}(\alpha); u\_{\mathcal{L},0}^{2}(\beta), u\_{\mathcal{R},0}^{2}(\beta); u\_{\mathcal{L},0}^{3}(\gamma), u\_{\mathcal{R},0}^{3}(\gamma) \right] \\\ \forall \ \alpha, \beta, \gamma \in [0, 1] \text{ and } 0 \le \alpha + \beta + \gamma \le 3. \end{array}$$

Taking the (α, β, γ)-cut of Equation (8), we have the following equations:

$$\begin{cases} \begin{aligned} u\_{L,n+1}^1(\alpha) &= -m\_L^1(\alpha)u\_{L,n}^1(\alpha) \\ u\_{R,n+1}^1(\alpha) &= -m\_{R,n}^1(\alpha)u\_{R,n}^1(\alpha) \\ u\_{L,n+1}^2(\beta) &= -m\_L^2(\beta)u\_{L,n}^2(\beta) \\ u\_{R,n+1}^2(\beta) &= -m\_{R,n}^2(\beta)u\_{R,n}^2(\beta) \\ u\_{L,n+1}^3(\gamma) &= -m\_L^3(\gamma)u\_{L,n}^3(\gamma) \\ u\_{R,n+1}^3(\gamma) &= -m\_{R,n}^3(\gamma)u\_{R,n}^3(\gamma) \end{aligned} \end{cases} \tag{54}$$

The general solutions of the above equations are as follows:

$$\begin{array}{l} \boldsymbol{u}\_{\rm L,n}(\boldsymbol{\alpha}) = \left(-m\_{\rm L}(\boldsymbol{\alpha})\right)^{n}\boldsymbol{u}\_{0} \\ \boldsymbol{u}\_{\rm R,n}(\boldsymbol{\alpha}) = \left(-m\_{\rm R}(\boldsymbol{\alpha})\right)^{n}\boldsymbol{u}\_{0} \\ \boldsymbol{u}\_{\rm L,n}(\boldsymbol{\beta}) = \left(-m\_{\rm L}(\boldsymbol{\beta})\right)^{n}\boldsymbol{u}\_{0} \\ \boldsymbol{u}\_{\rm R,n}(\boldsymbol{\beta}) = \left(-m\_{\rm R}(\boldsymbol{\beta})\right)^{n}\boldsymbol{u}\_{0} \\ \boldsymbol{u}\_{\rm L,n}(\boldsymbol{\gamma}) = \left(-m\_{\rm L}(\boldsymbol{\gamma})\right)^{n}\boldsymbol{u}\_{0} \\ \boldsymbol{u}\_{\rm R,n}(\boldsymbol{\gamma}) = \left(-m\_{\rm R}(\boldsymbol{\gamma})\right)^{n}\boldsymbol{u}\_{0} \end{array} \tag{55}$$

4.2.6. The Solution When the Initial Condition u0 and a > 0 Are Both Neutrosophic Numbers

$$\begin{array}{rcl} \text{Let} \qquad & [\![a]\!]\_{(a,\beta,\cdot\gamma)} & = & [a^1\_L(a), a^1\_R(a); a^2\_L(\beta), a^2\_R(\beta); a^3\_L(\gamma), a^3\_R(\gamma)] \quad \text{and} \quad [\![\widetilde{u}\_0]\!]\_{(a,\beta,\cdot\gamma)} & = \\ \left[u^1\_{L,0}(a), u^1\_{R,0}(a); u^2\_{L,0}(\beta), u^2\_{R,0}(\beta); u^3\_{L,0}(\gamma), u^3\_{R,0}(\gamma)\right] & & \\ \quad \forall \ \alpha, \beta, \gamma \in [0,1] \text{ and } 0 \le \alpha + \beta + \gamma \le 3. \end{array}$$

In this case, the solutions are given, following from Equation (50): when *n* is even:

$$\begin{cases} \begin{array}{l} u\_{L,n}^{1}(\alpha) = \left(a\_{L}^{1}(\alpha) \, a\_{R}^{1}(\alpha)\right)^{\frac{\mu}{2}} u\_{L,0}^{1}(\alpha) \\\ u\_{R,n}^{1}(\alpha) = \left(a\_{L}^{1}(\alpha) \, a\_{R}^{1}(\alpha)\right)^{\frac{\mu}{2}} u\_{R,0}^{1}(\alpha) \\\ u\_{L,n}^{2}(\beta) = \left(a\_{L}^{2}(\beta) \, a\_{R}^{2}(\beta)\right)^{\frac{\mu}{2}} u\_{L,0}^{2}(\beta) \\\ u\_{R,n}^{2}(\beta) = \left(a\_{L}^{2}(\beta) \, a\_{R}^{2}(\beta)\right)^{\frac{\mu}{2}} u\_{R,0}^{2}(\beta) \\\ u\_{L,n}^{3}(\gamma) = \left(a\_{L}^{3}(\gamma) \, a\_{R}^{3}(\gamma)\right)^{\frac{\mu}{2}} u\_{L,0}^{3}(\gamma) \\\ u\_{R,n}^{3}(\gamma) = \left(a\_{L}^{3}(\gamma) \, a\_{R}^{3}(\gamma)\right)^{\frac{\mu}{2}} u\_{R,0}^{3}(\gamma) \end{array} \tag{56}$$

when *n* is odd:

$$\begin{array}{l} u\_{L,n}^{1}(\alpha) = \left(a\_{L}^{1}(\alpha)\right)^{\frac{n-1}{2}} \left(a\_{R}^{1}(\alpha)\right)^{\frac{n+1}{2}} u\_{R,0}^{1}(\alpha) \\\ u\_{R,n}^{1}(\alpha) = \left(a\_{L}^{1}(\alpha)\right)^{\frac{n+1}{2}} \left(a\_{R}^{1}(\alpha)\right)^{\frac{n-1}{2}} u\_{L,0}^{1}(\alpha) \\\ u\_{L,n}^{2}(\beta) = \left(a\_{L}^{2}(\beta)\right)^{\frac{n-1}{2}} \left(a\_{R}^{2}(\beta)\right)^{\frac{n+1}{2}} u\_{R,0}^{2}(\beta) \\\ u\_{R,n}^{2}(\beta) = \left(a\_{L}^{2}(\beta)\right)^{\frac{n+1}{2}} \left(a\_{R}^{2}(\beta)\right)^{\frac{n-1}{2}} u\_{L,0}^{2}(\beta) \\\ u\_{L,n}^{3}(\gamma) = \left(a\_{L}^{3}(\gamma)\right)^{\frac{n-1}{2}} \left(a\_{R}^{3}(\gamma)\right)^{\frac{n+1}{2}} u\_{R,0}^{3}(\gamma) \\\ u\_{R,n}^{3}(\gamma) = \left(a\_{L}^{3}(\gamma)\right)^{\frac{n+1}{2}} \left(a\_{R}^{3}(\gamma)\right)^{\frac{n-1}{2}} u\_{L,0}^{3}(\gamma) \end{array} \tag{57}$$

4.2.7. The Solution When the Initial Condition u0 and a < 0 Are Both Neutrosophic Numbers

Let *a* = −*m* , *m* > 0. Let [*m*e] (α, β, γ) = h *m*1 *L* (α), *m*1 *R* (α); *m*2 *L* (β), *m*2 *R* (β); *m*3 *L* (γ), *m*3 *R* (γ) i and [e*u*0] (α, β, γ) = h *u* 1 *L*,0(α), *u* 1 *R*,0(α); *u* 2 *L*,0(β), *u* 2 *R*,0(β); *u* 3 *L*,0(γ), *u* 3 *R*,0(γ) i ∀ α , β, γ ∈ [0, 1] and 0 ≤ α + β + γ ≤ 3.

In a similar way, as seen in Equation (54), we have the following solutions.

The general solutions of the above equations are as follows:

$$\begin{cases} u\_{L,n}^1(\alpha) = (-m\_L(\alpha))^n u\_{L,0}^1(\alpha) \\ u\_{R,n}^1(\alpha) = (-m\_R(\alpha))^n u\_{R,0}^1(\alpha) \\ u\_{L,n}^2(\beta) = (-m\_L(\beta))^n u\_{L,0}^2(\beta) \\ u\_{R,n}^2(\beta) = (-m\_R(\beta))^n u\_{R,0}^2(\beta) \\ u\_{L,n}^3(\gamma) = (-m\_L(\gamma))^n u\_{L,0}^3(\gamma) \\ u\_{R,n}^3(\gamma) = (-m\_R(\gamma))^n u\_{R,0}^3(\gamma) \end{cases} \tag{58}$$

#### **5. Numerical Example**

**Example 1.** *Solve the di*ff*erence equation:*

$$u\_{n+1} = (2, 4, 6; 1, 4, 5; 2, 4, 5)u\_n \tag{59}$$

*with the initial condition* e*un*=0 = (50, 60, 70; 55, 60, 75; 50, 60, 80)

**Solution 1.** *If the* [e*un*] (α, β, γ) *is the (*α, β, γ*)-cut of a sequence of neutrosophic numbers, then its components are as follows:*

$$\begin{cases} u\_{L,n}^1(\alpha) = (2+2\alpha)^n (50+10\alpha) \\ u\_{R,n}^1(\alpha) = (6-2\alpha)^n (70-10\alpha) \\ u\_{L,n}^2(\beta) = (4-3\beta)^n (60-5\beta) \\ u\_{R,n}^2(\beta) = (4+\beta)^n (60+15\beta) \\ u\_{L,n}^3(\gamma) = (4-2\gamma)^n (60-10\gamma) \\ u\_{R,n}^3(\gamma) = (4+\gamma)^n (60+20\gamma) \end{cases} \tag{60}$$

**Remarks 2.** *We plot the solution for n* = 2*. From the above Table 2 and Figure 1, we see that u* 1 *L*,*n* (α) *is an increasing function and u* 1 *R*,*n* (α) *is a decreasing function, with respect to* α*. On the other hand, u* 2 *L*,*n* (β) *is a decreasing function and u* 2 *R*,*n* (β) *is an increasing function, with respect to* β*. Additionally, u* 3 *L*,*n* (γ) *is a decreasing function and u* 3 *R*,*n* (γ) *is an increasing function, with respect to* γ*. Therefore, using the concept of Definition 3.2, we call the solution a strong solution.*

u,୬ ଶ

γ

u,୬ ଷ

, , ,

**Table 2.** Solution for *n* = 2.

ଶ

() ,

ଷ (γ)

() ,

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(β) uୖ,୬

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n=5 u,୬ ଵ (α) uୖ,୬ ଵ (α) α u,୬ ଶ (β) uୖ,୬ ଶ (β) β u,୬ ଷ (γ) uୖ,୬ ଷ (γ) γ **Remarks 3.** *We plotted the solution for n* = 5*. From the above Table 3 and Figure 2, we see that u* 1 *L*,*n* (α) *is an increasing function and u* 1 *R*,*n* (α) *is a decreasing function, with respect to* α*. On the other hand, u* 2 *L*,*n* (β) *is a decreasing function and u* 2 *R*,*n* (β) *is an increasing function, with respect to* β*. Additionally, u* 3 *L*,*n* (γ) *is a decreasing function and u* 3 *R*,*n* (γ) *is an increasing function, with respect to* γ*. Therefore, using the concept of Definition 6, we call the solution a strong solution.*

= 2.

**Table 3.** Solution for *n* = 5.

, , ,

() ,

() ,

= 5.

() ,

() , ()

() ,

, , = 0.4 We interpret the solution for fixed α, β, γ = 0.4 and different *n* in Table 4 and Figure 3.

, , = 0.4 . **Table 4.** Solution for α, β, γ = 0.4 and different *n*.

= 5.

**Figure 3.** Graph for α, β, γ = 0.4 and different *n*.

, , = 0.4

**Example 2.** *Solve the di*ff*erence equation*:

[](ఈ,ఉ,ఊ) , ,

$$
\mu\_{n+1} - 4\mu\_n = 0 \tag{61}
$$

() = 4(50 + 10)

() = 4(70 − 10)

() = 4(60 − 5)

() = 4(60 + 15)

() = 4(60 − 10)

() = 4(60 + 20)

() = 4(70 − 10)

() = 4(50 + 10)

() = 4(60 + 15)

() = 4(60 − 5)

() = 4(60 + 20)

() = 4(60 − 10)

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*with initial condition* e*un*=0 = (50, 60, 70; 55, 60, 75; 50, 60, 80)

**Solution 2.** *If* [e*un*] (α, β, γ) *is the (*α, β, γ*)-cut of a sequence of neutrosophic numbers, then it's components are as follows:*

*when n is even:*

$$\begin{array}{l} u\_{L,n}^1(\alpha) = 4^n(50 + 10\alpha) \\ u\_{R,n}^1(\alpha) = 4^n(70 - 10\alpha) \\ u\_{L,n}^1(\beta) = 4^n(60 - 5\beta) \\ u\_{R,n}^1(\beta) = 4^n(60 + 15\beta) \\ u\_{L,n}^1(\gamma) = 4^n(60 - 10\gamma) \\ u\_{R,n}^1(\gamma) = 4^n(60 + 20\gamma) \end{array} \tag{62}$$

*when n is odd:*

$$\begin{array}{l} \mathfrak{u}\_{L,n}^{1}(\alpha) = 4^{n}(70 - 10\alpha) \\ \mathfrak{u}\_{R,n}^{1}(\alpha) = 4^{n}(50 + 10\alpha) \\ \mathfrak{u}\_{L,n}^{1}(\beta) = 4^{n}(60 + 15\beta) \\ \mathfrak{u}\_{R,n}^{1}(\beta) = 4^{n}(60 - 5\beta) \\ \mathfrak{u}\_{L,n}^{1}(\gamma) = 4^{n}(60 + 20\gamma) \\ \mathfrak{u}\_{R,n}^{1}(\gamma) = 4^{n}(60 - 10\gamma) \end{array} \tag{63}$$

*As previous examples, we easily interpret the solutions in a di*ff*erent manner.*

#### **6. Application of the Method in Actuarial Science**

Let us consider that a sum *S*0 is invested at a compound interest of *i* per unit amount and per unit of time and *St* is the amount at the end of time *t*. We then get the difference equation associated with the problem, which is:

$$\mathbf{S}\_{t+1} = \mathbf{S}\_t + i\mathbf{S}\_t = (1+i)\mathbf{S}\_t \tag{64}$$

If, for some reason, *i* may vary, we are interested to find the possible amount after a certain time interval.

For this problem, let us consider hypothetical data and solve it. Suppose a person has initially invested *St*=0 = 10000\$ in a firm, where they get about 4% interest (which may be considered a neutrosophic value).

As per Table 1, if we take the verbal phrase for a triangular neutrosophic number, we then set the interest rate as follows:

For the truth part: low as 3%, medium as 4%, high as 5%;

For the falsity portion: very low as 2%, medium as 4%, very high as 6%;

For the indeterminacy part: between low and very low 2.5%, medium 4%, between high and very high 5.5%,

i.e., we can takee*i* = (3, 4, 5; 2, 4, 6; 2.5, 4, 5.5)% per annum rate. We wish to predict the amount of money after 10 years.

Therefore, we get the fuzzy difference equation

$$\mathbf{S}\_{t+1} = \mathbf{S}\_t + i\mathbf{S}\_t = \begin{pmatrix} 1+\overline{i} \end{pmatrix} \mathbf{S}\_t \tag{65}$$

With the initial conditions *St*=0 = 10000\$ ande*i* = (3, 4, 5; 2, 4, 6; 2.5, 4, 5.5)%.

**Solution 3.** *Equation (65) is equivalent to*

$$S\_{t+1} = S\_t + iS\_t = (1 + (0.03, 0.04, 0.05; 0.02, 0.04, 0.06; 0.025, 0.04, 0.055))S\_{t+1}$$

*or*

$$S\_{t+1} = (1.03, 1.04, 1.05; 1.02, 1.04, 1.06; 1.025, 1.04, 1.055)S\_t \tag{66}$$

*with the initial condition St*=0 = 10000\$*.*

*The solution of (66) can be written using the concept of (19), as follows:*

$$\begin{array}{l} \mathcal{S}\_{L,t}^{1}(\alpha) = 10000(1.03 + 0.01\alpha)^{t} \\ \mathcal{S}\_{R,t}^{1}(\alpha) = 10000(1.05 - 0.01\alpha)^{t} \\ \mathcal{S}\_{L,t}^{2}(\beta) = 10000(1.04 - 0.02\beta)^{t} \\ \mathcal{S}\_{R,t}^{2}(\beta) = 10000(1.04 + 0.02\beta)^{t} \\ \mathcal{S}\_{L,t}^{3}(\gamma) = 10000(1.04 - 0.015\gamma)^{t} \\ \mathcal{S}\_{R,t}^{3}(\gamma) = 10000(1.04 + 0.015\gamma)^{t} \end{array} \tag{67}$$

**Remarks 4.** *(1) We plot the solution for t* = 10*. From the above Table 5 and Figure 4, we see that S* 1 *L*,*n* (α) *is an increasing function and S* 1 *R*,*n* (α) *is a decreasing function, with respect to* α*. On the other hand, S* 2 *L*,*n* (β) *is a decreasing function and S* 2 *R*,*n* (β) *is an increasing function, with respect to* β*. Additionally, S* 3 *L*,*n* (γ) *is a decreasing function and S* 3 *R*,*n* (γ) *is an increasing function, with respect to* γ*. Therefore, using the concept of Definition 3.2, we call the solution a strong solution. (2) From Table 5, we can see that we find the crisp solution at* α = 1*,* β, γ = 0 *(since, at* α = 1*,* β, γ = 0, *the neutrosophic number becomes a crisp number) and for t* = 10 *is equal to 14802.4428. Therefore, we can say that after 10 years, the most probable chance to get the money is 14802.4428\$.(3) If we consider* α = 0 *and* β, γ = 1*, i.e., in the case that we get the most uncertain solution interval, we observe that the truthiness of the solution belongs to the interval* [13439.1638, 16288.9463]*, the falsity belongs to the interval* [12189.9442, 17908.4770]*, and the indeterminacy belongs to the interval* [12800.8454, 17081.4446]*.*

**Table 5.** Solution for *t* = 10.

#### **7. Conclusions and Future Research Scope**

In this paper, we find the solution strategy for solving and analyzing homogeneous linear difference equations with neutrosophic numbers, i.e., we found the solutions of the homogeneous difference equations with initial conditions and coefficients, both as neutrosophic numbers. We demonstrate the solution of different cases using the neutrosophic characterization theorem, which is established in this paper. The strong and weak solution concepts are also applied to different results.

= 10

Moreover, the outcomes of the study are as follows:

For some limitations, we did not study the different perspectives of related research in the theory of difference equations with uncertainty in this present work. From this work, anyone can take motivation and find a new theory and results in the following field, as follows:

As a final argument, we can surely say that this research is very helpful to the research community who deals with discrete system modeling with uncertainty.

**Author Contributions:** Conceptualization, S.P.M.; Data curation, A.A. (Abdul Alamin); Formal analysis, S.A.; Resources, S.P.M. Software; Supervision, A.A. (Ali Ahmadian); Visualization, A.A. (Ali Ahmadian); Writing—original draft, A.A. (Abdul Alamin) and S.A.; Writing—review & editing, A.A. (Ali Ahmadian), S.S. and M.S. All authors have read and agreed to the published version of the manuscript.

**Funding:** This research received no external funding.

**Conflicts of Interest:** The authors declare no conflict of interest.

### **References**

© 2020 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

### *Article* **Neutrosophic Modeling of Talcott Parsons's Action and Decision-Making Applications for It**

#### **Cahit Aslan 1 , Abdullah Kargın 2,\* and Memet ¸Sahin 2**

**\*** Correspondence<EMAIL_ADDRESS>Tel.: +90-055-4270-6621

Received: 15 May 2020; Accepted: 3 July 2020; Published: 13 July 2020

**Abstract:** The grand theory of action of Parsons has an important place in social theories. Furthermore, there are many uncertainties in the theory of Parsons. Classical math logic is often insufficient to explain these uncertainties. In this study, we explain the grand theory of action of Parsons in neutrosociology for the first time. Thus, we achieve a more effective way of dealing with the uncertainties in the theory of Parsons as in all social theories. We obtain a similarity measure for single-valued neutrosophic numbers. In addition, we show that this measure of similarity satisfies the similarity measure conditions. By making use of this similarity measure, we obtain applications that allow finding the ideal society in the theory of Parsons within the theory of neutrosociology. In addition, we compare the results we obtained with the data in this study with the results of the similarity measures previously defined. Thus, we have checked the appropriateness of the decision-making application that we obtained.

