# New Operations of Totally Dependent-Neutrosophic Sets and Totally Dependent-Neutrosophic Soft Sets

^{1}

^{2}

^{3}

^{4}

^{5}

^{*}

## Abstract

**:**

## 1. Introduction

_{A}, I

_{A}, F

_{A}are 100% dependent, that is T

_{A}(x) + I

_{A}(x) + F

_{A}(x) ≤ 1 for any x in X.

## 2. Preliminaries and Motivation

#### 2.1. Some Basic Concepts

**Definition**

**1**

**[2].**Let X be a nonempty set (universe). An intuitionistic fuzzy set A on X is an object of the form:

_{A}(x), ν

_{A}(x))

**|**x ∈ X},

_{A}(x), ν

_{A}(x) ∈ [0, 1], μ

_{A}(x) + ν

_{A}(x) ≤ 1 for all x in X. μ

_{A}(x) ∈ [0, 1] is named the “degree of membership of x in A”, and ν

_{A}(x) is named the “degree of non-membership of x in A”.

**Definition**

**2**

**[6].**Assume that F(U) is the set of all fuzzy sets on U, and E is a set of parameters, A ⊆ E. If F is a mapping given by F:A → F(U), then the pair 〈F, A〉 is known as a fuzzy soft set over U.

**Definition**

**3**

**[26,27].**Let X be a nonempty set (universe). A totally dependent-neutrosophic set (or picture fuzzy set) A on X is an object of the form:

_{A}(x), η

_{A}(x), ν

_{A}(X))

**|**x ∈ X},

_{A}(x), η

_{A}(x), ν

_{A}(x) ∈ [0, 1], μ

_{A}(x) + η

_{A}(x) + ν

_{A}(x) ≤ 1, for all x in X. μ

_{A}(x) is named as the “degree of positive membership of x in A”, η

_{A}(x) is named as the “degree of neutral membership of x in A”, and ν

_{A}(x) is named the “degree of negative membership of x in A”.

**Definition**

**4**

**[26,28].**Assume that U is an initial universe set and E is a set of parameters, A ⊆ E. If F is a mapping given by F:A → TDNS(U), then the pair (F, A) is called a totally dependent-neutrosophic soft set (or picture fuzzy soft set) over U.

**Remark**

**1.**

- (1)
- A ⊆
_{1}B if ∀x ∈ X, μ_{A}(x) ≤ μ_{B}(x), η_{A}(x) ≥ η_{B}(x), ν_{A}(x) ≥ ν_{B}(x); - (2)
- A = B if A ⊆
_{1}B and B ⊆_{1}A; - (3)
- A ∪
_{1}B = {(x, max(μ_{A}(x), μ_{B}(x)), min(η_{A}(x), η_{B}(x)), min(ν_{A}(x), ν_{B}(x)))|x ∈ X}; - (4)
- A ∩
_{1}B = {(x, min(μ_{A}(x), μ_{B}(x)), max(η_{A}(x), η_{B}(x)), max(ν_{A}(x), ν_{B}(x)))|x ∈ X}; - (5)
- co(A) = A
^{c}= {(x, ν_{A}(x), η_{A}(x), μ_{A}(x))|x ∈ X}.

**Definition**

**5**

**[27].**For every two totally dependent-neutrosophic sets (TDNSs) A and B, type-2 inclusion relation, union, intersection operations, and the complement operation are defined as follows:

- (1)
- A ⊆
_{2}B if ∀x ∈ X, μ_{A}(x) ≤ μ_{B}(x), η_{A}(x) ≤ η_{B}(x), ν_{A}(x) ≥ ν_{B}(x); - (2)
- A = B if A ⊆
_{2}B and B ⊆_{2}A; - (3)
- A ∪
_{2}B = {(x, max(μ_{A}(x), μ_{B}(x)), min(η_{A}(x), η_{B}(x)), min(ν_{A}(x), ν_{B}(x)))**|**x ∈ X}; - (4)
- A ∩
_{2}B = {(x, min(μ_{A}(x), μ_{B}(x)), min(η_{A}(x), η_{B}(x)), max(ν_{A}(x), ν_{B}(x)))**|**x ∈ X}; - (5)
- co(A) = A
^{c}= {(x, ν_{A}(x), η_{A}(x), μ_{A}(x))**|**x ∈ X}.