**Keywords:** neutrosociology; modeling of grand theory of action of Talcott Parsons; single-valued neutrosophic number; measure of similarity; decision-making applications

#### **1. Introduction**

There are many uncertainties in the world. Classical math logic is usually insufficient to explain uncertainties. Thus, we are not always able to say for a situation or an event whether it is true or wrong in an absolute manner. For example, we cannot always say the weather is hot or cold. While the weather is hot according to some, it may be cold for others. Therefore, Smarandache obtained the neutrosophic logic and neutrosophic set to deal with uncertainties more objectively in 1998 [1]. 'T' is the membership degree, 'I' is the uncertainty degree and 'F' is the non-membership degree in the neutrosophic logic and neutrosophic sets. "T, I, F" are defined independently. In addition, a neutrosophic number has the form (T, I, F). Furthermore, neutrosophic logic is a generalization of fuzzy logic [2] and intuitionistic fuzzy logic [3] since fuzzy and intuitionistic fuzzy logic's membership, non–membership degrees are defined dependently. Thus, many researchers have obtained new structures and new applications on neutrosophic logic and sets [4–15].

In Section 2 of this study, we provide a literature review. In Section 3, we give related works. In Section 4, we include the definitions of neutrosophic sets [1], single-valued sets [6], similarity measures in [7] and [16], the theory of social action of Parsons [17], Hausdorff measure [18] and Hamming measure [18]. In Section 5, we re-model the social action theory of Parsons which is modeled in neutrosociology. In Section 6, we obtain a similarity measure for single-valued neutrosophic sets and prove that this measure meets the requirements of the similarity measure. In Section 7, we create the decision-making algorithm that we can choose the ideal society among the societies for the social action theory of Parsons in neutrosociology with the help of similarity measure in Section 6. In Section 8, we give sensitivity analysis for numeric example in Section 7; in Section 9, we give comparison methods. We compare the results we obtained with the data in this study with the results of the similarity measures previously defined. Thus, we have checked the appropriateness of the decision-making application we obtained; in Section 10, we discuss what we obtained in this study and make suggestions for studies that can be obtained by making use of this study; in Section 11, we give conclusions.

#### **2. Literature Review**

Similarity measure and decision-making practices emerge as an important application theory, especially after the definition of the fuzzy sets and neutrosophic sets. Many researchers have tried to deal with uncertainties by making new applications on neutrosophic sets, using similarity measures, TOPSIS method, VIKOR method, multicriteria method, Maximizing deviation method, decision tree methods, gray relational analysis method, etc. Recently, ¸Sahin et al. studied combined classic neutrosophic sets and double neutrosophic sets [19]; ¸Sahin et al. obtained decision-making applications for professional proficiencies in neutrosophic theory [16]; Uluçay et al. introduced decision-making applications for neutrosophic soft expert graphs [20]; Olgun et al. studied neutrosophic logic on the decision tree [21]; Wang et al. studied an extended VIKOR method with triangular fuzzy neutrosophic numbers [22]; Biswas et al. introduced TOPSIS method for decision–making applications [23]; ¸Sahin et al. obtained a maximizing deviation method in neutrosophic theory [24]; Biswas et al. studied gray relational analysis method for decision-making applications [25].

#### **3. Related Works**

Smarandache claims that sociopolitical events can be studied mathematically [4]. In addition, he claims that it is possible to design a tool to describe an equation, an operator, a mathematical structure or a social phoneme. Studying the past gives us an idea about the future, at least partially. For this reason, we need to construct neutrosophic theories that may describe the new possible types of social structures with a neutrosophic number form. Since the social word contains a high degree of subjectivity that causes a low level of unanimity, these theories necessarily address uncertainty. Most of the data we come across in the field of sociology may be vague, incomplete, contradictory, biased, hybrid, ignorant, redundant, etc. Therefore, they are neutrosophic in nature and neutrosophic sciences dealing with indeterminacy should be involved in the study of sociology [4].

For the very same reasons, Smarandache proposed a model to be used in neutrosophic studies. He states that a neutrosophic extension of an element x with a neutrosophic number form.

Parsons, who built his theory on methodological and meta-theoretical debates in the field of social science, also paid special attention to hermeneutic to explain the extent of the individual's voluntary involvement in action [26]. He made structural and functional explanations to maintain social balance and harmony [21]. While Parsons saw culture as values and norms that guide the actions of individuals in social life, he conceptualized the structure as a system of intertwined and independent parts [27]. According to Parsons, cultural objects are autonomous. He did this by distinguishing between the cultural and social systems. He also viewed society as a general system of action. In addition, many researchers have studied Parsons's social action theory [26–36].

In this study, Parsons's social action theory was aimed to re-model neutrosociology. As in all social theories, the social action theory of Parsons could not escape uncertainty [21]. Hence, the handling of it in neutrosociology theory would make this theory more useful. Therefore, we have obtained a similarity measure with single-valued neutrosophic numbers and included applications where this measure can be used as the neutrosophic equivalent of the ideal society in this theory.

#### **4. Preliminaries**

This section includes the definitions of neutrosophic sets [1], single-valued neutrosophic sets [6], similarity measures [7,16] and theory of social action of Parsons [17], Hausdorff measure [18] and Hamming measure [18].

**Definition 1** ([17])**.** *Parsons, who built his theory on methodological and meta-theoretical debates in the field of social science, also paid special attention to hermeneutic to explain the extent of the individual's voluntary involvement in action (so, which is neutrosophic). He made structural and functional explanations to maintain social balance and harmony. While Parsons saw culture as values and norms that guide the actions of individuals in social life, he conceptualized the structure as a system of intertwined and independent parts. According to Parsons, cultural objects are autonomous. He did this by distinguishing between the cultural and social systems. He also viewed society as a general system of action.*

**Definition 2** ([1])**.** *Let X be a universal set. Neutrosophic set S; is identified as S* = *{(x:TA*(*x*) *, IA*(*x*) *, FA*(*x*)>*, x* ∈ *X}. Where; on the condition that* 0 −≤ *TA*(*x*) + *IA*(*x*) + *FA*(*x*) ≤3 +*; the functions T:U* → ]0−,1+[ *is truth function, I:U* → ]0−,1+[ *is uncertain function and F:U* → ]0−,1+[ *is falsity function.*

**Definition 3** ([6])**.** *Let X be a universal set. Single-valued neutrosophic number set S; is identified as S* = *{(x:TA*(*x*) *, IA*(*x*) *, FA*(*x*)>*, x* ∈ *X}. Where; on condition that* 0≤ *TA*(*x*) + *IA*(*x*) + *FA*(*x*) ≤3*; the functions T:X* → [0,1] *is truth function, I:X* → [0,1] *is uncertainly function and F:X* → [0,1] *is falsity function.*

**Definition 4** ([6])**.** *Let A* = *{(x:*<*TA*(*x*) *, IA*(*x*) *, FA*(*x*)>*} and B* = *{(x:*<*TB*(*x*) *, IB*(*x*) *, FB*(*x*)>*} are single-valued neutrosophic numbers. If A* = *B; then TA*(*x*) = *TB*(*x*) *, IA*(*x*) = *IB*(*x*) *and FA*(*x*) = *FB*(*x*) *.*

**Definition 5** ([6])**.** *Let A* = *{(x:*<*TA*(*x*) *, IA*(*x*) *, FA*(*x*)>*} and B* = *{(x:*<*TB*(*x*) *, IB*(*x*) *, FB*(*x*)>*} are single-valued neutrosophic sets for x* ∈ *U. If A* < *B; then for* ∀ *x* ∈ *U; TA*(*x*) < *TB*(*x*) *, IA*(*x*) < *IB*(*x*) *and FA*(*x*) < *FB*(*x*) *.*

**Properties 1** ([7])**.** *Let A*1, *A*2 *and A*3 *are three single-valued neutrosophic numbers and S be a similarity measure. S provides the following conditions.*

i. 0 ≤ S(*A*1, *A*2) ≤ 1 ii. S(*A*1, *A*2) = S(*A*2, *A*1) iii. S(*A*1, *A*2) = 1 ⇔ *A*1 = *A*2 . iv. *If A*1 ≤ *A*2 ≤ *A*3 *then,* S(*A*1, *A*3) ≤ S(*A*1, *A*2).

**Definition 6** ([16])**.** *Let A*1 = <*T*1, *I*1, *F*1> *and A*2 = <*T*2, *I*2, *F*2> *be two single-valued neutrosophic numbers.*

$$S\_N(A\_1, A\_2) = $$

$$1 - (2/3) \lfloor \frac{\min\{|3(T\_1 - T\_2) - 2(F\_1 - F\_2)|, |F\_1 - F\_2|\}}{\lfloor \max\{|3(T\_1 - T\_2) - 2(F\_1 - F\_2)|, |F\_1 - F\_2|\}/5 \rfloor + 1} + 1$$

$$\frac{\min\{|4(T\_1 - T\_2) - 3(I\_1 - I\_2)|, |I\_1 - I\_2|\}}{\lfloor \max\{|4(T\_1 - T\_2) - 3(I\_1 - I\_2)|, |I\_1 - I\_2|\}/7 \rfloor + 1} +$$

$$\frac{\min\{|5(T\_1 - T\_2) - 2(F\_1 - F\_2) - 3(I\_1 - I\_2)|, |T\_1 - T\_2|\}}{\lfloor \max\{|5(T\_1 - T\_2) - 2(F\_1 - F\_2) - 3(I\_1 - I\_2)|, |T\_1 - T\_2|\}/10 + 1}$$

*is a similarity measure.*

**Definition 7** ([18])**.** *Let A*1 = <*T*1, *I*1, *F*1> *and A*2 = <*T*2, *I*2, *F*2> *be two single-valued neutrosophic numbers.*

$$S\_h(A\_1A\_2) = 1 - \max\{|T\_1 - T\_2|, |I\_1 - I\_2|, |F\_1 - F\_2|\}$$

*is a Hausdor*ff *similarity measure.*

**Definition 8** ([18])**.** *Let A*1 = <*T*1, *I*1, *F*1> *and A*2 = <*T*2, *I*2, *F*2> *be two single-valued neutrosophic numbers.*

$$S\_H(A\_1, A\_2) = 1 - \left( |T\_1 - T\_2| + |I\_1 - I\_2| + |F\_1 - F\_2| \right) / 3$$

*is a Hamming similarity measure.*

#### **5. Neutrosophic Modeling of Parson's Theory of Action**

According to the perfection of action categories of Parsons, it is inevitable to have deep doubts in every society and between layers of a particular society. However, "there is no ideal society in the sense that Marx defines, within each society the definition of ideal changes according to the place of a person within the society. By those who are at the top layer, the society is defined as ideal, by those at the lowest layer, it is far from being ideal, and by those in the mid-layer, who can sometimes be completely ignorant of what is an ideal society, it can be described as a fluctuating phenomenon depending on circumstances. Therefore, we always have a neutrosophic ideal society with an opposite and neutral triad. Naturally, this is valid for all societies since there are always people with more privileges than the others. Even in any a democratic society, some people have more privileges although they may form a small minority" [4].

Parsons developed a theory of action to explain how the macro and micro aspects of a particular social order show structural integrity together with the participation of its members. He took into account the voluntary participation of the individual in the social life on one hand, and structural continuity on the other. Here, it is assumed that the individual acts under the motivation of the social structure while taking action. According to him, social sciences should consider a trio considering the purposes, ends and ideals when examining actions.

#### *Grand Theory of Action*

The basic paradigm of Parson viewed society as a general system of action is based on the understanding of 'rational social action' of Weber [28]. However, according to Weber, sociology is a science that tries its interpretive understanding of social action to achieve a causal explanation of its course and its effects [36].

This interpretation is enriched from the perspective of the sociologist. Thus, social actions become neutrosophic. Others may agree, partially agree or disagree (1, 0, 0). Likewise, in the theory of Parson, the possibility of all members of society to participate in social values and norms that regulate, and guide human relations rather than individual activities is questionable, uncertain. Here we must see neutrosophic triplets.

According to Parson's theory, all social actions are based on five pattern variables. These:

Parsons believes that these variables classify expectations and the structure of relationships, making the intangible action theory more understandable. However, according to Parsons, pattern variables are twofold, and each pattern variable indicates a problem or riddle that must be solved by the actor before the action can be performed. At the same time, there is a wide variety between the traditional society and the modern society. However, these can be seen as binary for neutrosophic sociological analysis (1, 0), it is very difficult to determine which of the individual's behaviors are modern or traditional. Therefore, each of them should be considered as triple neutrosophic (1, 0, 0). The feminists' response to Parsons' family view can be given as an example. According to Parsons, the instrumental leadership role in the family structure in modern societies should be given to the spouse–father, on which the family's reputation and income are based [32]. However, according to feminists, this statement by Parsons is nothing more than the continuation of the status quo [35]. In addition, these pattern variables (stereotypes) do not say how people will behave when faced with role conflict, and we will once again encounter uncertainty. This uncertainty can only be answered by neutrosociology.

The society model that Parsons has compared to the biologic model of an organism is based on the understanding of "living systems" that continues in a balanced way. According to him, a change in any part of the social system leads to adaptive changes in other parts [33]. There are four main problems an all-action system must solve. These are adaptation, goal-attainment, integration and latent pattern maintenance (AGIL). In short, these are referred to as AGIL in Table 1.

**Table 1.** Pure adaptation, goal-attainment, integration and latent pattern maintenance (AGIL) model for all living systems [33].

"Adaptation" (A) is concerned with meeting the needs of the system from its environment and how resources are distributed within the system. Here, the system should provide sufficient resources from the environment and distribute it within itself. Social institutions are related to interrelated social rules and roles system that will meet social needs or functions and help solve social system problems. For example, economy, political order, law, religion, education and family are basic institutions for these. If a social system will continue to live, it needs structures and organizations that will function to adapt to its environment. The most dominant of these institutions is the economy. In "achieving the goal" (G), it is determined that the system reaches the specific target and which of these targets has priority. In other words, it should mobilize the resources and energies of the system and determine the priorities among them. "Integration" (I) refers to the coordination and harmony of parts of the system so that the system functions as a whole. To keep the system running, it must coordinate, correct, and regulate the relationships between the various actors or units in the system. "Latent pattern maintenance" (L) shows how to ensure the continuity of the action within the system according to a certain order or norm. The system should protect its values from deterioration and ensure the transfer of social values. Thus, it ensures the compliance of the members of the system. Especially family, religion, media and education have basic functions. Thanks to these, individuals gain a moral commitment to values shared socially [30].

The General Action Level is as follows in Table 2:

**Table 2.** General Action Level [30].

Ultimately we get this series: The social the system, the fiduciary the cognitive.

Let us rebuild this series neutrosociology: (1, 0, 0) (1, 0, 0) (1, 0, 0).

If we go back to the beginning, "Behavioral organic, Personality system, Cultural system and Social system" must work continuously to ensure social balance. This will be through "socialization" and "social control". If socialization "works", all members of the society will adhere to shared values, make appropriate choices between pattern variables, and do what is expected of them in harmony, integration and other issues. For example, people will marry and socialize their children (L), and the father in the family will gain bread as it should be (A) [35].

#### **6. A New Measurement of Similarity for Single-Valued Neutrosophic Numbers**

**Definition 9.** *Let A*1 = <*T*1, *I*1, *F*1>*, A*2 = <*T*2, *I*2, *F*2> *be two single-valued neutrosophic numbers. We define measure of similarity between A*1 *and A*2 *as follows*

$$\begin{split} \mathfrak{S}\_{N}(A\_{1},A\_{2}) &= \mathbf{1} - \begin{pmatrix} 2/3 \end{pmatrix} \begin{bmatrix} \frac{\min\{\sqrt{3(T\_{1}-T\_{2})^{2}+(I\_{1}-I\_{2})^{2}, \ [2(T\_{1}-T\_{2})-(I\_{1}-I\_{2})]/3 \}}{\max\{\sqrt{3(T\_{1}-T\_{2})^{2}+(I\_{1}-I\_{2})^{2}, \ [2(T\_{1}-T\_{2})-(I\_{1}-I\_{2})]/3 \}/2 \} + 1 \end{bmatrix} \\ &+ \frac{\min\{\sqrt{3(T\_{1}-T\_{2})^{2}+(F\_{1}-F\_{2})^{2}, \ [2(T\_{1}-T\_{2})-(F\_{1}-F\_{2})]/3 \}}{\max\{\sqrt{3(T\_{1}-T\_{2})^{2}+(F\_{1}-F\_{2})^{2}, \ [2(T\_{1}-T\_{2})-(F\_{1}-F\_{2})]/3 \}/2 \} + 1} \\ &+ \frac{\min\{\sqrt{2(T\_{1}-T\_{2})^{2}+(I\_{1}-I\_{2})^{2}+(F\_{1}-F\_{2})^{2}, \ [3(T\_{1}-T\_{2})-(I\_{1}-I\_{2})-(F\_{1}-F\_{2})]/5 \}}{\max\{\sqrt{2(T\_{1}-T\_{2})^{2}+(I\_{1}-I\_{2})^{2}+(F\_{1}-F\_{2})^{2}}, \ [3(T\_{1}-T\_{2})-(I\_{1}-I\_{2})-(F\_{1}-F\_{2})]/5 \} + 1} \end{split}$$

We show that the measure of similarity in Definition 9 meets the requirements in Properties 1.

**Theorem 1.** *Let SN be the measure of similarity in Definition 9. SN provides the following features.*

i. 0 ≤ *SN*(*A*1, *A*2) ≤ 1 ii. *SN*(*A*1, *A*2) = *SN*(*A*2, *A*1) iii. *SN*(*A*1, *A*2) = 1 *if and only if A*1 = *A*2*.* iv. *If A*1 ≤ *A*2 ≤ *A*3*, then SN*(*A*1, *A*3) ≤ *SN*(*A*1, *A*2)*.*

### **Proof: (i)** Since *A*1 and *A*2 are single-valued neutrosophic numbers, we have

*max*{ *min*{ q 3(*T*1−*T*2) 2+(*I*1−*I*2) 2 , |2(*T*1−*T*2)−(*I*1−*I*2)|/3} {*max*{ q 3(*T*1−*T*2) 2+(*I*1−*I*2) 2 , |2(*T*1−*T*2)−(*I*1−*I*2)|/3}/2}+1 } = 1/2, *min*{ *min*{ q 3(*T*1−*T*2) 2+(*I*1−*I*2) 2 , |2(*T*1−*T*2)−(*I*1−*I*2)|/3} {*max*{ q 3(*T*1−*T*2) 2+(*I*1−*I*2) 2 , |2(*T*1−*T*2)−(*I*1−*I*2)|/3}/2}+1 } = 0, *max*{ *min*{ q 3(*T*1−*T*2) 2+(*F*1−*F*2) 2 , |2(*T*1−*T*2)−(*F*1−*F*2)|/3} {*max*{ q 3(*T*1−*T*2) 2+(*F*1−*F*2) 2 , |2(*T*1−*T*2)−(*F*1−*F*2)|/3}/2}+1 } = 1/2, *min*{ *min*{ q 3(*T*1−*T*2) 2+(*F*1−*F*2) 2 , |2(*T*1−*T*2)−(*F*1−*F*2)|/3} {*max*{ q 3(*T*1−*T*2) 2+(*F*1−*F*2) 2 , |2(*T*1−*T*2)−(*F*1−*F*2)|/3}/2}+1 } = 0, *max*{ *min*{ q 2(*T*1−*T*2) 2+(*I*1−*I*2) 2+(*F*1−*F*2) 2 , |3(*T*1−*T*2)−(*I*1−*I*2)−(*F*1−*F*2)|/5} {*max*{ q 2(*T*1−*T*2) 2+(*I*1−*I*2) 2+(*F*1−*F*2) 2 , |3(*T*1−*T*2)−(*I*1−*I*2)−(*F*1−*F*2)|/5}/2}+1 } = 1/2, *min*{ *min*{ q 2(*T*1−*T*2) 2+(*I*1−*I*2) 2+(*F*1−*F*2) 2 , |3(*T*1−*T*2)−(*I*1−*I*2)−(*F*1−*F*2)|/5} {*max*{ q 2(*T*1−*T*2) 2+(*I*1−*I*2) 2+(*F*1−*F*2) 2 , |3(*T*1−*T*2)−(*I*1−*I*2)−(*F*1−*F*2)|/5}/2}+1 } = 0.

Therefore,

$$\begin{array}{rcl} \min \{ \mathbb{S}\_{\text{N}}(A\_1, A\_2) \} &=& 1 - 2/3(1/2 + 1/2 + 1/2) &=& 1 - 1 = 0, \\ \max \{ \mathbb{S}\_{\text{N}}(A\_1, A\_2) \} &=& 1 - 2/3(0 + 0 + 0) &=& 1 - 0 = 1. \end{array}$$

Hence, 0 ≤ *SN*(*A*1, *A*2) ≤1.

**(ii)**

*SN*(*A*1, *A*2) = 1 − (2/3)[ *min*{ q 3(*T*1−*T*2) 2+(*I*1−*I*2) 2 , |2(*T*1−*T*2)−(*I*1−*I*2)|/3} {*max*{ q 3(*T*1−*T*2) 2+(*I*1−*I*2) 2 , |2(*T*1−*T*2)−(*I*1−*I*2)|/3}/2}+1 + *min*{ q 3(*T*1−*T*2) 2+(*F*1−*F*2) 2 , |2(*T*1−*T*2)−(*F*1−*F*2)|/3} {*max*{ q 3(*T*1−*T*2) 2+(*F*1−*F*2) 2 , |2(*T*1−*T*2)−(*F*1−*F*2)|/3}/2}+1 + *min*{ q 2(*T*1−*T*2) 2+(*I*1−*I*2) 2+(*F*1−*F*2) 2 , |3(*T*1−*T*2)−(*I*1−*I*2)−(*F*1−*F*2)|/5} {*max*{ q 2(*T*1−*T*2) 2+(*I*1−*I*2) 2+(*F*1−*F*2) 2 , |3(*T*1−*T*2)−(*I*1−*I*2)−(*F*1−*F*2)|/5}/2}+1 ] = 1 − 2/3. { *min*{ q 3(*T*2−*T*1 ) 2+(*I*2−*I*1 ) 2 , |2(*T*2−*T*1 )−(*I*2−*I*1 )|/3} {*max*{ q 3(*T*2−*T*1 ) 2+(*I*2−*I*1 ) 2 , |2(*T*2−*T*1 )−(*I*2−*I*1 )|/3}/2}+1 + *min*{ q 3(*T*2−*T*12) 2+(*F*2−*F*1 ) 2 , |2(*T*2−*T*1 )−(*F*2−*F*1 )|/3} {*max*{ q 3(*T*2−*T*1 ) 2+(*F*2−*F*1 ) 2 , |2(*T*2−*T*1 )−(*F*2−*F*1 )|/3}/2}+1 + *min*{ q 2(*T*2−*T*1 ) 2+(*I*2−*I*1 ) 2+(*F*2−*F*1 ) 2 , |3(*T*2−*T*1 )−(*I*2−*I*1 )−(*F*2−*F*1 )|/5} {*max*{ q 2(*T*2−*T*1 ) 2+(*I*2−*I*1 ) 2+(*F*2−*F*1 ) 2 , |3(*T*2−*T*1 )−(*I*2−*I*1 )−(*F*2−*F*1 )|/5}/2}+1 } = *SN*(*A*2 , *A*1).

#### **(iii)** We assume that

$$\begin{split} S\_N(A\_1, A\_2) &= 1 - (2/3) \lceil \frac{\min\{\sqrt{3(T\_1 - T\_2)^2 + (I\_1 - I\_2)^2}, \left| 2(T\_1 - T\_2) - (I\_1 - I\_2) \right|/3\}}{\lceil \max\{\sqrt{3(T\_1 - T\_2)^2 + (I\_1 - I\_2)^2}, \left| 2(T\_1 - T\_2) - (I\_1 - I\_2) \right|/3\}/2 \rceil + 1} \\ &+ \frac{\min\{\sqrt{3(T\_1 - T\_2)^2 + (F\_1 - F\_2)^2}, \left| 2(T\_1 - T\_2) - (F\_1 - F\_2) \right|/3\}}{\lceil \max\{\sqrt{3(T\_1 - T\_2)^2 + (F\_1 - F\_2)^2}, \left| 2(T\_1 - T\_2) - (F\_1 - F\_2) \right|/3\}/2 \} + 1} \\ &+ \frac{\min\{\sqrt{2(T\_1 - T\_2)^2 + (I\_1 - I\_2)^2}, \left| 2(T\_1 - T\_2) - (I\_1 - I\_2) - (F\_1 - F\_2) \right|/3\}/2 \}}{\lceil \max\{\sqrt{2(T\_1 - T\_2)^2 + (I\_1 - I\_2)^2}, \left| 3(T\_1 - T\_2) - (I\_1 - I\_2) - (F\_1 - F\_2) \right|/3\} + 1} \end{split}$$

Therefore,

$$\begin{split} -\left(\mathsf{Z}\left/\mathsf{Z}\right)\left[\frac{\min\{\sqrt{3(T\_{1}-T\_{2})^{2}+(I\_{1}-I\_{2})^{2}},\left|2(T\_{1}-T\_{2})-(I\_{1}-I\_{2})\right|/3\right\}}{\left|\max\{\sqrt{3(T\_{1}-T\_{2})^{2}+(I\_{1}-I\_{2})^{2}},\left|2(T\_{1}-T\_{2})-(I\_{1}-I\_{2})\right|/3\right\}+1}\right. \\ +\left.\frac{\min\{\sqrt{3(T\_{1}-T\_{2})^{2}+(F\_{1}-F\_{2})^{2}},\left|2(T\_{1}-T\_{2})-(F\_{1}-F\_{2})\right|/3\}}{\left|\max\{\sqrt{3(T\_{1}-T\_{2})^{2}+(F\_{1}-F\_{2})^{2}},\left|2(T\_{1}-T\_{2})-(F\_{1}-F\_{2})\right|/3\}/2\right. \\ +\left.\frac{\min\{\sqrt{2(T\_{1}-T\_{2})^{2}+(I\_{1}-I\_{2})^{2}}+(F\_{1}-F\_{2})^{2},\left|3(T\_{1}-T\_{2})-(I\_{1}-I\_{2})-(F\_{1}-F\_{2})\right|/5\}\right. \\ +\left.\frac{\min\{\sqrt{2(T\_{1}-T\_{2})^{2}+(I\_{1}-I\_{2})^{2}}+(F\_{1}-F\_{2})^{2},\left|3(T\_{1}-T\_{2})-(I\_{1}-I\_{2})-(F\_{1}-F\_{2})\right|/5\}\right. \end{split}$$

So,

$$\begin{array}{c} \min\{\sqrt{3(T\_1 - T\_2)^2 + (I\_1 - I\_2)^2}, \left|2(T\_1 - T\_2) - (I\_1 - I\_2)\right|/3\rangle\\ \max\{\sqrt{3(T\_1 - T\_2)^2 + (I\_1 - I\_2)^2}, \left|2(T\_1 - T\_2) - (I\_1 - I\_2)\right|/3\rangle/2\rangle + 1\\ \min\{\sqrt{3(T\_1 - T\_2)^2 + (F\_1 - F\_2)^2}, \left|2(T\_1 - T\_2) - (F\_1 - F\_2)\right|/3\rangle\}}{\max\{\sqrt{3(T\_1 - T\_2)^2 + (F\_1 - F\_2)^2}, \left|2(T\_1 - T\_2) - (F\_1 - F\_2)\right|/3\rangle/2\} + 1} \ = & 0,\\ \min\{\sqrt{2(T\_1 - T\_2)^2 + (I\_1 - I\_2)^2}, \left|3(T\_1 - T\_2) - (I\_1 - I\_2) - (I\_1 - I\_2) - (F\_1 - F\_2)\right|/5\rangle} \\ \max\{\sqrt{2(T\_1 - T\_2)^2 + (I\_1 - I\_2)^2}, \left|3(T\_1 - T\_2) - (I\_1 - I\_2) - (F\_1 - F\_2)\right|/5\rangle\} + 1 \end{array}$$

Therefore,

$$\begin{array}{c|c} \min\{\sqrt{3(T\_1 - T\_2)^2 + (I\_1 - I\_2)^2}, & \left|2(T\_1 - T\_2) - (I\_1 - I\_2)\right|/3\rangle = 0, \\\min\{\sqrt{3(T\_1 - T\_2)^2 + (F\_1 - F\_2)^2}, & \left|2(T\_1 - T\_2) - (F\_1 - F\_2)\right|/3\rangle = 0, \\\min\{\sqrt{2(T\_1 - T\_2)^2 + (I\_1 - I\_2)^2 + (F\_1 - F\_2)^2}, & \left|3(T\_1 - T\_2) - (I\_1 - I\_2) - (F\_1 - F\_2)\right|/5\rangle = 0. \end{array}$$

Now, we write all the cases that can make these statements 0 one-by-one.

(a) We assume that

$$\sqrt{2\left(T\_1 - T\_2\right)^2 + \left(I\_1 - I\_2\right)^2 + \left(F\_1 - F\_2\right)^2} = \ 0. \tag{1}$$

Therefore, it is

$$\left(2(T\_1 - T\_2)^2 + \left(I\_1 - I\_2\right)^2 + \left(F\_1 - F\_2\right)^2\right) = 0.1$$

Here, it is obtained that *T*1 − *T*2 = 0, *I*1 − *I*2 = 0 and *F*1 − *F*2 = 0. Hence, we get *T*1 = *T*2, *I*1 = *I*2 and *F*1 = *F*2. By Definition 4, *A*1 = *A*2.

(b) Let

$$\sqrt{\Im(T\_1 - T\_2)^2 + \left(I\_1 - I\_2\right)^2} = \ 0,\tag{2}$$

$$
\sqrt{3(T\_1 - T\_2)^2 + (F\_1 - F\_2)^2} = 0.
\tag{3}
$$

By (2) and (3), we obtain 3(*T*1 − *T*2) 2 + (*I*1 − *I*2) 2 = 0 and 3(*T*1 − *T*2) 2 + (*F*1 − *F*2) 2 = 0. Therefore, we obtain *T*1 − *T*2 = 0, *I*1 − *I*2 = 0 and *F*1 − *F*2 = 0. Hence, we obtain *T*1 = *T*2, *I*1 = *I*2

and *F*1 = *F*2. By Definition 4, *A*1 = *A*2.

(c) We assume that

$$\sqrt{\Im(T\_1 - T\_2)^2 + \left(I\_1 - I\_2\right)^2} = \begin{array}{c} 0, \\ \text{(4)} \end{array} \tag{4}$$

$$\left| 2(T\_1 - T\_2) - (F\_1 - F\_2) \right| \\
= 0,\tag{5}$$

By (4), we have

$$T\_1 - T\_2 = 0 \text{ and } I\_1 - I\_2 = \begin{array}{c} 0. \\ \end{array} \tag{6}$$

Hence, we obtain that *F*1 − *F*2 = 0 by (5) and (6). Hence, *T*1 = *T*2, *I*1 = *I*2 and *F*1 = *F*2. By Definition 4, we get *A*1 = *A*2. (d) We assume that

$$\left| 2(T\_1 - T\_2) - (I\_1 - I\_2) \right| / 3 = 0,\tag{7}$$

$$\left| 2(T\_1 - T\_2) - (F\_1 - F\_2) \right| / 3 = 0,\tag{8}$$

$$\left| \Im(T\_1 - T\_2) - (I\_1 - I\_2) - (F\_1 - F\_2) \right| / 5 = 0. \tag{9}$$

By (7) and (8), we obtain

$$T\_1 - T\_2 = I\_1 - I\_2 = F\_1 - F\_2.\tag{10}$$

Hence, *T*1 − *T*2 = 0 by (9) and (10).

Hence, *T*1 = *T*2, *I*1 = *I*2 and *F*1 = *F*2. By Definition 4, *A*1 = *A*2.

We assume that *A*1 = *A*2. Therefore, by Definition 4, it is *T*1 = *T*2, *I*1 = *I*2, *F*1 = *F*2. Because of this, we have

$$\begin{split} S\_{N}(A\_{1},A\_{2}) &= \ 1 - (2/3) \lfloor \frac{\min\{\sqrt{3(T\_{1}-T\_{2})^{2}+(I\_{1}-I\_{2})^{2}}, \left|2(T\_{1}-T\_{2})-(I\_{1}-I\_{2})\right|/3\}}{\lfloor \max\{\sqrt{3(T\_{1}-T\_{2})^{2}+(I\_{1}-I\_{2})^{2}}, \left|2(T\_{1}-T\_{2})-(I\_{1}-I\_{2})\right|/3\} \rangle} \\ &+ \frac{\min\{\sqrt{3(T\_{1}-T\_{2})^{2}+(F\_{1}-F\_{2})^{2}}, \left|2(T\_{1}-T\_{2})-(F\_{1}-F\_{2})\right|/3\}}{\lfloor \max\{\sqrt{3(T\_{1}-T\_{2})^{2}+(F\_{1}-F\_{2})^{2}}, \left|2(T\_{1}-T\_{2})-(F\_{1}-F\_{2})\right|/3\}/2\} \\ &+ \frac{\min\{\sqrt{2(T\_{1}-T\_{2})^{2}+(I\_{1}-I\_{2})^{2}}+(F\_{1}-F\_{2})^{2}, \left|3(T\_{1}-T\_{2})-(I\_{1}-I\_{2})-(F\_{1}-F\_{2})\right|/3\}}{\lfloor \max\{\sqrt{2(T\_{1}-T\_{2})^{2}+(I\_{1}-I\_{2})^{2}}+(F\_{1}-F\_{2})^{2}, \left|3(T\_{1}-T\_{2})-(I\_{1}-I\_{2})-(F\_{1}-F\_{2})\right|/3\}}\ = \ 0 \end{split}$$

*Symmetry* **2020**, *12*, 1166

**(iv)** We assume that *A*1 ≤ *A*2 ≤ *A*3. By Definition 5, it is *T*1 ≤ *T*2 ≤ *T*3, *I*1 ≥ *I*2 ≥ *I*3, *F*1 ≥ *F*2 ≥ *F*3. Hence, we obtain that

$$\begin{split} \min & \left| \sqrt{3(T\_1 - T\_2)^2 + (I\_1 - I\_2)^2}, \left| 2(T\_1 - T\_2) - (I\_1 - I\_2) \right| / 3 \right| \le 1, \\ & \max \{ \sqrt{3(T\_1 - T\_2)^2 + (I\_1 - I\_2)^2}, \left| 2(T\_1 - T\_2) - (I\_1 - I\_2) \right| / 3 \} / 2 \} \le 1, \\ & \min \{ \sqrt{3(T\_1 - T\_3)^2 + (I\_1 - I\_3)^2}, \left| 2(T\_1 - T\_3) - (I\_1 - I\_3) \right| / 3 \} \le 1, \\ & \max \{ \sqrt{3(T\_1 - T\_3)^2 + (I\_1 - I\_3)^2}, \left| 2(T\_1 - T\_3) - (I\_1 - I\_3) \right| / 3 \} / 2 \} \le 1. \end{split}$$

Therefore, we have

$$\begin{array}{c|c} \min\{\sqrt{3(T\_1 - T\_2)^2 + (I\_1 - I\_2)^2}, & \left|2(T\_1 - T\_2) - (I\_1 - I\_2)\right|/3\rangle \le\\ \min\{\sqrt{3(T\_1 - T\_3)^2 + (I\_1 - I\_3)^2}, & \left|2(T\_1 - T\_3) - (I\_1 - I\_3)\right|/3\rangle, \\ \max\{\sqrt{3(T\_1 - T\_2)^2 + (I\_1 - I\_2)^2}, & \left|2(T\_1 - T\_2) - (I\_1 - I\_2)\right|/3\rangle/2\rangle \le\\ \max\{\sqrt{3(T\_1 - T\_3)^2 + (I\_1 - I\_3)^2}, & \left|2(T\_1 - T\_3) - (I\_1 - I\_3)\right|/3\rangle/2\}. \end{array}$$

Hence,

$$\frac{\min\{\sqrt{3(T\_1 - T\_2)^2 + (I\_1 - I\_2)^2}, \left| 2(T\_1 - T\_2) - (I\_1 - I\_2) \right|/3\}}{\max\{\sqrt{3(T\_1 - T\_2)^2 + (I\_1 - I\_2)^2}, \left| 2(T\_1 - T\_2) - (I\_1 - I\_2) \right|/3\}/2\} + 1} \le$$

$$\frac{\min\{\sqrt{3(T\_1 - T\_2)^2 + (I\_1 - I\_2)^2}, \left| 2(T\_1 - T\_2) - (I\_1 - I\_2) \right|/3\}}{\max\{\sqrt{3(T\_1 - T\_2)^2 + (I\_1 - I\_2)^2}, \left| 2(T\_1 - T\_2) - (I\_1 - I\_2) \right|/3\}/2\} + 1} \tag{11}$$

In addition,

$$\begin{aligned} \min\{\sqrt{3(T\_1 - T\_2)^2 + (F\_1 - F\_2)^2}, \left| 2(T\_1 - T\_2) - (F\_1 - F\_2) \right|/3 \} &\le 1, \\\max\{\sqrt{3(T\_1 - T\_2)^2 + (F\_1 - F\_2)^2}, \left| 2(T\_1 - T\_2) - (F\_1 - F\_2) \right|/3 \}/2 &\le 1, \\\min\{\sqrt{3(T\_1 - T\_3)^2 + (F\_1 - F\_3)^2}, \left| 2(T\_1 - T\_3) - (F\_1 - F\_3) \right|/3 \} &\le 1, \\\max\{\sqrt{3(T\_1 - T\_3)^2 + (F\_1 - F\_3)^2}, \left| 2(T\_1 - T\_3) - (F\_1 - F\_3) \right|/3 \}/2 &\le 1. \end{aligned}$$

Therefore, we obtain that

$$\begin{array}{l} \min\{\sqrt{3(T\_1 - T\_2)^2 + (F\_1 - F\_2)^2}, \sqrt{2(T\_1 - T\_2) - (F\_1 - F\_2)} \big| / 3\} \le\\ \min\{\sqrt{3(T\_1 - T\_3)^2 + (F\_1 - F\_3)^2}, \sqrt{2(T\_1 - T\_3) - (F\_1 - F\_3)} \big| / 3\},\\ \max\{\sqrt{3(T\_1 - T\_2)^2 + (F\_1 - F\_2)^2}, \sqrt{2(T\_1 - T\_2) - (F\_1 - F\_2)} \big| / 3\} \le\\ \max\{\sqrt{3(T\_1 - T\_3)^2 + (F\_1 - F\_3)^2}, \left| 2(T\_1 - T\_3) - (F\_1 - F\_3) \right| / 3\} / 2\}. \end{array}$$

Hence,

$$\frac{\min\{\sqrt{3(T\_1 - T\_2)^2 + (F\_1 - F\_2)^2}, \left| 2(T\_1 - T\_2) - (F\_1 - F\_2) \right|/3\}}{\max\{\sqrt{3(T\_1 - T\_2)^2 + (F\_1 - F\_2)^2}, \left| 2(T\_1 - T\_2) - (F\_1 - F\_2) \right|/3\}/2\} \le$$

$$\frac{\min\{\sqrt{3(T\_1 - T\_2)^2 + (F\_1 - F\_2)^2}, \left| 2(T\_1 - T\_2) - (F\_1 - F\_2) \right|/3\}}{\max\{\sqrt{3(T\_1 - T\_2)^2 + (F\_1 - F\_2)^2}, \left| 2(T\_1 - T\_2) - (F\_1 - F\_2) \right|/3\}/2\} . \tag{12}$$