**Remark**

**2.**

**Proposition**

**1**

**[27].**For every TDNS’s A, B, and C, the following assertions are true:

- (1)
- If A ⊆
_{2}B and B ⊆_{2}C, then A ⊆_{2}C; - (2)
- (A
^{c})^{c}= A; - (3)
- A ∩
_{2}B = B ∩_{2}A, A ∪_{2}B = B ∪_{2}A; - (4)
- (A ∩
_{2}B) ∩_{2}C = A ∩_{2}(B ∩_{2}C), (A ∪_{2}B) ∪_{2}C = A ∪_{2}(B ∪_{2}C); - (5)
- (A ∩
_{2}B) ∪_{2}C = (A ∪_{2}C) ∩_{2}(B ∪_{2}C), (A ∪_{2}B) ∩_{2}C = (A ∩_{2}C) ∪_{2}(B ∩_{2}C); - (6)
- (A ∩
_{2}B)^{c}= A^{c}∪_{2}B^{c}, (A ∪_{2}B)^{c}= A^{c}∩_{2}B^{c}.

**Definition**

**6**

**[31].**Assume that α = (μ

_{α}, η

_{α}, ν

_{α}, ρ

_{α}) is a totally dependent-neutrosophic number (picture fuzzy number), where μ

_{α}+ η

_{α}+ ν

_{α}≤ 1 and ρ

_{α}= 1 − μ

_{α}− η

_{α}− ν

_{α}. The mapping S(α) = μ

_{α}− ν

_{α}is called the score function, and the mapping H(α) = μ

_{α}+ η

_{α}+ ν

_{α}is called the accuracy function, where S(α) ∈ [−1, 1], H(α) ∈ [0, 1]. Moreover, for any two totally dependent-neutrosophic numbers (picture fuzzy number) α and β,

- (1)
- when S(α) > S(β), we say that α is superior to β, and it is expressed by α ≻ β;
- (2)
- when S(α) = S(β), then
- (i)
- when H(α) = H(β), we say that α is equivalent to β, and it is expressed by α ∼ β;
- (ii)
- when H(α) > H(β), we say that α is superior to β, and it is expressed by α ≻ β.

**Definition**

**7**

**[32].**Let (M, ∨, ∧,

^{−}, 0, 1) be a universal algebra. Then (M, ∨, ∧,

^{−}, 0, 1) is called a generalized De Morgan algebra (or GM-algebra), if (M, ∨, ∧, 0, 1) is a bounded lattice and the unary operation satisfies the identities:

- (1)
- (x
^{−})^{−}= x; - (2)
- (x ∧ y)
^{−}= x^{−}∨ y^{−}; - (3)
- 1
^{−}= 0.

#### 2.2. On Inclusion Relations of Totally Dependent-Neutrosophic Sets (Picture Fuzzy Sets)

_{1}, x

_{2}, x

_{3}), that is, the first component of x is expressed by x

_{1}, the second component of x is expressed by x

_{2}, and the third component of x is expressed by x

_{3}

**.**Moreover, the units of D* are expressed by 1

_{D}

_{*}= (1, 0, 0) and 0

_{D}

_{*}= (0, 0, 1), respectively.

_{2}B if and only if (∀x ∈ X) (μ

_{A}(x), η

_{A}(x), ν

_{A}(x)) ≤

_{2}(μ

_{B}(x), η

_{B}(x), ν

_{B}(x)).

_{2}B = {(max(μ

_{A}(x), μ

_{B}(x)), min(η

_{A}(x), η

_{B}(x)), min(ν

_{A}(x), ν

_{B}(x)))

**|**x ∈ X}

= {(μ

_{A}(x), η

_{A}(x), ν

_{A}(x)) ∨

_{2}(μ

_{B}(x), η

_{B}(x), ν

_{B}(x))

**|**x ∈ X};

_{2}B = {(min(μ

_{A}(x), μ

_{B}(x)), min(η

_{A}(x), η

_{B}(x)), max(ν

_{A}(x), ν

_{B}(x)))

**|**x ∈ X}

= {(μ

_{A}(x), η

_{A}(x), ν

_{A}(x)) ∧

_{2}(μ

_{B}(x), η

_{B}(x), ν

_{B}(x))

**|**x ∈ X};

**Example**

**1.**

_{2}y = (0.4, 0.3, 0.1), x ∧

_{2}y = (0.3, 0.3, 0.2).