In addition,

$$\begin{array}{l} \min\{\sqrt{2(T\_1 - T\_2)^2 + (I\_1 - I\_2)^2 + (F\_1 - F\_2)^2}, \left| 3(T\_1 - T\_2) - (I\_1 - I\_2) - (F\_1 - F\_2) \right| / 5\} \le 1\\\max\{\sqrt{2(T\_1 - T\_2)^2 + (I\_1 - I\_2)^2 + (F\_1 - F\_2)^2}, \left| 3(T\_1 - T\_2) - (I\_1 - I\_2) - (F\_1 - F\_2) \right| / 5\} / 3 \le 1, \\\min\{\sqrt{2(T\_1 - T\_3)^2 + (I\_1 - I\_3)^2 + (F\_1 - F\_3)^2}, \left| 3(T\_1 - T\_3) - (I\_1 - I\_3) - (F\_1 - F\_3) \right| / 5\} \le 1\\\max\{\sqrt{2(T\_1 - T\_3)^2 + (I\_1 - I\_3)^2 + (F\_1 - F\_3)^2}, \left| 3(T\_1 - T\_3) - (I\_1 - I\_3) - (F\_1 - F\_3) \right| / 5\} / 3 \le 1, \end{array}$$

Hence, we have

$$\begin{array}{c|c} \min\{\sqrt{2(T\_1 - T\_2)^2 + (I\_1 - I\_2)^2 + (F\_1 - F\_2)^2}, & \left|3(T\_1 - T\_2) - (I\_1 - I\_2) - (F\_1 - F\_2)\right|/5\rangle \le\\ \min\{\sqrt{2(T\_1 - T\_3)^2 + (I\_1 - I\_3)^2 + (F\_1 - F\_3)^2}, & \left|3(T\_1 - T\_3) - (I\_1 - I\_3) - (F\_1 - F\_3)\right|/5\rangle,\\ \max\{\sqrt{2(T\_1 - T\_2)^2 + (I\_1 - I\_2)^2 + (F\_1 - F\_2)^2}, & \left|3(T\_1 - T\_2) - (I\_1 - I\_2) - (F\_1 - F\_2)\right|/5\rangle/2 \le 1, \\ \max\{\sqrt{2(T\_1 - T\_3)^2 + (I\_1 - I\_3)^2 + (F\_1 - F\_3)^2}, & \left|3(T\_1 - T\_3) - (I\_1 - I\_3) - (F\_1 - F\_3)\right|/5\rangle/2. \end{array}$$

Hence,

$$\begin{split} \frac{\min\{\sqrt{2(T\_1 - T\_2)^2 + (I\_1 - I\_2)^2 + (F\_1 - F\_2)^2}, \left| 3(T\_1 - T\_2) - (I\_1 - I\_2) - (F\_1 - F\_2) \right|/5\rangle}{\max\{\sqrt{2(T\_1 - T\_2)^2 + (I\_1 - I\_2)^2 + (F\_1 - F\_2)^2}, \left| 3(T\_1 - T\_2) - (I\_1 - I\_2) - (F\_1 - F\_2) \right|/5\rangle/2\} + 1} \\ &+ \frac{\min\{\sqrt{2(T\_1 - T\_3)^2 + (I\_1 - I\_3)^2 + (F\_1 - F\_3)^2}, \left| 3(T\_1 - T\_3) - (I\_1 - I\_3) - (F\_1 - F\_3) \right|/5\rangle}{\max\{\sqrt{2(T\_1 - T\_3)^2 + (I\_1 - I\_3)^2 + (F\_1 - F\_3)^2}, \left| 3(T\_1 - T\_3) - (I\_1 - I\_3) - (F\_1 - F\_3) \right|/5\rangle/2\}}. \end{split} \tag{13}$$

By (11), (12) and (13), we have

$$\begin{split} & \quad \mathbf{1} - (2/3) \left[ \frac{\min\{\sqrt{3(T\_{1}-T\_{3})^{2}+(I\_{1}-I\_{3})^{2}}, \left[2(T\_{1}-T\_{3})-(I\_{1}-I\_{3})\right]/3\right]}{\max\{\sqrt{3(T\_{1}-T\_{3})^{2}+(I\_{1}-I\_{3})^{2}}, \left[2(T\_{1}-T\_{3})-(I\_{1}-I\_{3})\right]/3//2\} + 1} \\ & \quad + \frac{\min\{\sqrt{3(T\_{1}-T\_{3})^{2}+(F\_{1}-F\_{3})^{2}}, \left[2(T\_{1}-T\_{3})-(F\_{1}-F\_{3})\right]/3\right\}}{\max\{\sqrt{3(T\_{1}-T\_{3})^{2}+(F\_{1}-F\_{3})^{2}}, \left[2(T\_{1}-T\_{3})-(F\_{1}-F\_{3})\right]/3//2\} + 1} \\ & \quad + \frac{\min\{\sqrt{2(T\_{1}-T\_{3})^{2}+(I\_{1}-I\_{3})^{2}}, \left[2(T\_{1}-T\_{3})-(F\_{1}-F\_{3})\right]/3, \left[2(T\_{1}-T\_{3})-(F\_{1}-I\_{3})\right]/5\} |}{\max\{\sqrt{2(T\_{1}-T\_{3})^{2}+(I\_{1}-I\_{3})^{2}}, \left[3(T\_{1}-T\_{3})-(I\_{1}-I\_{3})-(F\_{1}-F\_{3})\right]/5\}/2\} \end{split} \leq \\ & \begin{split} 1 - (2/3) \left[ \frac{\min\{\sqrt{3(T\_{1}-T\_{2})^{2}+(I\_{1}-I\_{2})^{2}}, \left[3(T\_{1}-T\_{3})-(F\_{1}-F\_{3})\right]/5\$$

Hence, we get *SN*(*A*1, *A*3) ≤ *SN*(*A*1, *A*2) as desired.

#### **7. Decision-Making Applications for Neutrosophic Modeling of Talcott Parsons's Action**

In this section, we give an algorithm for applications that allow us to find the ideal society in the grand theory of action of Parsons by taking advantage of the similarity measure in Definition 9. In addition, we give a numeric example to this algorithm.

#### *7.1. Algorithm*

**1. Step:** To find out which societies are closer to the ideal society, the criteria to be considered are determined. The criteria of the ideal society in the grand theory of action of Parsons [17] are taken as below:

Let the set of these criteria be C = {*c*1, *c*2, . . . , *c*5}.

**2. Step:** Let the set of weighted values of the criteria be W = {*w*1, *w*2, . . . , *wm*} and let the weighted values be taken as below:

the weighted value of the criterion *c*1 is *w*1, the weighted value of the criterion *c*2 is *w*2, the weighted value of the criterion *c*3 is *w*3, the weighted value of the criterion *c*4 is *w*4 and the weighted value of the criterion *c*5 is *w*5.

> In addition, it must be P*m i* = 1 *wi* = 1 and *w*1, *w*2, . . . , *wm* ∈ [0,1].

In this study, we will take the weighted value of each criterion as equal. If necessary, different weighted values can be selected for each criterion.

**3. Step:** Each society that will be taken into ideal society assessment should be evaluated by sociologists determined as a single-valued neutrosophic number. Let T = {*t*1, *t*2, . . . , *tn*} be set of societies. Symbolic representation of societies as single-valued neutrosophic sets are denoted as:

*t*1= {*c*1:< *Tt*1 (*c*1 ) , *It*1 (*c*1 ) , *Ft*1 (*c*1 )>, *c*2:< *Tt*1 (*c*2) , *It*1 (*c*2) , *Ft*1 (*c*2)>, . . . , *c*5:< *Tt*1 (*c*5) , *It*1 (*c*5) , *Ft*1 (*c*5)>; *ci* ∈ C (i = 1, 2, . . . , 5)}, *t*2= {*c*1:< *Tt*2(*c*1 ) , *It*2(*c*1 ) , *Ft*2(*c*1 )>, *c*2:< *Tt*2(*c*2) , *It*2(*c*2) , *Ft*2(*c*2)>, . . . , *c*5:< *Tt*2(*c*5) , *It*2(*c*5) , *Ft*2(*c*5)>; *ci* ∈ C (i = 1, 2, . . . , 5)}, *t*3= {*c*1:< *Tt*3(*c*1 ) , *It*3(*c*1 ) , *Ft*3(*c*1 )>, *c*2:< *Tt*3(*c*2) , *It*3(*c*2) , *Ft*3(*c*2)>, . . . , *c*5:< *Tt*3(*c*5) , *It*3(*c*5) , *Ft*3(*c*5)>; *ci* ∈ C (i = 1, 2, . . . , 5)}, *tn*= {*c*1:< *Ttn*(*c*1 ) , *Itn*(*c*1 ) , *Ftn*(*c*1 )>, *c*2:< *Ttn*(*c*2) , *Itn*(*c*2) , *Ftn*(*c*2)>, . . . , *c*5:< *Ttn*(*c*5) , *Itn*(*c*5) , *Ftn*(*c*5)>; *ci* ∈ C (i = 1, 2, . . . , 5)}.

Here, *c*1, *c*2, . . . , *c*5 are the criteria in Step 1. Thus, each society will be obtained as a single-valued neutrosophic number according to the given criteria.

**4. Step**: To compare how close the societies are to ideal society in the theory of Parsons, an imaginary perfect society is determined. Perfect society under the similarity measure we have obtained should be as

I = {*c*1 :< 1, 0, 0 >, *x*2 :< 1, 0, 0 >, . . . , *c*5 : < 1, 0, 0 >; *ci* ∈ C (i = 1, 2, . . . , 5) }.

Hence, we will accept the existence of an imaginary society that includes 100% truth, 0% uncertainty and 0% falsity according to each criterion.

**5. Step:** We express the societies given as a single-valued neutrosophic set in step 3 in a table according to criteria. Thus, we will obtain Table 3.

**Table 3.** Criteria table of societies.

**6. Step:** We will process each criterion values given for each society separately and each criterion values of the perfect society I in Step 4 separately with similarity measure. Hence, we will obtain Table 4.

**Table 4.** Similarity table for each social criteria to perfect society criteria.

**7. Step:** In this step, we will obtain a weighted similarity table (Table 5).

**Table 5.** Weighted similarity table for each social criteria to perfect society criteria.

In this study, this step is not needed since we take the same weighted value of each criterion. More precisely, Tables 4 and 5 will be the same since the weighted values are equal. This step can be used if necessary.

#### **8. Step:**

In this last step, we will obtain a similarity value table (Table 6) by applying *SNk*(*tk* , I) = P*n i* = 1 *wi* .*SN*(*Ici* , *tkc i* ).

**Table 6.** Similarity value table of societies to the perfect society.

See Figure 1.

**Figure 1.** Diagram of the algorithm.

*7.2. Numeric Example*

Using the steps in Section 7.1, we show how close the 4 societies are to the ideal society. **1. Step:** Let the criteria of an ideal society in the theory of Parsons be as it is in Step 1 of Section 7.1;

ହ Let C = {*c*1, *c*2, . . . , *c*5} be the set of criteria.

ଵ ଶ ହ ଵ ଶ **2. Step:** In this example, we will take the weight values of each criterion equal so that *w*1 = *w*2 = . . . = *w*5 = 0.2.

ହ ଵ ଶ ଷ ସ **3. Step:** Let the set of societies be T = {*t*1, *t*2, *t*3, *t*4}. We assume that the single-valued neutrosophic set with evaluation of societies by sociologists according to the criteria in Step 1 will be as below:

ଵ ଵ ଶ ଷ ସ ହ ଶ ଵ ଶ ଷ ସ ହ *t*1= {*c*1:< 0.6, 0.2, 0.1 >, *c*2:< 0.7, 0.2, 0.1 >, *c*3:< 0.4, 0.1, 0.2 >, *c*4:< 0.8, 0.1, 0 >, *c*5 :< 0.5, 0.1, 0.2 >} *t*2= {*c*1:< 0.5, 0.2, 0.3 >, *c*2:< 0.6, 0.1, 0.3 >, *c*3:< 0.8, 0.1, 0.2 >, *c*4:< 0.4, 0.1, 0.4 >, *c*5 :< 0.9, 0, 0.1 >} *t*3= {*c*1:< 0.5, 0.2, 0.1 >, *c*2:< 0.8, 0.1, 0.1 >, *c*3:< 0.8, 0.1, 0 >, *c*4:< 0.7, 0.2, 0.1 >, *c*5 :< 0.7, 0.2, 0.3 >} *t*4= {*c*1:< 0.7, 0.2, 0.1 >, *c*2:< 0.6, 0.2, 0.2 >, *c*3:< 0.7, 0.2, 0.1 >, *c*4:< 0.7, 0.1, 0.2 >, *c*5 :< 0.8, 0.1, 0.1 >}

ଷ ଵ ଶ ଷ ସ ହ

#### **4. Step:** Let the dream perfect society that we compare societies be

I = {*c*1 :< 1, 0, 0 >, *c*2 :< 1, 0, 0 >, *c*3 : < 1, 0, 0 >, *c*3 :< 1, 0, 0 >, *c*4 :< 1, 0, 0 >, *c*5 : < 1, 0, 0 >}.

**5. Step:** Let us express the societies as a single-valued neutrosophic set in Step 3 in Table 7.

**Table 7.** The criteria table of societies.

**6. Step:** Using the similarity measure, we obtain the similarity table (Table 8) which is the similarity of the criteria of societies to the criteria of the perfect society.

**Table 8.** The similarity table of the criteria of societies to the criteria of the perfect society.

**7. Step:** In this example, there is no need to make any changes in Table 8 since we take the weighted value of each criterion as equal.

**8. Step:** In this step, we obtain similarity values of the societies in Table 8 to the perfect society.

Now, we obtain the similarity values of the societies in Table 9 and we obtain Table 10 by dividing the values in Table 9 by 5, taking the weighted values as equal for each society on 5 criteria and, hence, getting the results in the range [0,1].

**Table 9.** The similarity value table of the societies to the perfect society.

**Table 10.** The similarity rate of the societies to the perfect society.

In addition, the similarity value of each society to the perfect society in Table 10 is obtained. The result of the evaluation is given. Thus, societies closest to the perfect society are obtained as *t***3**, *t***4**, *t***2** and *t***1** respectively.

#### **8. Sensitivity Analysis**

In 7.1 Numeric example, we take the weighted values of criteria W = {*w*1, *w*2, . . . , *wm*} equal such that

the weighted value of *c*1 criteria *w*1 = 0.2 the weighted value of *c*2 criteria *w*2 = 0.2 the weighted value of *c*3 criteria *w*3 = 0.2 the weighted value of *c*4 criteria *w*4 = 0.2 the weighted value of *c*5 criteria *w*5 = 0.2

Thus, societies closest to the perfect society are obtained as *t***3**, *t***4**, *t***2**, *t***1** respectively.

(a) If we take the W = {*w*1 = 0.1, *w*2 = 0.3, *w*3 = 0.2, *w*4 = 0.2, *w*5 = 0.2}, then we obtain that societies closest to the perfect society are obtained as *t*3, *t*4, *t*2 and *t*1 respectively (Table 11).

**Table 11.** The similarity rate of the societies to the perfect society for W = {*w*1 = 0.1, *w*2 = 0.3, *w*3 = 0.2, *w*4 = 0.2, *w*5 = 0.2}.

Thus, we obtain the same result with the Numeric Example 7.1.

(b) If we take the W = {*w*1 = 0.2, *w*2 = 0.2, *w*3 = 0.3, *w*4 = 0.1, *w*5 = 0.2}, then we obtain that societies closest to the perfect society are obtained as *t*3, *t*4, *t*2 and *t*1 respectively respectively (Table 12).

**Table 12.** The similarity rate of the societies to the perfect society for W = {*w*1 = 0.2, *w*2 = 0.2, *w*3 = 0.3, *w*4 = 0.1, *w*5 = 0.2}.

Thus, we obtain same result with Numeric Example 7.1.

(c) If we take the W = {*w*1 = 0.3, *w*2 = 0.1, *w*3 = 0.2, *w*4 = 0.2, *w*5 = 0.2}, then we obtain that societies closest to the perfect society are obtained as *t*4, *t*3, *t*2 and *t*1 respectively respectively (Table 13).

**Table 13.** The similarity rate of the societies to the perfect society for W = {*w*1 = 0.3, *w*2 = 0.1, *w*3 = 0.2, *w*4 = 0.2, *w*5 = 0.2}.

Thus, we obtain a different result from Numeric Example 7.1.

(d) If we take the W = {*w*1 = 0.2, *w*2 = 0.2, *w*3 = 0.1, *w*4 = 0.3, *w*5 = 0.2}, then we obtain that societies closest to the perfect society are obtained as *t*4, *t*1, *t*2 and *t*3 respectively respectively (Table 14).

**Table 14.** The similarity rate of the societies to the perfect society for W = {*w*1 = 0.2, *w*2 = 0.2, *w*3 = 0.1, *w*4 = 0.3, *w*5 = 0.2}.

Thus, we obtain a different result with Numeric Example 7.1.

(e) If we take the W = {*w*1 = 0.2, *w*2 = 0.2, *w*3 = 0.2, *w*4 = 0.3, *w*5 = 0.1}, then we obtain that societies closest to the perfect society are obtained as *t*3, *t*4, *t*1, and *t*2 respectively respectively (Table 15).

**Table 15.** The similarity rate of the societies to the perfect society for W = {*w*1 = 0.2, *w*2 = 0.2, *w*3 = 0.2, *w*4 = 0.3, *w*5 = 0.1}.

Thus, we obtain a different result from Numeric Example 7.1.

(f) If we take the W = {*w*1 = 0.2, *w*2 = 0.2, *w*3 = 0.2, *w*4 = 0.1, *w*5 = 0.3}, then we obtain that societies closest to the perfect society are obtained as *t*4, *t*2, *t*3, and *t*1 respectively respectively (Table 16).

**Table 16.** The similarity rate of the societies to the perfect society for W = {*w*1 = 0.2, *w*2 = 0.2, *w*3 = 0.2, *w*4 = 0.1, *w*5 = 0.3}.

Thus, we obtain a different result from Numeric Example 7.1. Now, we give results in (a), (b), (c), (d), (e) and (f) in Table 17.

**Table 17.** Ideal societies according to weighted values.

As seen in Table 17, if we take W = {*w*1 = 0.2, *w*2 = 0.2, *w*3 = 0.2, *w*4 = 0.1, *w*5 = 0.3} or W = {*w*1 = 0.1, *w*2 = 0.3, *w*3 = 0.2, *w*4 = 0.2, *w*5 = 0.2}, then we obtain same result with Numeric Example 7.1. In other cases, we obtain different results from Numeric Example 7.1.

#### **9. Study Comparison Methods**

In this section, we have compared the obtained results of the data in our Example 1 with the results of the similarity measures, Hausdorff measure [18], Hamming measure [18] and the previously defined similarity measure [16].

**1. If we use the similarity measure in Definition 6 [16] for Example 1, we obtain Table 18 as a result.**

**Table 18.** The similarity rate according to similarity measure, in Definition 6 [16], of the societies to the perfect society.

Thus, societies closest to the perfect society are obtained as *t***3**, *t***4**, *t***1** and *t***2** respectively according to similarity measure in Definition 6 [16].

#### **2. If we use the Hausdor**ff **measure [18] for Example 1, we obtain Table 19 as a result.**

**Table 19.** The similarity rate according to Hausdorff measure, in Definition 7 [18], of the societies to the perfect society.

Thus, societies closest to the perfect society are obtained as *t***3** = *t***4**, *t***2** and *t***1** respectively according to Hausdorff similarity measure in Definition 7 [18].

#### **3. If we use the Hamming measure [18] for Example 1, we obtain Table 20 as a result.**

**Table 20.** The similarity rate according to Hamming similarity measure, in Definition 8 [18], of the societies to the perfect society.

Thus, societies closest to the perfect society are obtained as *t***3**, *t***4**, *t***1** and *t***2** respectively according to Hamming similarity measure, in Definition 8 [18].

As a result,

according to our similarity measure, the perfect society is obtained as *t***3**, *t***4**, *t***2**, *t***1** respectively; according to similarity measure [16], the perfect society is obtained as *t***3**, *t***4**, *t***1**, *t***2** respectively; according to Hausdorff measure [18], the perfect society is obtained as *t***3** = *t***4**, *t***2**, *t***1** respectively; according to Hamming measure [18], the perfect society is obtained as *t***3**, *t***4**, *t***1**, *t***2** respectively.

#### **10. Discussions**

In this study, we explained the grand theory of action of Parsons, which has an important place in social theories, for the first time in neutrosociology. Thus, like all social theories, we have achieved a more effective way of dealing with uncertainties in the theory of Parsons. In addition, we have obtained a similarity measure for single-valued neutrosophic numbers. By making use of this similarity measure, we have obtained applications that allow finding the ideal society in the theory of Parsons within the theory of neutrosociology. Hence, we have added a new structure to neutrosophic theory, neutrosociology theory. In addition, by utilizing this study, other social theories can be explained in neutrosociology. Thus, the uncertainties encountered can be dealt with more easily. In addition, by using neutrosophic numbers and sets related to other social theories, new similarity measures can be obtained, and the consistency of these measures can be checked.

#### **11. Conclusions**

In Section 9, we obtained different results for the similarity measure [16]; Hausdorff measure [18]; and Hamming measure [18]. In addition, we give a comparison in Table 21.

**Author Contributions:** In this article, each author contributed equally. C.A. obtained Neutrosophic Modeling of Grand Theory of Action of Talcott Parsons; A.K. introduced similarity measure and algorithm; M. ¸S. obtained examples and organized the study. All authors have read and agreed to the published version of the manuscript.

**Funding:** This research received no external funding.

**Conflicts of Interest:** The authors declare no conflicts of interest.

#### **References**

© 2020 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## **Combination of the Single-Valued Neutrosophic Fuzzy Set and the Soft Set with Applications in Decision-Making**

**Ahmed Mostafa Khalil 1 , Dunqian Cao 2,\* , Abdelfatah Azzam 3,4 and Florentin Smarandache 5 and Wedad R. Alharbi 6**

Received: 16 July 2020; Accepted: 8 August 2020; Published: 14 August 2020

**Abstract:** In this article, we propose a novel concept of the single-valued neutrosophic fuzzy soft set by combining the single-valued neutrosophic fuzzy set and the soft set. For possible applications, five kinds of operations (e.g., subset, equal, union, intersection, and complement) on single-valued neutrosophic fuzzy soft sets are presented. Then, several theoretical operations of single-valued neutrosophic fuzzy soft sets are given. In addition, the first type for the fuzzy decision-making based on single-valued neutrosophic fuzzy soft set matrix is constructed. Finally, we present the second type by using the AND operation of the single-valued neutrosophic fuzzy soft set for fuzzy decision-making and clarify its applicability with a numerical example.

**Keywords:** single-valued neutrosophic fuzzy set; soft set; Algorithm 1; Algorithm 2; decision-making

#### **1. Introduction**

Many areas (e.g., physics, social sciences, computer sciences, and medicine) work with vague data that require fuzzy sets [1], intuitionistic fuzzy sets [2], picture fuzzy sets [3], and other mathematical tools. Molodtsov [4] presented a novel approach termed "soft set theory", which plays a very significant role in different fields. Therefore, several researchers have developed some methods and operations of soft set theory. For instance, Maji et al. [5] introduced some notions of and operations on soft sets. In addition, Maji et al. [6] gave an application of soft sets to solve fuzzy decision-making. Maji et al. [7] proposed the notion of fuzzy soft sets, followed by studies on inverse fuzzy soft sets [8], belief interval-valued soft sets [9], interval-valued intuitionistic fuzzy soft sets [10], interval-valued picture fuzzy soft sets [11], interval-valued neutrosophic soft sets [12], and generalized picture fuzzy soft sets [13].  Karaaslan and Hunu [18] defined the notion of type-2 single-valued neutrosophic sets and gave several distance measure methods: Hausdorff, Hamming, and Euclidean distances for Type-2 single-valued neutrosophic sets. Al-Quran

et al. [19] presented the notion of fuzzy parameterized complex neutrosophic soft expert sets and gave a novel approach by transforming from the complex case to the real case for decision-making. Qamar and Hassan [20] proposed a novel approach to Q-neutrosophic soft sets and studied several operations of Q-neutrosophic soft sets. Further, they generalized Q-neutrosophic soft expert sets based on uncertainty for decision-making [21]. On the other hand, Uluçay et al. [22] presented the concept of generalized neutrosophic soft expert sets and applied a novel algorithm for multiple-criteria decision-making. Zhang et al. [23] gave novel algebraic operations of totally dependent neutrosophic sets and totally dependent neutrosophic soft sets. In 2018, Smarandache [24] generalized the soft set to the hypersoft set by transforming the function *F* into a multi-argument function.

Fuzzy sets are used to tackle uncertainty using the membership grade, whereas neutrosophic sets are used to tackle uncertainty using the truth, indeterminacy, and falsity membership grades, which are considered as independent. As the motivation of this article, we present a novel notion of the single-valued neutrosophic fuzzy soft set, which can be seen as a novel single-valued neutrosophic fuzzy soft set model, which gives rise to some new concepts. Since neutrosophic fuzzy soft sets have some difficulties in dealing with some real-life problems due to the nonstandard interval of neutrosophic components, we introduce the single-valued neutrosophic fuzzy soft set (i.e., the single-valued neutrosophic set has a symmetric form, since the membership (T) and nonmembership (F) are symmetric with each other, while indeterminacy (I) is in the middle), which is considered as an instance of neutrosophic fuzzy soft sets. The structural operations (e.g., subset, equal, union, intersection, and complement) on single-valued neutrosophic fuzzy soft sets, and several fundamental properties of the five operations above are introduced. Lastly, two novel approaches (i.e., Algorithms 1 and 2) to fuzzy decision-making depending on single-valued neutrosophic fuzzy soft sets are discussed, in addition to a numerical example to show the two approaches we have developed.

The rest of this article is arranged as follows. Section 2 briefly introduces several notions related to fuzzy sets, neutrosophic sets, single-valued neutrosophic sets, neutrosophic fuzzy sets, single-valued neutrosophic fuzzy sets, soft sets, fuzzy soft sets, and neutrosophic soft sets. Section 3 discusses single-valued neutrosophic fuzzy soft sets (along with their basic operations and structural properties). Section 4 gives two algorithms for single-valued neutrosophic fuzzy soft sets for decision-making. Lastly, the conclusions are given in Section 5.

#### **2. Preliminaries**

In the following, we present a short survey of seven definitions which are necessary to this paper.

#### *2.1. Fuzzy Set*

**Definition 1** (cf. [1])**.** *Assume that X (i.e., X* = {*x*1, *x*2, ..., *xp*}) *is a set of elements and µ*(*xp*) *is a membership function of element xp* ∈ *X*. *Then*

*(1) The following mapping (called fuzzy set), is given by*

$$
\mu: \mathbf{x} \longrightarrow [0, 1]
$$

*and* [0, 1] *X is a set of whole fuzzy subset over X*. *(2) Let*

$$\mu = \left\{ \frac{\mu(\mathbf{x}\_1)}{\mathbf{x}\_1}, \frac{\mu(\mathbf{x}\_2)}{\mathbf{x}\_2}, \dots, \frac{\mu(\mathbf{x}\_p)}{\mathbf{x}\_p} \, \middle| \, \mathbf{x}\_p \in X \right\} \in [0, 1]^{X\_1}$$

*and*

$$\nu = \left\{ \frac{\nu(\mathbf{x\_1})}{\mathbf{x\_1}}, \frac{\nu(\mathbf{x\_2})}{\mathbf{x\_2}}, \dots, \frac{\nu(\mathbf{x\_p})}{\mathbf{x\_p}} \, \middle| \, \mathbf{x\_p} \in X \right\} \in [0, 1]^X.$$

*Then*

*Symmetry* **2020**, *12*, 1361

*(1) The union µ* ∪ *ν*, *is defined as*

$$\mu \cup \nu = \left\{ \frac{\mu(\mathbf{x\_1}) \lor \nu(\mathbf{x\_1})}{\mathbf{x\_1}}, \frac{\mu(\mathbf{x\_2}) \lor \nu(\mathbf{x\_2})}{\mathbf{x\_2}}, \dots, \frac{\mu(\mathbf{x\_p}) \lor \nu(\mathbf{x\_p})}{\mathbf{x\_p}} \, \middle| \, \mathbf{x\_p} \in \mathcal{X} \right\}.$$

*(2) The intersection µ* ∩ *ν*, *is defined as*

$$\mu \cap \nu = \left\{ \frac{\mu(\mathbf{x\_1}) \wedge \nu(\mathbf{x\_1})}{\mathbf{x\_1}}, \frac{\mu(\mathbf{x\_2}) \wedge \nu(\mathbf{x\_2})}{\mathbf{x\_2}}, \dots, \frac{\mu(\mathbf{x\_p}) \wedge \nu(\mathbf{x\_p})}{\mathbf{x\_p}} \, \middle| \, \mathbf{x\_p} \in \mathcal{X} \right\}.$$

*2.2. Neutrosophic Set and Single-Valued Neutrosophic Set*

**Definition 2** (cf. [25,26])**.** *Assume that X (i.e., X* = {*x*1, *x*2, ..., *xp*}) *is a set of elements and*

$$\Phi = \left\{ \frac{\left(T\_{\Phi}(\mathbf{x}\_{p}), I\_{\Phi}(\mathbf{x}\_{p}), F\_{\Phi}(\mathbf{x}\_{p})\right)}{\mathbf{x}\_{p}} \; \middle| \; \mathbf{x}\_{p} \in \mathcal{X}, \; \mathbf{0} \le T\_{\Phi}(\mathbf{x}\_{p}) + I\_{\Phi}(\mathbf{x}\_{p}) + F\_{\Phi}(\mathbf{x}\_{p}) \le \mathfrak{I} \right\}.$$

**Definition 3** (cf. [27])**.** *Assume that X (i.e., X* = {*x*1, *x*2, ..., *xp*}) *is a set of elements and*

$$\Phi = \left\{ \frac{\left(T\_{\hat{\Phi}}(\mathbf{x}\_{\mathcal{P}}), I\_{\hat{\Phi}}(\mathbf{x}\_{\mathcal{P}}), F\_{\hat{\Phi}}(\mathbf{x}\_{\mathcal{P}}), \mu(\mathbf{x}\_{\mathcal{P}})\right)}{\mathbf{x}\_{\mathcal{P}}} \; \middle| \; \mathbf{x}\_{\mathcal{P}} \in \mathcal{X}, \; 0 \le T\_{\Phi}(\mathbf{x}\_{\mathcal{P}}) + I\_{\Phi}(\mathbf{x}\_{\mathcal{P}}) + F\_{\Phi}(\mathbf{x}\_{\mathcal{P}}) \le 3 \right\}.$$

**Definition 4** (cf. [27])**.** *Let* Φˆ , Ψˆ ∈ (SVNFS) *X*, *where*

$$\Phi = \left\{ \frac{\left(T\_{\hat{\Phi}}(\mathbf{x}\_{p}), I\_{\hat{\Phi}}(\mathbf{x}\_{p}), F\_{\hat{\Phi}}(\mathbf{x}\_{p}), \mu(\mathbf{x}\_{p})\right)}{\mathbf{x}\_{p}} \; \middle| \; \mathbf{x}\_{p} \in \mathcal{X}, \; 0 \le T\_{\hat{\Phi}}(\mathbf{x}\_{p}) + I\_{\hat{\Phi}}(\mathbf{x}\_{p}) + F\_{\hat{\Phi}}(\mathbf{x}\_{p}) \le 3 \right\}$$

*and*

$$\Psi = \left\{ \frac{\left(T'\_{\hat{\mathbf{Y}}}(\mathbf{x}\_{p}), I'\_{\hat{\mathbf{Y}}}(\mathbf{x}\_{p}), F'\_{\hat{\mathbf{Y}}}(\mathbf{x}\_{p}), \mu'(\mathbf{x}\_{p})\right)}{\mathbf{x}\_{p}} \; \middle| \; \mathbf{x}\_{p} \in \mathcal{X}, \; \mathbf{0} \le T\_{\Psi}(\mathbf{x}\_{p}) + I\_{\Psi}(\mathbf{x}\_{p}) + F\_{\Psi}(\mathbf{x}\_{p}) \le \mathfrak{I} \right\}.$$

*The following operations (i.e., complement, inclusion, equal, union, and intersection) are defined by*

$$\mathbf{A}^c(1) \quad \hat{\Phi}^c = \left\{ \frac{\left( \mathbf{F}\_{\Phi}(\mathbf{x}\_p), 1 - I\_{\Phi}(\mathbf{x}\_p), T\_{\Phi}(\mathbf{x}\_p), 1 - \mu(\mathbf{x}\_p) \right)}{\mathbf{x}\_p} \; \middle| \; \mathbf{x}\_p \in \mathcal{X} \right\}.$$

*Symmetry* **2020**, *12*, 1361

*(2)* Φˆ ⊆ Ψˆ ⇐⇒ *T*Φˆ (*xp*) ≤ *T* ′ Ψˆ (*xp*), *I*Φˆ (*xp*) ≥ *I* ′ Ψˆ (*xp*), *F*Φˆ (*xp*) ≥ *F* ′ Ψˆ (*xp*) *and µ*(*xp*) ≤ *µ* ′ (*xp*) (∀*xp* ∈ *X*).

$$(3)\quad\Phi=\Psi\Longleftrightarrow\Phi\subseteq\Psi\text{ and }\Psi\subseteq\Phi.$$

$$\begin{split} (4) \quad \boldsymbol{\Phi} \cup \boldsymbol{\Psi} &= \left\{ \frac{\left( \mathbf{F}\_{\boldsymbol{\Phi}}(\mathbf{x}\_{p}) \lor \mathbf{F}\_{\boldsymbol{\Phi}}'(\mathbf{x}\_{p}), I\_{\boldsymbol{\Phi}}(\mathbf{x}\_{p}) \land I\_{\boldsymbol{\Phi}}'(\mathbf{x}\_{p}), T\_{\boldsymbol{\Phi}}(\mathbf{x}\_{p}) \land T\_{\boldsymbol{\Psi}}'(\mathbf{x}\_{p}), \boldsymbol{\mu}(\mathbf{x}\_{p}) \lor \boldsymbol{\mu}'(\mathbf{x}\_{p}) \right)}{\mathbf{x}\_{p}} \; \middle| \; \mathbf{x}\_{p} \in \mathcal{X} \right\}. \\ (5) \quad \boldsymbol{\Phi} \cap \boldsymbol{\Psi} &= \left\{ \frac{\left( \mathbf{F}\_{\boldsymbol{\Phi}}(\mathbf{x}\_{p}) \land \mathbf{F}\_{\boldsymbol{\Phi}}'(\mathbf{x}\_{p}), I\_{\boldsymbol{\Phi}}(\mathbf{x}\_{p}) \lor I\_{\boldsymbol{\Phi}}'(\mathbf{x}\_{p}), T\_{\boldsymbol{\Phi}}(\mathbf{x}\_{p}) \lor T\_{\boldsymbol{\Phi}}'(\mathbf{x}\_{p}), \boldsymbol{\mu}(\mathbf{x}\_{p}) \land \boldsymbol{\mu}'(\mathbf{x}\_{p}) \right)}{\mathbf{x}\_{p}} \; \middle| \; \mathbf{x}\_{p} \in \mathcal{X} \right\}. \end{split}$$

*xp*

*2.4. Soft Set, Fuzzy Soft Set, and Neutrosophic Soft Set*

**Definition 5** (cf. [4,7,28])**.** *Assume that X (i.e., X* = {*x*1, *x*2, ..., *xp*}) *is a set of elements and I (i.e., I* = {*i*1, *i*2, ..., *iq*}*) is a set of parameters, where* (*p*, *q* ∈ *N*, *N are natural numbers*)*. Then*

*(1) The following mapping (called a soft set), is given by*

$$\mathcal{S}: I \to P(X)\_{\prime\prime}$$

*where P*(*X*) *is a set of all subsets over X*.

*(2) The following mapping (called a fuzzy soft set), is given by*

$$
\tilde{\mathcal{S}} : I \to [0,1]^X \prime
$$

*where* [0, 1] *X is a set of whole fuzzy subset over X.*

*(3) The following mapping (called a neutrosophic soft set), is given by*

$$\overline{\mathcal{S}} : I \to (\mathbb{N} \mathbb{B})\_X$$

*where* (NS) *X is a set of whole neutrosophic subset over X.*

**Example 1.** *Assume that the two brothers Mr. Z and Mr. M plan to go the car dealership office to purchase a new car. Suppose that the car dealership office contains types of new cars X* = {*x*1, *x*2, *x*3, *x*4} *and I* = {*i*1, *i*2, *i*3} *characterize three parameters, where i*1 *is "cheap", i*2 *is "expensive", and i*3 *is "beautiful". Then*

*(1) By Definition 5(1) we can describe the soft sets as S*(*i*1 ) = {*x*1, *x*3}, *S*(*i*2) = {*x*3, *x*4}, *and S*(*i*3) = {*x*2}. *Therefore,*

$$\mathcal{S} = \left\{ \frac{\{\mathbf{x}\_1, \mathbf{x}\_3\}}{i\_1}, \frac{\{\mathbf{x}\_3, \mathbf{x}\_4\}}{i\_2}, \frac{\{\mathbf{x}\_2\}}{i\_3} \right\}.$$

*(2) It is obvious to replace the crisp number 0 or 1 by a membership of fuzzy information. Therefore, by Definition 5(2) we can describe the fuzzy soft sets by S*e (*i*1 ) = n 0.3 *x*1 , 0.4 *x*2 , 0.6 *x*3 , 0.5 *x*4 o , *S*e (*i*2) = n 0.6 *x*1 , 0.9 *x*2 , 0.1 *x*3 , 0.2 *x*4 o , *S*e (*i*3) = n 0.7 *x*1 , 0.5 *x*2 , 0.2 *x*3 , 0.9 *x*4 o . *Then,*

$$\widetilde{S} = \left\{ \frac{\left\{ \frac{0.3}{\underline{\times}\_1}, \frac{0.4}{\underline{\times}\_2}, \frac{0.6}{\underline{\times}\_3}, \frac{0.5}{\underline{\times}\_4} \right\}}{\dot{i}\_1}, \frac{\left\{ \frac{0.6}{\underline{\times}\_1}, \frac{0.9}{\underline{\times}\_2}, \frac{0.1}{\underline{\times}\_3}, \frac{0.2}{\underline{\times}\_4} \right\}}{\dot{i}\_2}, \frac{\left\{ \frac{0.7}{\underline{\times}\_1}, \frac{0.5}{\underline{\times}\_2}, \frac{0.2}{\underline{\times}\_3}, \frac{0.9}{\underline{\times}\_4} \right\}}{\dot{i}\_3} \right\}.$$

*(3) By Definition 5(3) we can describe the neutrosophic soft sets as*

$$
\widetilde{S}\_{(i\_1)} = \left\{ \frac{(0.3, 0.7, 0.5)}{\varkappa\_1}, \frac{(0.1, 0.8, 0.5)}{\varkappa\_2}, \frac{(0.2, 0.6, 0.8)}{\varkappa\_3}, \frac{(0.4, 0.7, 0.6)}{\varkappa\_4} \right\},
$$

$$
\widetilde{S}\_{(i\_2)} = \left\{ \frac{(0.3, 0.7, 0.5)}{\varkappa\_1}, \frac{(0.1, 0.8, 0.5)}{\varkappa\_2}, \frac{(0.2, 0.6, 0.8)}{\varkappa\_3}, \frac{(0.5, 0.8, 0.3)}{\varkappa\_4} \right\},
$$

*and*

$$\widetilde{S}\_{(i\_{\mathfrak{I}})} = \left\{ \frac{(0.3, 0.7, 0.5)}{\mathfrak{x}\_1}, \frac{(0.1, 0.8, 0.5)}{\mathfrak{x}\_2}, \frac{(0.2, 0.6, 0.8)}{\mathfrak{x}\_3}, \frac{(0.8, 0.9, 0.2)}{\mathfrak{x}\_4} \right\}.$$

#### **3. Single-Valued Neutrosophic Fuzzy Soft Set**

In the following, we propose the concept of a single-valued neutrosophic fuzzy soft set and study some definitions, propositions, and examples.