_{1}y is not an upper bound of x and y. Moreover,

_{2}(x ∧

_{2}y) = (0.3, 0.3, 0.1) ≠ x.

_{2}and ∧

_{2}.

**Example**

**2.**

_{2}y, x ∨

_{2}y = (0.4, 0.35, 0.1) ≠ y.

_{2}y $\overline{)\Rightarrow}$ x ∨

_{2}y = y.

**Definition**

**8.**

_{3}defined as the following is called a type-3 inclusion relation: A ⊆

_{3}B when and only when

_{A}(x) < μ

_{B}(x), ν

_{A}(x) ≥ ν

_{B}(x)), or (μ

_{A}(x) = μ

_{B}(x), ν

_{A}(x) > ν

_{B}(x)),

or (μ

_{A}(x) = μ

_{B}(x), ν

_{A}(x) = ν

_{B}(x) and η

_{A}(x) ≤ η

_{B}(x)).

_{3}is built on the following order relation on D* (see [29], it is named a type-3 order relation):

**Remark**

**3.**

_{3}”. The strict proof process of the basic properties of the order relation “≤

_{3}” is not given in the literature [29], and these proofs are presented in this article (see next section). In addition, if x ≤

_{3}y and y ≤

_{3}x are not true for x, y ∈ D*, then x is not comparable to y, which is expressed by $x|{|}_{{\le}_{3}}y$.

## 3. On Type-3 Ordering Relation

_{3}) is a partial ordered set. Then, we prove that D* makes up a lattice through type-3 intersection and type-3 union operations.

**Proposition**

**2.**

_{3}) is a partial ordered set.

**Proof.**

- (1)
- By the definition of ≤
_{3}, we have x ≤_{3}x. - (2)
- Assume that x ≤
_{3}y and y ≤_{3}x, then- Case 1:
- x
_{1}< y_{1}and x_{3}≥ y_{3}. According to the definition of y ≤_{3}x, we can get x_{1}≥ y_{1}, which is contradictory. - Case 2:
- x
_{1}= y_{1}and x_{3}> y_{3}. According to the definition of y ≤_{3}x, we can get x_{3}≤ y_{3}, which is contradictory. - Case 3:
- x
_{1}= y_{1}and x_{3}= y_{3}. From x ≤_{3}y, we have x_{2}≤ y_{2}; also from y ≤_{3}x, we have x_{2}≥ y_{2}. Thus, x_{2}= y_{2}.

It follows that, (x ≤_{3}y and y ≤_{3}x) ⇒ (x_{1}= y_{1}, x_{2}= y_{2}and x_{3}= y_{3}) ⇒ x = y. - (3)
- Assume that x ≤
_{3}y, y ≤_{3}z, then- Case 1:
- (x
_{1}< y_{1}, x_{3}≥ y_{3}) and (y_{1}< z_{1}, y_{3}≥ z_{3}). It follows that x_{1}< z_{1}and x_{3}≥ z_{3}, thus x ≤_{3}z. - Case 2:
- (x
_{1}= y_{1}, x_{3}> y_{3}) and (y_{1}= z_{1}, y_{3}> z_{3}). It follows that x_{1}= z_{1}and x_{3}> z_{3}, thus x ≤_{3}z. - Case 3:
- (x
_{1}= y_{1}, x_{3}= y_{3}, x_{2}≤ y_{2}) and (y_{1}= z_{1}, y_{3}= z_{3}, y_{2}≤ z_{2}). It follows that x_{1}= z_{1}, x_{3}= z_{3}and x_{2}≤ z_{2}, thus x ≤_{3}z. - Case 4:
- (x
_{1}< y_{1}, x_{3}≥ y_{3}) and (y_{1}= z_{1}, y_{3}≥ z_{3}). It follows that x_{1}< z_{1}and x_{3}≥ z_{3}, thus x ≤_{3}z. - Case 5:
- (x
_{1}< y_{1}, x_{3}≥ y_{3}) and (y_{1}= z_{1}, y_{3}= z_{3}, y_{2}≤ z_{2}). It follows that x_{1}< z_{1}, x_{3}≥ z_{3}and y_{2}≤ z_{2}, thus x ≤_{3}z. - Case 6:
- (x
_{1}= y_{1}, x_{3}≥ y_{3}) and (y_{1}< z_{1}, y_{3}≥ z_{3}). It follows that x_{1}< z_{1}and x_{3}≥ z_{3}, thus x ≤_{3}z. - Case 7:
- (x
_{1}= y_{1}, x_{3}≥ y_{3}) and (y_{1}= z_{1}, y_{3}= z_{3}, y_{2}≤ z_{2}). It follows that x_{1}= z_{1}, x_{3}≥ z_{3}and y_{2}≤ z_{2}, thus x ≤_{3}z. - Case 8:
- (x
_{1}= y_{1}, x_{3}= y_{3}, x_{2}≤ y_{2}) and (y_{1}< z_{1}, y_{3}≥z_{3}). It follows that x_{1}< z_{1}, x_{3}≥ z_{3}and x_{2}≤ y_{2}, thus x ≤_{3}z. - Case 9:
- (x
_{1}= y_{1}, x_{3}= y_{3}, x_{2}≤ y_{2}) and (y_{1}= z_{1}, y_{3}>z_{3}), then x_{1}= z_{1}, x_{3}>z_{3}and x_{2}≤ y_{2}, thus x ≤_{3}z.