**Definition 6.** *Assume that X (i.e., X* = {*x*1, *x*2, ..., *xp*}) *is a set of elements, I (i.e., I* = {*i*1, *i*2, ..., *iq*}*) is a set of parameters, and* S *X I is called a soft universe. A single-valued neutrosophic fuzzy soft set* Φˆ (*iq*) *over X*, *denoted by* (SVNFS) *X I, is defined by*

$$\Phi\_{(i\_{\emptyset})} = \left\{ \frac{(T\_{\Phi\_{(i\mathbf{q})}}(\mathbf{x}\_p), I\_{\Phi\_{(i\mathbf{q})}}(\mathbf{x}\_p), F\_{\Phi\_{(i\mathbf{q})}}(\mathbf{x}\_p), \mu(\mathbf{x}\_p))}{\mathbf{x}\_p} \; \middle| \; i\_{\mathbf{q}} \in I, \; \mathbf{x}\_p \in \mathcal{X}, \; 0 \le T\_{\Phi\_{(i\mathbf{q})}}(\mathbf{x}\_p) + I\_{\Phi\_{(i\mathbf{q})}}(\mathbf{x}\_p) + F\_{\Phi\_{(i\mathbf{q})}}(\mathbf{x}\_p) \le 3 \right\}, \; \mathbf{x}\_p \in \mathcal{X}$$

*where p*, *q* ∈ *N* (*N are natural numbers*) *and µ*(*xp*) ∈ [0, 1]. *For each parameter iq* ∈ *I and for each xp* ∈ *X*, *T*Φˆ (*iq*) (*xp*) ∈ [0, 1] *(i.e., the degree of truth membership), I*Φˆ (*iq*) (*xp*) ∈ [0, 1] *(i.e., the degree of indeterminacy membership), and F*Φˆ (*iq*) (*xp*) ∈ [0, 1] *(i.e., the degree of falsity membership).*

**Example 2.** *Assume that X* = {*x*1, *x*2, *x*3} *are three kinds of novel cars and I* = {*i*1, *i*2, *i*3} *are three parameters, where i*1 *is "cheap", i*2 *is "expensive", and i*3 *is "beautiful". Let µ* ∈ [0, 1] *X and* Φˆ (*iq*) ∈ (SVNFS) *X I are defined as follows* (*q* = 1, 2, 3)*:*

$$\begin{aligned} \Phi\_{(i\_1)} &= \left\{ \frac{(0.3, 0.7, 0.5, 0.2)}{\varkappa\_1}, \frac{(0.1, 0.8, 0.5, 0.5)}{\varkappa\_2}, \frac{(0.2, 0.6, 0.8, 0.7)}{\varkappa\_3} \right\}, \\\Phi\_{(i\_2)} &= \left\{ \frac{(0.9, 0.4, 0.5, 0.7)}{\varkappa\_1}, \frac{(0.3, 0.7, 0.5, 0.4)}{\varkappa\_2}, \frac{(0.8, 0.2, 0.6, 0.8)}{\varkappa\_3} \right\}, \\\Phi\_{(i\_3)} &= \left\{ \frac{(0.6, 0.3, 0.5, 0.6)}{\varkappa\_1}, \frac{(0.3, 0.5, 0.6, 0.4)}{\varkappa\_2}, \frac{(0.7, 0.1, 0.6, 0.3)}{\varkappa\_3} \right\}. \end{aligned}$$

*Additionally, we can write by matrix form as*

$$
\Phi = \begin{pmatrix}
\begin{array}{c|cc}
\text{I} & \text{x}\_1 & \text{x}\_2 & \text{x}\_3 \\
\hline
i\_1 & (0.3, 0.7, 0.5, 0.2) & (0.1, 0.8, 0.5, 0.5) & (0.2, 0.6, 0.8, 0.7) \\
i\_2 & (0.9, 0.4, 0.5, 0.7) & (0.3, 0.7, 0.5, 0.4) & (0.8, 0.2, 0.6, 0.8) \\
i\_3 & (0.6, 0.3, 0.5, 0.6) & (0.3, 0.5, 0.6, 0.4) & (0.7, 0.1, 0.6, 0.3)
\end{array}
\end{pmatrix}
$$

.

**Definition 7.** *Let* Φˆ (*iq*) , Ψˆ (*iq*) ∈ (SVNFS) *X I over* S *X I and µ*, *µ* ′ ∈ [0, 1] *X*, *where*

$$\Phi\_{(i\_{\mathbb{Q}})} = \left\{ \frac{(\mathbf{T}\_{\Phi\_{(i\_{\mathbb{Q}})}}(\mathbf{x}\_p), I\_{\Phi\_{(i\_{\mathbb{Q}})}}(\mathbf{x}\_p), \mathbf{F}\_{\Phi\_{(i\_{\mathbb{Q}})}}(\mathbf{x}\_p), \mu(\mathbf{x}\_p))}{\mathbf{x}\_p} \; \middle| \; i\_{\mathbb{q}} \in I, \; \mathbf{x}\_p \in \mathcal{X}, \; \mathbf{0} \le \mathbf{T}\_{\Phi\_{(i\_{\mathbb{Q}})}}(\mathbf{x}\_p) + I\_{\Phi\_{(i\_{\mathbb{Q}})}}(\mathbf{x}\_p) + \mathbf{F}\_{\Phi\_{(i\_{\mathbb{Q}})}}(\mathbf{x}\_p) \le \mathbf{3} \right\} $$

*and*

$$\Psi\_{(i\_{\mathbb{Q}})} = \left\{ \frac{\left(T\_{\Psi\_{(i\_{\mathbb{Q}})}}^{\prime}(\mathbf{x}\_{p}), I\_{\Psi\_{(i\_{\mathbb{Q}})}}^{\prime}(\mathbf{x}\_{p}), F\_{\Psi\_{(i\_{\mathbb{Q}})}}^{\prime}(\mathbf{x}\_{p}), \mu^{\prime}(\mathbf{x}\_{p})\right)}{\mathbf{x}\_{p}} \; \middle| \; i\_{\mathbb{q}} \in I, \; \mathbf{x}\_{\mathbb{P}} \in \mathcal{X}, \; \mathbf{0} \le T\_{\Psi\_{(i\_{\mathbb{Q}})}}^{\prime}(\mathbf{x}\_{p}) + I\_{\Psi\_{(i\_{\mathbb{Q}})}}^{\prime}(\mathbf{x}\_{p}) + F\_{\Psi\_{(i\_{\mathbb{Q}})}}^{\prime}(\mathbf{x}\_{p}) \le \mathfrak{F} \right\}.$$

*Then,* Φˆ (*iq*) ⋐ Ψˆ (*iq*) *(i.e.,* Φˆ (*iq*) *is a single-valued neutrosophic fuzzy soft subset of* Ψˆ (*iq*) *) if*

$$\begin{aligned} (1) \quad &\mu(\mathbf{x}\_p) \le \mu'(\mathbf{x}\_p) \,\forall \mathbf{x}\_p \in \mathbf{X};\\ (2) \quad &\text{For all } \mathbf{i}\_q \in I, \mathbf{x}\_p \in \mathbf{X}, \mathbf{T}\_{\Phi\_{(i\_q)}}(\mathbf{x}\_p) \le \mathbf{T}'\_{\Phi\_{(i\_q)}}(\mathbf{x}\_p), I\_{\Phi\_{(i\_q)}}(\mathbf{x}\_p) \ge I'\_{\Phi\_{(i\_q)}}(\mathbf{x}\_p), \mathbf{F}\_{\Phi\_{(i\_q)}}(\mathbf{x}\_p) \ge \mathbf{F}'\_{\Phi\_{(i\_q)}}(\mathbf{x}\_p). \end{aligned}$$

**Example 3.** *(Continued from Example 2). Let* Ψˆ (*iq*) ∈ (SVNFS) *X I be defined as follows* (*q* = 1, 2, 3)*:*

$$
\Psi = \begin{pmatrix}
\mathbf{I} & \mathbf{x}\_1 & \mathbf{x}\_2 & \mathbf{x}\_3 \\
\hline
i\_1 & (0.4, 0.6, 0.4, 0.4) & (0.2, 0.7, 0.3, 0.5) & (0.3, 0.4, 0.7, 1) \\
i\_2 & (1, 0.3, 0.5, 0.8) & (0.4, 0.6, 0.4, 0.6) & (0.9, 0.2, 0.4, 0.9) \\
i\_3 & (0.7, 0.2, 0.4, 0.7) & (0.4, 0.5, 0.6, 0.6) & (0.8, 0.1, 0.5, 0.5)
\end{pmatrix}
$$

.

*Thus,* Φˆ (*iq*) ⋐ Ψˆ (*iq*) (∀*iq* ∈ *I*).

**Definition 8.** *Let* Φˆ (*iq*) , Ψˆ (*iq*) ∈ (SVNFS) *X I over* S *X I and µ*, *µ* ′ ∈ [0, 1] *X*, *where*

$$\Phi\_{(i\_{\!\!\!\!})} = \left\{ \frac{(\mathbf{T}\_{\Phi\_{(i\!\!\!\!})}(\mathbf{x}\_p), I\_{\Phi\_{(i\!\!\!)}}(\mathbf{x}\_p), \mathbf{F}\_{\Phi\_{(i\!\!\!)}}(\mathbf{x}\_p), \mu(\mathbf{x}\_p))}{\mathbf{x}\_p} \; \middle| \; i\_{\!\!\!} \in I, \; \mathbf{x}\_p \in \mathbf{X}, \; \mathbf{0} \le T\_{\Phi\_{(i\!\!\!\!}}(\mathbf{x}\_p) + I\_{\Phi\_{(i\!\!\!)}}(\mathbf{x}\_p) + \mathbf{F}\_{\Phi\_{(i\!\!\!)}}(\mathbf{x}\_p) \le 3 \right\}$$

*and*

$$\Psi\_{(i\_{\emptyset})} = \left\{ \frac{(T'\_{\Psi\_{(i\_{\emptyset})}}(\mathbf{x}\_{\mathcal{P}}), I'\_{\Psi\_{(i\_{\emptyset})}}(\mathbf{x}\_{\mathcal{P}}), F'\_{\Psi\_{(i\_{\emptyset})}}(\mathbf{x}\_{\mathcal{P}}), \mu'(\mathbf{x}\_{\mathcal{P}}))}{\mathbf{x}\_{\mathcal{P}}} \; \middle| \; i\_{\emptyset} \in I, \; \mathbf{x}\_{\mathcal{P}} \in \mathcal{X}, \; \mathbf{0} \le T'\_{\Psi\_{(i\_{\emptyset})}}(\mathbf{x}\_{\mathcal{P}}) + I'\_{\Psi\_{(i\_{\emptyset})}}(\mathbf{x}\_{\mathcal{P}}) + F'\_{\Psi\_{(i\_{\emptyset})}}(\mathbf{x}\_{\mathcal{P}}) \le \mathbf{3} \right\}.$$

*Then,* Φˆ (*iq*) = Ψˆ (*iq*) *(i.e.,* Φˆ (*iq*) *is a single-valued neutrosophic fuzzy soft equal to* Ψˆ (*iq*) *) if* Φˆ (*iq*) ⋐ Ψˆ (*iq*) *and* Φˆ (*iq*) ⋑ Ψˆ (*iq*) .

**Definition 9.** *Let* Φˆ (*iq*) ∈ (SVNFS) *X I over* S *X I and µ* ∈ [0, 1] *X*, *where*

$$\Phi\_{(i\_{\!\!\!\!})} = \left\{ \frac{(T\_{\Phi\_{(i\_{\!\!\!\!\!})}(\mathbf{x}\_{\mathcal{P}}), I\_{\Phi\_{(i\_{\!\!\!\!})}}(\mathbf{x}\_{\mathcal{P}}), F\_{\Phi\_{(i\_{\!\!\!\!})}}(\mathbf{x}\_{\mathcal{P}}), \mu(\mathbf{x}\_{\mathcal{P}}))}{\mathbf{x}\_{\mathcal{P}}} \; \middle| \; i\_{\!\!\!} \in I, \; \mathbf{x}\_{\mathcal{P}} \in \mathcal{X}, \; \mathbf{0} \le T\_{\Phi\_{(i\_{\!\!\!\!\!})}}(\mathbf{x}\_{\mathcal{P}}) + I\_{\Phi\_{(i\_{\!\!\!\!})}}(\mathbf{x}\_{\mathcal{P}}) + F\_{\Phi\_{(i\_{\!\!\!\!})}}(\mathbf{x}\_{\mathcal{P}}) \le 3 \right\}$$

*over* S *X I* . *Then,*

*(1)* Φˆ (*iq*) *is called a single-valued neutrosophic fuzzy soft null set (denoted by* ∅ˆ (*iq*) ), *defined as*

$$\bigotimes\_{(i\_{\!\!\!/})} = \left\{ \frac{(0,1,1,0)}{\varkappa\_p} | i\_{\!\!\!/} \in I \lrcorner \varkappa\_p \in X \right\}.$$

*(2)* Φˆ (*iq*) *is called a single-valued neutrosophic fuzzy soft universal set (denoted by X*ˆ (*iq*) ), *defined as*

$$\hat{X}\_{(i\_q)} = \left\{ \frac{(1,0,0,1)}{\mathfrak{x}\_p} | i\_q \in I \,\,\mathfrak{x}\_p \in X \right\}.$$

**Example 4.** *(Continued from Example 2). Then,* ∅ˆ (*iq*) , *X*ˆ (*iq*) ∈ (SVNFS) *X I are defined as follows:*

$$
\begin{array}{c}
\mathcal{O} = \begin{pmatrix}
I & \mathfrak{x}\_{1} & \mathfrak{x}\_{2} & \mathfrak{x}\_{3} \\
\hline
i\_{1} & (0,1,1,0) & (0,1,1,0) & (0,1,1,0) \\
i\_{2} & (0,1,1,0) & (0,1,1,0) & (0,1,1,0) \\
i\_{3} & (0,1,1,0) & (0,1,1,0) & (0,1,1,0)
\end{pmatrix}
\end{array}
$$

*and*

$$
\hat{X} = \begin{pmatrix}
I & \mathfrak{x}\_1 & \mathfrak{x}\_2 & \mathfrak{x}\_3 \\
\hline
i\_1 & (1,0,0,1) & (1,0,0,1) & (1,0,0,1) \\
i\_2 & (1,0,0,1) & (1,0,0,1) & (1,0,0,1) \\
i\_3 & (1,0,0,1) & (1,0,0,1) & (1,0,0,1)
\end{pmatrix}.
$$

**Definition 10.** *Let* Φˆ (*iq*) , Ψˆ (*iq*) ∈ (SVNFS) *X I over* S *X I and µ*, *µ* ′ ∈ [0, 1] *X*, *where*

$$\hat{\Phi}\_{(i\_{\emptyset})} = \left\{ \frac{(T\_{\Phi\_{(i\_{\emptyset})}}(\mathbf{x}\_{p}), I\_{\Phi\_{(i\_{\emptyset})}}(\mathbf{x}\_{p}), F\_{\Phi\_{(i\_{\emptyset})}}(\mathbf{x}\_{p}), \mu(\mathbf{x}\_{p}))}{\mathbf{x}\_{p}} \; \middle| \; i\_{\emptyset} \in I, \; \mathbf{x}\_{p} \in \mathcal{X}, \; 0 \le T\_{\Phi\_{(i\_{\emptyset})}}(\mathbf{x}\_{p}) + I\_{\Phi\_{(i\_{\emptyset})}}(\mathbf{x}\_{p}) + F\_{\Phi\_{(i\_{\emptyset})}}(\mathbf{x}\_{p}) \le 3 \right\}$$

*and*

$$\Psi\_{(i\_{\emptyset})} = \left\{ \frac{(T'\_{\Psi\_{(i\_{\emptyset})}}(\mathbf{x}\_{\mathcal{P}}), I'\_{\Psi\_{(i\_{\emptyset})}}(\mathbf{x}\_{\mathcal{P}}), F'\_{\Psi\_{(i\_{\emptyset})}}(\mathbf{x}\_{\mathcal{P}}), \mu'(\mathbf{x}\_{\mathcal{P}}))}{\mathbf{x}\_{\mathcal{P}}} \; \middle| \; i\_{\emptyset} \in I, \; \mathbf{x}\_{\mathcal{P}} \in \mathcal{X}, \; \mathbf{0} \le T'\_{\Psi\_{(i\_{\emptyset})}}(\mathbf{x}\_{\mathcal{P}}) + I'\_{\Psi\_{(i\_{\emptyset})}}(\mathbf{x}\_{\mathcal{P}}) + F'\_{\Psi\_{(i\_{\emptyset})}}(\mathbf{x}\_{\mathcal{P}}) \le \mathbf{3} \right\}.$$

*Then,*

*(1) The union* Φˆ (*iq*) ⋒ Ψˆ (*iq*) *is defined as*

$$\Phi\_{\left(\begin{smallmatrix}\mathbf{q}\end{smallmatrix}\right)}\left\uplus\Psi\_{\left(\begin{smallmatrix}\mathbf{q}\end{smallmatrix}\right)} = \left\{ \frac{\left(T\_{\Phi\_{\left(\begin{smallmatrix}\mathbf{q}\end{subarray}\right)}\left(\mathbf{x}\_{p}\right)\circ T\_{\Phi\_{\left(\begin{smallmatrix}\mathbf{q}\end{smallmatrix}\right)}}\left(\mathbf{x}\_{p}\right),I\_{\Phi\_{\left(\begin{smallmatrix}\mathbf{q}\end{smallmatrix}\right)}}\left(\mathbf{x}\_{p}\right)\*I\_{\Phi\_{\left(\begin{smallmatrix}\mathbf{q}\end{smallmatrix}\right)}}^{}\left(\mathbf{x}\_{p}\right),F\_{\Phi\_{\left(\begin{smallmatrix}\mathbf{q}\end{smallmatrix}\right)}}\left(\mathbf{x}\_{p}\right),\mu\left(\mathbf{x}\_{p}\right)\circ\mu^{\left(\mathbf{x}\_{p}\right)}\right) \right\} \left|i\_{\mathbf{q}}\in I,\mathbf{x}\_{p}\in X\right\rangle \right\}.$$

*(2) The intersection* Φˆ (*iq*) ⋓ Ψˆ (*iq*) *is defined as*

$$\Phi\_{(i\_{\mathbb{Q}})} \otimes \Psi\_{(i\_{\mathbb{Q}})} = \left\{ \frac{(\mathbf{T}\_{\Phi\_{(i\_{\mathbb{Q}})}}(\mathbf{x}\_{\mathbb{P}}) \ast \mathbf{T}\_{\Phi\_{(i\_{\mathbb{Q}})}}'(\mathbf{x}\_{\mathbb{P}}), \mathbf{I}\_{\Phi\_{(i\_{\mathbb{Q}})}}(\mathbf{x}\_{\mathbb{P}}) \diamond I\_{\Phi\_{(i\_{\mathbb{Q}})}}'(\mathbf{x}\_{\mathbb{P}}), \mathbf{F}\_{\Phi\_{(i\_{\mathbb{Q}})}}(\mathbf{x}\_{\mathbb{P}}) \diamond \mathbf{F}\_{\Phi\_{(i\_{\mathbb{Q}})}}'(\mathbf{x}\_{\mathbb{P}}), \mu(\mathbf{x}\_{\mathbb{P}}) \ast \mu'(\mathbf{x}\_{\mathbb{P}})}{\mathbf{x}\_{\mathbb{P}}} \Big| i\_{\mathbb{q}} \in I, \mathbf{x}\_{\mathbb{P}} \in X \right\}.$$

**Example 5.** *(Continued from Examples 2 and 3). For α*, *β* ∈ [0, 1], *let the t-norm (i.e., given as α* ∗ *β* = *α* ∧ *β) and the t-conorm (i.e., given as α* ◦ *β* = *α* ∨ *β). Then,*

$$
\Phi \uplus \Psi = \begin{pmatrix}
\begin{array}{c|cc}
\text{I} & \text{x}\_1 & \text{x}\_2 & \text{x}\_3 \\
\hline
i\_1 & (0.4, 0.6, 0.4, 0.4) & (0.2, 0.7, 0.3, 0.5) & (0.3, 0.4, 0.7, 1) \\
i\_2 & (1, 0.3, 0.5, 0.8) & (0.4, 0.6, 0.4, 0.6) & (0.9, 0.2, 0.4, 0.9) \\
i\_3 & (0.7, 0.2, 0.4, 0.7) & (0.4, 0.5, 0.6, 0.6) & (0.8, 0.1, 0.5, 0.5)
\end{array}
\end{pmatrix}
$$

*and*

$$
\Phi \uplus \Psi = \begin{pmatrix}
\begin{array}{c|cc}
\text{I} & \text{x}\_1 & \text{x}\_2 & \text{x}\_3 \\
\hline
i\_1 & (0.3, 0.7, 0.5, 0.2) & (0.1, 0.8, 0.5, 0.5) & (0.2, 0.6, 0.8, 0.7) \\
i\_2 & (0.9, 0.4, 0.5, 0.7) & (0.3, 0.7, 0.5, 0.4) & (0.8, 0.2, 0.6, 0.8) \\
i\_3 & (0.6, 0.3, 0.5, 0.6) & (0.3, 0.5, 0.6, 0.4) & (0.7, 0.1, 0.6, 0.3)
\end{array}
\end{pmatrix}
$$

.