_{3}y and y ≤

_{3}z) ⇒ x ≤

_{3}z.

_{3}) is a partial ordered set. □

**Remark**

**4.**

**Proposition**

**3.**

_{3}y = inf(x, y), x ∨

_{3}y = sup(x, y), and (D*, ≤

_{3}) is a lattice.

**Proof.**

_{3}y or y ≤

_{3}x, then, by the definition of “∧

_{3}”, x ∧

_{3}y is the largest lower bound of x, y, i.e., x ∧

_{3}y = inf(x, y). Moreover, suppose that x ≤

_{3}y or y ≤

_{3}x, then x ∨

_{3}y is the smallest upper bound of x, y, i.e., x ∨

_{3}y = sup(x, y).

_{3}” and “∨

_{3}”, we have:

_{3}y = inf(x, y): Let

_{1}, z

_{2}, z

_{3}) = (min(x

_{1}, y

_{1}), 1 − min(x

_{1}, y

_{1}) − max(x

_{3}, y

_{3}), max(x

_{3}, y

_{3})).

_{1}≥ min(x

_{1}, y

_{1}) = z

_{1}, x

_{3}≤ max(x

_{3}, y

_{3}) = z

_{3}.

_{1}> z

_{1}and x

_{3}≤ z

_{3}, then z ≤

_{3}x.

_{1}= z

_{1}and x

_{3}< z

_{3}, then z ≤

_{3}x.

_{1}= z

_{1}and x

_{3}= z

_{3}, then y

_{1}≥ x

_{1}, y

_{3}≤ x

_{3}, and x ≤

_{3}y or y ≤

_{3}x. This is contradictory to the assumed condition $x|{|}_{{\le}_{3}}y$.

_{3}x. In the same way, we can obtain z ≤

_{3}y. That is, z is a lower bound of x and y.

_{1}, a

_{2}, a

_{3}) ∈ D* such that a ≤

_{3}x and a ≤

_{3}y.