**Proposition 1.** *Let* ∅ˆ (*iq*) , *X*ˆ (*iq*) , Φˆ (*iq*) ∈ (SVNFS) *X I over* S *X I and µ* ∈ [0, 1] *X*. *Then the following hold:*

$$(\mathbf{5}) \quad \Phi\_{(i\_{\mathbf{\dot{q}}})} \uplus \mathbf{\hat{X}\_{(i\_{\mathbf{\dot{q}}})}} = \mathbf{\hat{X}\_{(i\_{\mathbf{\dot{q}}})}},$$

*(6)* Φˆ (*iq*) ⋓ *X*ˆ (*iq*) = Φˆ (*iq*)

**Proof.** Follows from Definitions 9 and 10.

.

**Proposition 2.** *Let* Φˆ (*iq*) , Ψˆ (*iq*) , Γˆ (*iq*) ∈ (SVNFS) *X I over* S *X I and µ*, *µ* ′ , *µ* ′′ ∈ [0, 1] *X*. *Then the following hold:*

**Proof.** Follows from Definition 10.

**Proposition 3.** *Let* Φˆ (*iq*) , Ψˆ (*iq*) ∈ (SVNFS) *X I over* S *X I* , *µ*, *µ* ′ ∈ [0, 1] *X*, *and* Ψˆ (*iq*) ⋐ Φˆ (*iq*) . *Then the following hold:*

*(1)* Φˆ (*iq*) ⋒ Ψˆ (*iq*) = Φˆ (*iq*) ; *(2)* Φˆ (*iq*) ⋓ Ψˆ (*iq*) = Ψˆ (*iq*) .

**Proof.** Follows from Definitions 7 and 10.

Next, we propose a definition, example, remark, and two propositions on the complement of (SVNFS) *X I* over S *X I* .

**Definition 11.** *Let* Φˆ (*iq*) ∈ (SVNFS) *X I over* S *X I and µ* ∈ [0, 1] *X*, *where*

$$\Phi\_{(l\_q)} = \left\{ \frac{(T\_{\Phi\_{(l\_q)}}(\mathbf{x}\_p), I\_{\Phi\_{(l\_q)}}(\mathbf{x}\_p), F\_{\Phi\_{(l\_q)}}(\mathbf{x}\_p), \mu(\mathbf{x}\_p))}{\mathbf{x}\_p} \; \middle| \; i\_q \in I\_r \; \mathbf{x}\_p \in \mathcal{X}, \; 0 \le T\_{\Phi\_{(l\_q)}}(\mathbf{x}\_p) + I\_{\Phi\_{(l\_q)}}(\mathbf{x}\_p) + F\_{\Phi\_{(l\_q)}}(\mathbf{x}\_p) \le 3 \right\}.$$

*Then, the complement* Φˆ *c* (*iq*) *of* Φˆ (*iq*) *is defined as*

$$\Phi\_{(i\_{\!\!\!\!})}^{\varepsilon} = \left\{ \frac{\left(F\_{\Phi\_{(i\_{\!\!\!\!\!})}}(\mathbf{x}\_{p}), \mathbf{1} - I\_{\Phi\_{(i\_{\!\!\!\!})}}(\mathbf{x}\_{p}), T\_{\Phi\_{(i\_{\!\!\!\!})}}(\mathbf{x}\_{p}), \mathbf{1} - \mu(\mathbf{x}\_{p})\right)}{\mathbf{x}\_{p}} \, \Big|\, i\_{\!\!\!} \in I, \ \mathbf{x}\_{p} \in X\right\}.$$

**Example 6.** *(Continued from Example 2). The complement* Φˆ *c* (*iq*) *of* Φˆ (*iq*) *is calculated by*

$$
\hat{\Phi}^{\mathcal{L}} = \begin{pmatrix}
\begin{array}{c|cccc}
\hline
I & \text{x}\_1 & \text{x}\_2 & \text{x}\_3 \\
\hline
\dot{\iota}\_1 & (0.5, 0.3, 0.3, 0.8) & (0.5, 0.2, 0.1, 0.5) & (0.8, 0.4, 0.2, 0.3) \\
\dot{\iota}\_2 & (0.5, 0.6, 0.9, 0.3) & (0.5, 0.3, 0.3, 0.6) & (0.6, 0.8, 0.8, 0.2) \\
\dot{\iota}\_3 & (0.5, 0.7, 0.6, 0.4) & (0.6, 0.5, 0.3, 0.6) & (0.6, 0.9, 0.7, 0.7)
\end{array}
\end{pmatrix}.
$$

**Proposition 4.** *Let* ∅ˆ (*iq*) , *X*ˆ (*iq*) , Φˆ (*iq*) ∈ (SVNFS) *X I over* S *X I* , *and µ* ∈ [0, 1] *X*. *Then, the following hold:*

**Proof.** Follows from Definitions 9 and 11.

**Remark 1.** *The equality of* Φˆ (*iq*) ⋒ Φˆ *c* (*iq*) = *X*ˆ (*iq*) *and* Φˆ (*iq*) ⋓ Φˆ *c* (*iq*) = ∅ˆ (*iq*) *does not hold by the following example.*

**Example 7.** *(Continued from Examples 2 and 6). Then,* Φˆ *c* (*iq*) *of* Φˆ (*iq*) *is calculated by*

$$
\hat{\Phi} \uplus \hat{\Phi}^{\epsilon} = \begin{pmatrix}
1 & x\_1 & x\_2 & x\_3 \\
\hline
i\_1 & (0.5, 0.3, 0.3, 0.8) & (0.5, 0.2, 0.1, 0.5) & (0.8, 0.4, 0.2, 0.3) \\
i\_1 & (0.5, 0.6, 0.9, 0.3) & (0.5, 0.3, 0.3, 0.6) & (0.6, 0.8, 0.8, 0.2) \\
i\_1 & (0.5, 0.7, 0.6, 0.4) & (0.6, 0.5, 0.3, 0.6) & (0.6, 0.9, 0.7, 0.7)
\end{pmatrix}
$$

*and*

$$
\hat{\Phi}\circledast\hat{\Phi}^{\mathcal{L}} = \begin{pmatrix}
\begin{array}{c|cc}
\text{ } & \text{x}\_1 & \text{x}\_2 & \text{x}\_3 \\
\hline
i\_1 & (0.3, 0.7, 0.5, 0.2) & (0.1, 0.8, 0.5, 0.5) & (0.2, 0.6, 0.8, 0.7) \\
i\_2 & (0.9, 0.4, 0.5, 0.7) & (0.3, 0.7, 0.5, 0.4) & (0.8, 0.2, 0.6, 0.8) \\
i\_3 & (0.6, 0.3, 0.5, 0.6) & (0.3, 0.5, 0.6, 0.4) & (0.7, 0.1, 0.6, 0.3)
\end{array}
\end{pmatrix}.
$$

*This shows that* Φˆ (*iq*) ⋒ Φˆ *c* (*iq*) 6= *X*ˆ (*iq*) *and* Φˆ (*iq*) ⋓ Φˆ *c* (*iq*) 6= ∅ˆ (*iq*) .

**Proposition 5.** *Let* Φˆ (*iq*) , Ψˆ (*iq*) ∈ (SVNFS) *X I over* S *X I and µ*, *µ* ′ ∈ [0, 1] *X*. *Then, the following hold:*

$$(\mathfrak{I}) \quad (\mathfrak{\mathring{\Phi}}\_{(i\_q)} \uplus \Psi\_{(i\_q)})^c = \mathfrak{\mathring{\Phi}}\_{(i\_q)}^c \uplus \Psi\_{(i\_q)}^c;$$

$$(\mathfrak{\mathring{\Phi}}) \quad (\mathfrak{\mathring{\Phi}}\_{\text{---}} \circ \mathfrak{\mathring{\Psi}}\_{\text{---}})^c \quad \mathfrak{\mathring{\Phi}}\_{\text{---}} \circ \mathfrak{\mathring{\Phi}}\_{\text{---}};$$

$$(\mathcal{Q}) \quad (\mathfrak{d}\_{(i\_q)} \oplus \Psi\_{(i\_q)})^c = \mathfrak{d}\_{(i\_q)}^c \oplus \Psi\_{(i\_q)}^c.$$

**Proof.** Consider *a* ∗ *b* = *a* ∧ *b* (t-norm) and *α* ◦ *β* = *α* ∨ *β* (t-conorm) (∀*α*, *β* ∈ [0, 1]). We have

$$\begin{split}(\mathbf{1}) &= \left(\Phi\_{\mathbf{\tilde{\mathbf{\mathbf{d}}}\_{({\mathbf{r}})}}}(\mathbf{p})\,\Psi\_{\mathbf{\tilde{\mathbf{d}}}\_{({\mathbf{r}})}}\right)^{c}({\mathbf{r}}\_{\mathbf{p}})\right) \\ &= \left(\left(\frac{\left(\mathbf{\tilde{\mathbf{\mathbf{d}}}\_{({\mathbf{d}}\_{({\mathbf{r}})}}}({\mathbf{r}}\_{\mathbf{p}})\circ\mathbf{T}\_{\mathbf{\tilde{\mathbf{d}}}\_{({\mathbf{r}})}}({\mathbf{r}}\_{\mathbf{p}})\,\mathbf{L}\_{\mathbf{\tilde{\mathbf{d}}}\_{({\mathbf{r}})}}({\mathbf{r}}\_{\mathbf{p}})\circ\mathbf{L}\_{\mathbf{\tilde{\mathbf{d}}}\_{({\mathbf{r}})}}({\mathbf{r}}\_{\mathbf{p}})\circ\mathbf{T}\_{\mathbf{\tilde{\mathbf{d}}}\_{({\mathbf{r}})}}({\mathbf{r}}\_{\mathbf{p}})\circ\mathbf{T}\_{\mathbf{\tilde{\mathbf{d}}}\_{({\mathbf{r}})}}^{c}({\mathbf{r}}\_{\mathbf{p}})\,\mu({\mathbf{r}}\_{\mathbf{p}})\,\mu^{\dagger}({\mathbf{r}}\_{\mathbf{p}})\right)\,\Big|\_{\mathbf{r}}\,\mathbf{d}\_{1}\times I\_{r}\times\mathbf{X}\Big)\right)^{c}}{\mathbf{x}\_{\mathbf{p}}} \\ &= \left(\frac{\left(\mathbf{f}\_{\mathbf{\tilde{\mathbf{d}}}\_{({\mathbf{d}}\_{({\mathbf{r}})}}({\mathbf{r}}\_{\mathbf{p}})\circ\mathbf{H}\_{{\mathbf{d}}}^{r}({\mathbf{r}}\_{\mathbf{p}}),1-{}({\mathbf{d}}\_{{\mathbf{d}}\_{({\mathbf{r}})}}({\mathbf{r}}\_{\mathbf{p}})\wedge\mathbf{T}\_{\mathbf{\tilde{\mathbf{d}}}\_{({$$

$$(\mathbf{2}) \quad (\Phi\_{(i\_q)} \circledast \Psi\_{(i\_q)})^c(\mathfrak{x}\_p).$$

= *T*Φˆ (*iq* ) (*xp*) ∗ *T* ′ Ψˆ (*iq* ) (*xp*), *I*Φˆ (*iq* ) (*xp*) ◦ *I* ′ Ψˆ (*iq* ) (*xp*), *F*Φˆ (*iq* ) (*xp*) ◦ *F* ′ Ψˆ (*iq* ) (*xp*), *µ*(*xp*) ∗ *µ* ′ (*xp*) *xp iq* ∈ *I*, *xp* ∈ *X c* = *F*Φˆ (*iq* ) (*xp*) ◦ *F* ′ Ψˆ (*iq* ) (*xp*), 1 − (*I*Φˆ (*iq* ) (*xp*) ◦ *I* ′ Ψˆ (*iq* ) (*xp*)), *T*Φˆ (*iq* ) (*xp*) ∗ *T* ′ Ψˆ (*iq* ) (*xp*), 1 − (*µ*(*xp*) ∗ *µ* ′ (*xp*)) *xp iq* ∈ *I*, *xp* ∈ *X* = *F*Φˆ (*iq* ) (*xp*) ∨ *F* ′ Ψˆ (*iq* ) (*xp*), 1 − (*I*Φˆ (*iq* ) (*xp*) ∨ *I* ′ Ψˆ (*iq* ) (*xp*)), *T*Φˆ (*iq* ) (*xp*) ∧ *T* ′ Ψˆ (*iq* ) (*xp*), 1 − (*µ*(*xp*) ∧ *µ* ′ (*xp*)) *xp iq* ∈ *I*, *xp* ∈ *X* = *F*Φˆ (*iq* ) (*xp*) ∨ *F* ′ Ψˆ (*iq* ) (*xp*), 1 − *I*Φˆ (*iq* ) (*xp*) ∧ 1 − *I* ′ Ψˆ (*iq* ) (*xp*), *T*Φˆ (*iq* ) (*xp*) ∧ *T* ′ Ψˆ (*iq* ) (*xp*), 1 − *µ*(*xp*) ∨ 1 − *µ* ′ (*xp*) *xp iq* ∈ *I*, *xp* ∈ *X* = *F*Φˆ (*iq* ) (*xp*) ◦ *F* ′ Ψˆ (*iq* ) (*xp*), 1 − *I*Φˆ (*iq* ) (*xp*) ∗ 1 − *I* ′ Ψˆ (*iq* ) (*xp*), *T*Φˆ (*iq* ) (*xp*) ∗ *T* ′ Ψˆ (*iq* ) (*xp*), 1 − *µ*(*xp*) ◦ 1 − *µ* ′ (*xp*) *xp iq* ∈ *I*, *xp* ∈ *X* = *F*Φˆ (*iq*) (*xp*), 1 − *I*Φˆ (*iq*) (*xp*), *T*Φˆ (*iq*) (*xp*), 1 − *µ*(*xp*) *xp iq* ∈ *I*, *xp* ∈ *X* ⋒ *F* ′ Ψˆ (*iq*) (*xp*), 1 − *I* ′ Ψˆ (*iq*) (*xp*), *T* ′ Ψˆ (*iq*) (*xp*), 1 − *µ* ′ (*xp*) *xp iq* ∈ *I*, *xp* ∈ *X* = Φˆ *c* (*iq*) (*xp*) ⋒ Ψˆ *c* (*iq*) (*xp*).

#### **4. Two Algorithms of Single-Valued Neutrosophic Fuzzy Soft Sets for Decision-Making**

Depending on single-valued neutrosophic fuzzy soft sets, in the following, we introduce two new approaches for fuzzy decision-making problems.

Next, we construct Algorithm 1 as the first type for decision-making (i.e., the first application of a single-valued neutrosophic fuzzy soft set).

**Algorithm 1:** Determine the optimal decision based on a single-valued neutrosophic fuzzy soft set matrix.

**First step:** Input the single-valued neutrosophic fuzzy soft set Φˆ (*iq*) ∈ (SVNFS) *X I* as follows:

$$\hat{\Phi}\_{(l\_{\boldsymbol{q}})} = \left\{ \frac{(T\_{\hat{\Phi}\_{(\boldsymbol{q})}}(\mathbf{x}\_{\mathcal{P}}), I\_{\hat{\Phi}\_{(\boldsymbol{q})}}(\mathbf{x}\_{\mathcal{P}}), F\_{\hat{\Phi}\_{(\boldsymbol{q})}}(\mathbf{x}\_{\mathcal{P}}), \mu(\mathbf{x}\_{\mathcal{P}}))}{\mathbf{x}\_{\mathcal{P}}} \; \middle| \; i\_{\boldsymbol{q}} \in I, \; \mathbf{x}\_{\mathcal{P}} \in \mathcal{X}, \; \mathbf{0} \le T\_{\hat{\Phi}\_{(\boldsymbol{q})}}(\mathbf{x}\_{\mathcal{P}}) + I\_{\hat{\Phi}\_{(\boldsymbol{q})}}(\mathbf{x}\_{\mathcal{P}}) + F\_{\hat{\Phi}\_{(\boldsymbol{q})}}(\mathbf{x}\_{\mathcal{P}}) \le \mathbf{3} \right\}, \; \phi\_{\mathcal{P}} \in \mathcal{X} \; \phi\_{\mathcal{P}}, \; \phi\_{\mathcal{P}} \in \mathcal{X}$$

to be evaluated by a group of experts *n* to element *x* on parameter *i*, where *T*Φˆ (*iq*) (*xp*) ∈ [0, 1] (i.e., the degree of truth membership), *I*Φˆ (*iq*) (*xp*) (i.e., the degree of indeterminacy membership), *F*Φˆ (*iq*) (*xp*) (i.e., the degree of falsity membership), and *µ*(*xp*) ∈ [0, 1].

**Second step:** Input the single-valued neutrosophic fuzzy soft set in matrix form (written as M*q*×*p*, *p*, *q* ∈ *N*):

M*q*×*p* = *T*Φˆ (*i*1 ) (*x*1), *I*Φˆ (*i*1 ) (*x*1), *F*Φˆ (*i*1 ) (*x*1), *µ*(*x*1) *T*Φˆ (*i*1 ) (*x*2), *I*Φˆ (*i*1 ) (*x*2), *F*Φˆ (*i*1 ) (*x*2), *µ*(*x*2) · · · *T*Φˆ (*i*2 ) (*xp*), *I*Φˆ (*i*2 ) (*xp*), *F*Φˆ (*i*2 ) (*xp*), *µ*(*xp*) *T*Φˆ (*i*2 ) (*x*1), *I*Φˆ (*i*2 ) (*x*1), *F*Φˆ (*i*2 ) (*x*1), *µ*(*x*1) *T*Φˆ (*i*2 ) (*x*2), *I*Φˆ (*i*2 ) (*x*2), *F*Φˆ (*i*2 ) (*x*2), *µ*(*x*2) · · · *T*Φˆ (*i*2 ) (*xp*), *I*Φˆ (*i*2 ) (*xp*), *F*Φˆ (*i*2 ) (*xp*), *µ*(*xp*) *T*Φˆ (*i*3 ) (*x*1), *I*Φˆ (*i*3 ) (*x*1), *F*Φˆ (*i*3 ) (*x*1), *µ*(*x*1) *T*Φˆ (*i*3 ) (*x*2), *I*Φˆ (*i*3 ) (*x*2), *F*Φˆ (*i*3 ) (*x*2), *µ*(*x*2) · · · *T*Φˆ (*i*3 ) (*xp*), *I*Φˆ (*i*3 ) (*xp*), *F*Φˆ (*i*3 ) (*xp*), *µ*(*xp*) . . . . . . . . . . . . *T*Φˆ (*iq*) (*x*1), *I*Φˆ (*iq*) (*x*1), *F*Φˆ (*iq*) (*x*1), *µ*(*x*1) *T*Φˆ (*iq*) (*x*2), *I*Φˆ (*iq*) (*x*2), *F*Φˆ (*iq*) (*x*2), *µ*(*x*2) · · · *T*Φˆ (*iq*) (*xp*), *I*Φˆ (*iq*) (*xp*), *F*Φˆ (*iq*) (*xp*), *µ*(*xp*) .

**Third step:** Calculate the center matrix (i.e.,

*δ*Φˆ (*iq*) (*xp*) = (*T*Φˆ (*iq*) (*xp*) + *I*Φˆ (*iq*) (*xp*) + *F*Φˆ (*iq*) (*xp*)) − *µ*(*xp*)):

$$
\mathbb{C}\_{q \times p} = \begin{pmatrix}
\delta\_{\hat{\Phi}\_{(i\_1)}}(\mathbf{x}\_1) & \delta\_{\hat{\Phi}\_{(i\_1)}}(\mathbf{x}\_2) & \cdots & \delta\_{\hat{\Phi}\_{(i\_1)}}(\mathbf{x}\_p) \\
\delta\_{\hat{\Phi}\_{(i\_2)}}(\mathbf{x}\_1) & \delta\_{\hat{\Phi}\_{(i\_2)}}(\mathbf{x}\_2) & \cdots & \delta\_{\hat{\Phi}\_{(i\_2)}}(\mathbf{x}\_p) \\
\vdots & \vdots & \ddots & \vdots \\
\delta\_{\hat{\Phi}\_{(i\_q)}}(\mathbf{x}\_1) & \delta\_{\hat{\Phi}\_{(i\_q)}}(\mathbf{x}\_2) & \cdots & \delta\_{\hat{\Phi}\_{(i\_q)}}(\mathbf{x}\_p)
\end{pmatrix}.
$$

**Fourth step:** Calculate the *d max*(*xj*) (maximum decision), *d min*(*xj*) (minimum decision), and *S*(*xj*) (score) of elements *xj* (*j* = 1, 2, · · · , *p*):

$$d^{\max}(\mathbf{x}\_{\dot{\jmath}}) = \sum\_{i=1}^{q} \left(\mathbf{1} - \delta\_{\Phi\_{(\dot{\jmath})}}(\mathbf{x}\_{\dot{\jmath}})\right)^2 \mathbf{d}^{\min}(\mathbf{x}\_{\dot{\jmath}}) = \sum\_{i=1}^{q} (\delta\_{\Phi\_{(i\_{\dot{\jmath}})}}(\mathbf{x}\_{\dot{\jmath}}))^2$$

(to understand the motivation behind this method, let *ρ* be the Euclidean metric on *R q* , 0 = (0, · · · , 0) *T* ∈ *R q* , 1 = (1, · · · , 1) *T* ∈ *R q* , and *θ j* = (*θ*1,*xj* , *θ*2,*xj* , · · · , *θq*,*xj* ) *T* ∈ *R q* . Thus *S*(*xj*) = [*ρ*(*θ j* , 1)]2 + [*ρ*(*θ j* , 0)]2 (*j* = 1, 2, · · · , *p*)). **Fifth step:** Obtain the decision *p* satisfying

$$\mathfrak{x}\_p = \max\left\{ S(\mathfrak{x}\_1), S(\mathfrak{x}\_2), \dots, S(\mathfrak{x}\_{\mathfrak{f}}) \right\}.$$

Now, we show the principle and steps of the above Algorithm 1 by using the following example.

**Example 8.** *An investment company wants to choose some investment projects to make full use of idle funds. There are five alternatives X* = {*z*1, *z*2, *z*3, *z*4, *z*5} *that can be selected: two internet education projects (denoted as z*1 *and z*2*) and three film studio investments (represented as z*3, *z*4, *z*5*). According to the project investment books, the decision-makers evaluate the five alternatives from the following three parameters I* = {*i*1, *i*2, *i*3}*, where i*1 *is "human resources", i*2 *is "social benefits", and i*3 *is "expected benefits". The data of the single-valued neutrosophic fuzzy soft set* Φˆ (*iq*) ∈ (SVNFS) *X I is given by*

.

*Now, we will explain the practical meaning of alternatives X by taking the alternative z*1 *as an example: the single-valued neutrosophic fuzzy soft set* Φˆ (*i*1 ) (*z*1) = (0.3, 0.7, 0.5, 0.2) *is the evaluation by four expert groups; the single-valued neutrosophic fuzzy soft value 0.3 (meaning that 30*% *say yes in the first expert group) in* Φˆ (*i*1 ) (*z*1)*, the single-valued neutrosophic fuzzy soft value 0.7 (meaning 70% say no in the second expert group) in* Φˆ (*i*1 ) (*z*1), *the single-valued neutrosophic fuzzy soft value 0.5 (meaning 50*% *say yes in the third expert group) in* Φˆ (*i*1 ) (*z*1)*, and fuzzy value 0.2 (meaning 20% say no in the fourth expert group) in* Φˆ (*i*1 ) (*z*1). *Then, the single-valued neutrosophic fuzzy soft set in matrix form* M3×5 *in the second step of Algorithm 1 is given by*

$$\mathcal{M}\_{3 \times 5} = \begin{pmatrix} (0.3, 0.7, 0.5, 0.2) & (0.9, 0.4, 0.5, 0.7) & (0.6, 0.3, 0.5, 0.6) \\ (0.1, 0.8, 0.5, 0.5) & (0.3, 0.7, 0.5, 0.4) & (0.3, 0.5, 0.6, 0.4) \\ (0.2, 0.6, 0.8, 0.7) & (0.8, 0.2, 0.6, 0.8) & (0.7, 0.1, 0.6, 0.3) \\ (0.5, 0.6, 0.5, 0.2) & (0.3, 0.7, 0.2, 0.5) & (0.8, 0.9, 0.6, 0.4) \\ (0.4, 0.7, 0.9, 0.1) & (0.7, 0.8, 0.8, 0.3) & (0.7, 0.8, 0.9, 0.6) \end{pmatrix}.$$

*Thus, we obtain the following center matrix C*3×5 *of* M3×5 *in the third step of Algorithm 1:*

$$\mathbf{C}\_{3 \times 5} = \begin{pmatrix} 1.3 & 1.1 & 0.8 \\ 0.9 & 1.1 & 1 \\ 0.9 & 0.8 & 1.1 \\ 1.4 & 0.7 & 1.9 \\ 1.9 & 2 & 1.8 \end{pmatrix}$$

.

*By calculating, we get dmax*(*zj*)*, dmin*(*zj*)*, and S*(*zj*) *of elements zj (j* = 1, 2, 3, 4, 5*):*

$$d^{\max}(z\_1) = 0.14, d^{\max}(z\_2) = 0.02, d^{\max}(z\_3) = 0.06, d^1(z\_4) = 1.06, d^{\max}(z\_5) = 2.45;$$

$$d^{\min}(z\_1) = 3.54, d^{\min}(z\_2) = 3.02, d^{\min}(z\_3) = 2.66, d^{\min}(z\_4) = 6.06, d^{\min}(z\_5) = 10.85;$$

$$S(z\_1) = 3.68, S(z\_2) = 3.04, S(z\_3) = 2.72, S(z\_4) = 7.12, S(z\_5) = 13.3.$$

*Finally, we can see from the fifth step that z*5 *is the best decision.*

Now, we present Algorithm 2 as a second type for a decision-making problem (i.e., a second application of the single-valued neutrosophic fuzzy soft set) as follows:

**Algorithm 2:** Determine the optimal decision based on AND operation of two single-valued neutrosophic fuzzy soft sets.

**First step:** Input the single-valued neutrosophic fuzzy soft sets Φˆ (*iq*) ∈ (SVNFS) *X I* and Ψˆ (*jq*) ∈ (SVNFS) *X J*, defined, respectively, as follows:

$$\Phi\_{(i\_{\mathbb{Q}})} = \left\{ \frac{(T\_{\Phi\_{(i\_{\mathbb{Q}})}}(\mathbf{x}\_p), I\_{\Phi\_{(i\_{\mathbb{Q}})}}(\mathbf{x}\_p), F\_{\Phi\_{(i\_{\mathbb{Q}})}}(\mathbf{x}\_p), \mu(\mathbf{x}\_p))}{\mathbf{x}\_p} \; \middle| \; i\_{\mathbb{q}} \in I, \; \mathbf{x}\_p \in \mathcal{X}, \; \mathbf{0} \le T\_{\Phi\_{(i\_{\mathbb{Q}})}}(\mathbf{x}\_p) + I\_{\Phi\_{(i\_{\mathbb{Q}})}}(\mathbf{x}\_p) + F\_{\Phi\_{(i\_{\mathbb{Q}})}}(\mathbf{x}\_p) \le \mathbf{3} \right\}, \; \mathbf{0} \le T\_{\Phi\_{(i\_{\mathbb{Q}})}} \; \mathbf{x}\_p, \; \mathbf{0} \le T\_{\Phi\_{(i\_{\mathbb{Q}})}}(\mathbf{x}\_p), \; \mathbf{0} \le T\_{\Phi\_{(i\_{\mathbb{Q}})}}(\mathbf{x}\_p) + \mathbf{1}\_p, \; \mathbf{x}\_p \in \mathcal{X}, \; \mathbf{0} \le T\_{\Phi\_{(i\_{\mathbb{Q}})}}(\mathbf{x}\_p) + \mathbf{1}\_p, \; \mathbf{0} \le T\_{\Phi\_{(i\_{\mathbb{Q}})}}(\mathbf{x}\_p) \le \mathbf{0} \right\}$$

to be evaluated by a group of experts *n* to element *x* on parameter *i*, where *T*Φˆ (*iq*) (*xp*) ∈ [0, 1] (i.e., the degree of truth membership), *I*Φˆ (*iq*) (*xp*) (i.e., the degree of indeterminacy membership), *F*Φˆ (*iq*) (*xp*) (i.e., the degree of falsity membership), and *µ*(*xp*) ∈ [0, 1],

$$\Psi\_{(j\_{\boldsymbol{q}})} = \left\{ \frac{(T'\_{\Psi\_{({\boldsymbol{q}})}}(\mathbf{x}\_{p}), I'\_{\Psi\_{({\boldsymbol{q}})}}(\mathbf{x}\_{p}), F'\_{\Psi\_{({\boldsymbol{q}})}}(\mathbf{x}\_{p}), \boldsymbol{\mu}'(\mathbf{x}\_{p}))}{\mathbf{x}\_{p}} \; \Big| \; j\_{\boldsymbol{q}} \in \mathcal{J}, \; \mathbf{x}\_{p} \in \mathcal{X}, \; \mathbf{0} \le \mathcal{T}'\_{\Psi\_{({\boldsymbol{q}})}}(\mathbf{x}\_{p}) + I'\_{\Psi\_{({\boldsymbol{q}})}}(\mathbf{x}\_{p}) + \mathbf{F}'\_{\Psi\_{({\boldsymbol{q}})}}(\mathbf{x}\_{p}) \le \mathbf{3} \right\}$$

to be evaluated by a group of experts *n* to element *x* on parameter *j*, where *T*Ψˆ ′ (*jq*) (*xp*) ∈ [0, 1] (i.e., the degree of truth membership), *I*Ψˆ ′ (*jq*) (*xp*) (i.e., the degree of indeterminacy membership), *F*Ψˆ ′ (*jq*) (*xp*) (i.e., the degree of falsity membership), and *µ*(*xp*) ∈ [0, 1].

**Second step:** Define and calculate the AND operation of two single-valued neutrosophic fuzzy soft sets Φˆ (*iq*) ∈ (SVNFS) *X I* and Ψˆ (*jq*) ∈ (SVNFS) *X J*, denoted by (Φˆ ∧Ψˆ )(*iq*,*jq*) (∀*i* ∈ *I*, *j* ∈ *J*), defined as

$$(\boldsymbol{\Phi}\boldsymbol{\nabla}\boldsymbol{\hat{\nabla}}\boldsymbol{\Psi})\_{(i\_{\emptyset})\_{\emptyset}} = \left\{ \frac{(\boldsymbol{T}\_{\boldsymbol{\Phi}\_{(i\_{\emptyset})}}(\boldsymbol{\boldsymbol{x}}\_{p}) \wedge \boldsymbol{T}\_{\boldsymbol{\Psi}\_{(i\_{\emptyset})}}^{\boldsymbol{\prime}}(\boldsymbol{x}\_{p}), \boldsymbol{I}\_{\boldsymbol{\Phi}\_{(i\_{\emptyset})}}(\boldsymbol{x}\_{p}) \vee \boldsymbol{I}\_{\boldsymbol{\Psi}\_{(i\_{\emptyset})}}^{\boldsymbol{\prime}}(\boldsymbol{x}\_{p}), \boldsymbol{F}\_{\boldsymbol{\Phi}\_{(i\_{\emptyset})}}(\boldsymbol{x}\_{p}) \vee \boldsymbol{F}\_{\boldsymbol{\Psi}\_{(i\_{\emptyset})}}^{\boldsymbol{\prime}}(\boldsymbol{x}\_{p}), \boldsymbol{\mu}(\boldsymbol{x}\_{p}) \wedge \boldsymbol{\mu}^{\boldsymbol{\prime}}(\boldsymbol{x}\_{p}) \right)}{\boldsymbol{x}\_{p}} \left| \boldsymbol{i}\_{\emptyset} \in I, \boldsymbol{j}\_{\emptyset} \in \boldsymbol{I}, \boldsymbol{x}\_{p} \in X \right\}.$$

**Third step:** Define and write the truth membership (Φˆ ∧Ψˆ ) *T* (*iq*,*jq*) , the indeterminacy membership (Φˆ ∧Ψˆ ) *I* (*iq*,*jq*) , and the falsity membership (Φˆ ∧Ψˆ ) *F* (*iq*,*jq*) , respectively, as follows:

$$(\Phi \Box \Psi)^{T}\_{(i\_{q},j\_{q})} = \left\{ \frac{(T\_{\Phi\_{(i\_{q})}}(\mathbf{x}\_{p}) \wedge T^{\prime}\_{\Psi\_{(i\_{q})}}(\mathbf{x}\_{p}), \mu(\mathbf{x}\_{p}) \wedge \mu^{\prime}(\mathbf{x}\_{p}))}{\mathbf{x}\_{p}} \, \middle| \, i\_{q} \in I, j\_{q} \in I, \mathbf{x}\_{p} \in \mathcal{X} \right\},$$

$$(\Phi \Box \Psi)^{T}\_{(i\_{q},j\_{q})} = \left\{ \frac{(I\_{\Phi\_{(i\_{q})}}(\mathbf{x}\_{p}) \vee I^{\prime}\_{\Psi\_{(j\_{q})}}(\mathbf{x}\_{p}), \mu(\mathbf{x}\_{p}) \wedge \mu^{\prime}(\mathbf{x}\_{p}))}{\mathbf{x}\_{p}} \, \middle| \, i\_{q} \in I, j\_{q} \in I, \mathbf{x}\_{p} \in \mathcal{X} \right\},$$

and

$$(\Phi \nabla \Psi)^{F}\_{(i\_{\vec{q}},j\_{\vec{q}})} = \left\{ \frac{(F\_{\Phi\_{(i\_{\vec{q}})}}(\mathbf{x}\_{p}) \lor F^{\prime}\_{\Psi\_{(j\_{\vec{q}})}}(\mathbf{x}\_{p}), \mu(\mathbf{x}\_{p}) \land \mu^{\prime}(\mathbf{x}\_{p}))}{\mathbf{x}\_{p}} \, \middle| \, \mathbf{i}\_{\vec{q}} \in I\_{\prime} \, j\_{\vec{q}} \in J\_{\prime} \, \mathbf{x}\_{p} \in \mathbf{X} \right\}.$$

**Fourth step:** Define and compute the max-matrices of (Φˆ ∧Ψˆ ) *T* (*iq*,*jq*) , (Φˆ ∧Ψˆ ) *I* (*iq*,*jq*) , and (Φˆ ∧Ψˆ ) *F* (*iq*,*jq*) , respectively, for every *xp* ∈ *X* as follows (*p* = 1, 2, · · · , *N*):

$$(\boldsymbol{\Phi} \nabla \boldsymbol{\Psi})^{T}\_{(\boldsymbol{i}\_{\hat{q}}, \boldsymbol{j}\_{\hat{q}})}(\boldsymbol{x}\_{p}) = \frac{1}{2} \Big( (T\_{\hat{\boldsymbol{\Phi}}\_{(\boldsymbol{i}\_{\hat{q}})}}(\boldsymbol{x}\_{p}) \wedge T'\_{\boldsymbol{\Psi}\_{(\boldsymbol{j}\_{\hat{q}})}}(\boldsymbol{x}\_{p})) + (\boldsymbol{\mu}(\boldsymbol{x}\_{p}) \wedge \boldsymbol{\mu}'(\boldsymbol{x}\_{p})) \Big),$$

$$(\boldsymbol{\Phi} \nabla \boldsymbol{\Psi})^{T}\_{(\boldsymbol{i}\_{\hat{q}}, \boldsymbol{j}\_{\hat{q}})}(\boldsymbol{x}\_{p}) = \left( (I\_{\hat{\boldsymbol{\Phi}}\_{(\boldsymbol{i}\_{\hat{q}})}}(\boldsymbol{x}\_{p}) \vee I'\_{\boldsymbol{\Psi}\_{(\boldsymbol{i}\_{\hat{q}})}}(\boldsymbol{x}\_{p})) \times (\boldsymbol{\mu}(\boldsymbol{x}\_{p}) \wedge \boldsymbol{\mu}'(\boldsymbol{x}\_{p})) \right).$$

and

$$(\Phi \overleftarrow{\wedge} \Psi)^{F}\_{(i\_{q}j\_{q})}(\mathbf{x}\_{p}) = \left( (F\_{\Phi\_{(i\_{q})}}(\mathbf{x}\_{p}) \vee F'\_{\Psi\_{(j\_{q})}}(\mathbf{x}\_{p})) - (\mu(\mathbf{x}\_{p}) \wedge \mu'(\mathbf{x}\_{p})) \right)^{2}.$$

#### **Algorithm 2:** *Cont.*

**Fifth step:** Calculate and write the max-decision *τT* (i.e., *τT* : *X* → *R*), *τI* (i.e., *τI* : *X* → *R*), and *τF* (i.e., *τF* : *X* → *R*) of (Φˆ ∧Ψˆ ) *T* (*iq*,*jq*) , (Φˆ ∧Ψˆ ) *I* (*iq*,*jq*) , and (Φˆ ∧Ψˆ ) *F* (*iq*,*jq*) , respectively, for every *xp* ∈ *X* as follows (*p* = 1, 2, · · · , *N*):

$$\tau\_T(\mathbf{x}\_p) = \sum\_{(i,j)\in I\times J} \delta\_T(\mathbf{x}\_p)(i,j), \quad \tau\_I(\mathbf{x}\_p) = \sum\_{(i,j)\in I\times J} \delta\_F(\mathbf{x}\_p)(i,j), \quad \text{and} \quad \sum\_{(i,j)\in I\times J} \delta\_F(\mathbf{x}\_p)(i,j).$$

where

*δT*(*xp*)(*i*, *j*) = (Φˆ ∧Ψˆ ) *T* (*iq*,*jq*) (*xp*), (Φˆ ∧Ψˆ ) *T* (*iq*,*jq*) (*xp*) = max{(Φˆ ∧Ψˆ ) *T* (*uq*,*vq*) (*xp*) : (*u*, *v*) ∈ *I* × *J*} 0, otherwise *δI*(*xp*)(*i*.*j*) = (Φˆ ∧Ψˆ ) *I* (*iq*,*jq*) (*xp*), (Φˆ ∧Ψˆ ) *I* (*iq*,*jq*) (*xp*) = max{(Φˆ ∧Ψˆ ) *I* (*uq*,*vq*) (*xp*) : (*u*, *v*) ∈ *I* × *J*} 0, otherwise *δF*(*xp*)(*i*.*j*) = (Φˆ ∧Ψˆ ) *F* (*iq*,*jq*) (*xp*), (Φˆ ∧Ψˆ ) *F* (*iq*,*jq*) (*xp*) = max{(Φˆ ∧Ψˆ ) *F* (*uq*,*vq*) (*xp*) : (*u*, *v*) ∈ *I* × *J*} 0, otherwise

,

,

.

**Sixth step:** Calculate the score *S*(*xp*) of element *xp* as follows (*p* = 1, 2, · · · , *N*):

$$S(\mathfrak{x}\_p) = \mathfrak{x}\_T(\mathfrak{x}\_p) + \mathfrak{x}\_I(\mathfrak{x}\_p) + \mathfrak{x}\_F(\mathfrak{x}\_p).$$

**Seventh step:** Obtain the decision *p* satisfying

$$\mathfrak{x}\_p = \max\left\{ S(\mathfrak{x}\_1), S(\mathfrak{x}\_2), \dots, S(\mathfrak{x}\_{\mathfrak{j}}) \right\}.$$

Now, we show the principle and steps of the above Algorithm 2 using the following example.

**Example 9.** *(Continued from Example 11). Suppose that an investment company also adds three different parameters J* = {*j*1, *j*2, *j*3}*, where j*1 *is "marketing management", j*2 *is "productivity of capital", and j*3 *is "interest rates". The data of the single-valued neutrosophic fuzzy soft set* Ψˆ (*jq*) ∈ (SVNFS) *X J is given by*