- Case 1:
- (a
_{1}< x_{1}, a_{3}≥ x_{3}) and (a_{1}< y_{1}, a_{3}≥ y_{3}). It follows that a_{1}< min(x_{1}, y_{1}) = z_{1}and a_{3}≥ max(x_{3}, y_{3}) = z_{3}, thus a ≤_{3}z. - Case 2:
- (a
_{1}= x_{1}, a_{3}> x_{3}) and (a_{1}= y_{1}, a_{3}> y_{3}). It follows that a_{1}= min(x_{1}, y_{1}) = z_{1}and a_{3}> max(x_{3}, y_{3}) = z_{3}, thus a ≤_{3}z. - Case 3:
- (a
_{1}= x_{1}, a_{3}= x_{3}, a_{2}≤ x_{2}) and (a_{1}= y_{1}, a_{3}= y_{3}, a_{2}≤ y_{2}). It follows that a_{1}= min(x_{1}, y_{1}) = z_{1}, a_{3}= max(x_{3}, y_{3}) = z_{3}and a_{2}≤ min(x_{2}, y_{2}). Since a_{1}+ a_{2}+ a_{3}≤ 1, so a_{2}≤ 1 − min(x_{1}, y_{1}) − max(x_{3}, y_{3}) = z_{2}, thus a ≤_{3}z. - Case 4:
- (a
_{1}= x_{1}, a_{3}> x_{3}) and (a_{1}< y_{1}, a_{3}≥ y_{3}). It follows that a_{1}≤ min(x_{1}, y_{1}) and a_{3}≥ max(x_{3}, y_{3}). If (a_{1}< min(x_{1}, y_{1}), a_{3}≥ max(x_{3}, y_{3})) or (a_{1}= min(x_{1}, y_{1}), a_{3}> max(x_{3}, y_{3})), then a ≤_{3}z; If a_{1}= min(x_{1}, y_{1}) and a_{3}= max(x_{3}, y_{3}), from this and the hidden condition a_{1}+ a_{2}+ a_{3}≤ 1, we get a_{2}≤ 1 − min(x_{1}, y_{1}) − max(x_{3}, y_{3}) = z_{2}, hence a ≤_{3}z. - Case 5:
- (a
_{1}< x_{1}, a_{3}≥ x_{3}) and (a_{1}= y_{1}, a_{3}> y_{3}). It follows that a_{1}≤ min(x_{1}, y_{1}) = z_{1}and a_{3}≥ max(x_{3}, y_{3}) = z_{3}. Similar to Case 4, we can get a ≤_{3}z. - Case 6:
- (a
_{1}< x_{1}, a_{3}≥ x_{3}) and (a_{1}= y_{1}, a_{3}= y_{3}, a_{2}≤ y_{2}). It follows that y_{1}< x_{1}and y_{3}≥ x_{3}, so y ≤_{3}x, it is a contradiction with hypothesis $x|{|}_{{\le}_{3}}y$. - Case 7:
- (a
_{1}= x_{1}, a_{3}> x_{3}) and (a_{1}= y_{1}, a_{3}= y_{3}, a_{2}≤ y_{2}). It follows that y_{1}= x_{1}and y_{3}> x_{3}, so y ≤_{3}x, which is a contradiction with hypothesis $x|{|}_{{\le}_{3}}y$. - Case 8:
- (a
_{1}= x_{1}, a_{3}= x_{3}, a_{2}≤ x_{2}) and (a_{1}< y_{1}, a_{3}≥ y_{3}). It follows that x_{1}< y_{1}and x_{3}≥ y_{3}, so x ≤_{3}y, which is a contradiction with hypothesis $x|{|}_{{\le}_{3}}y$. - Case 9:
- (a
_{1}= x_{1}, a_{3}= x_{3}, a_{2}≤ x_{2}) and (a_{1}= y_{1}, a_{3}> y_{3}). It follows that x_{1}= y_{1}and x_{3}> y_{3}, so x ≤_{3}y, which is a contradiction with hypothesis $x|{|}_{{\le}_{3}}y$.

_{3}z. That is, z = (min(x

_{1}, y

_{1}), 1 − min(x

_{1}, y

_{1}) − max(x

_{3}, y

_{3}), max(x

_{3}, y

_{3})) is the largest lower bound of x, y.

_{3}y = sup(x, y): Let

_{1}, w

_{2}, w

_{3}) = (max(x

_{1}, y

_{1}), 0, min(x

_{3}, y

_{3})).

_{1}≤ max(x

_{1}, y

_{1}) = w

_{1}, x

_{3}≥ min(x

_{3}, y

_{3}) = w

_{3}.

_{1}< w

_{1}and x

_{3}≥ w

_{3}, then x ≤

_{3}w.

_{1}= w

_{1}and x

_{3}> w

_{3}, then x ≤

_{3}w.

_{1}= w

_{1}and x

_{3}= w

_{3}, then y

_{1}≤ x

_{1}, y

_{3}≥ x

_{3}, so y ≤

_{3}x or x ≤

_{3}y, which is contradictory to the assumed condition $x|{|}_{{\le}_{3}}y$. Thus, x ≤

_{3}w.

_{3}w. Hence, w is an upper bound of x and y.

_{1}, a

_{2}, a

_{3}) ∈ D* such that x ≤

_{3}a, y ≤

_{3}a.

- Case 1:
- (x
_{1}< a_{1}, x_{3}≥ a_{3}) and (y_{1}< a_{1}, y_{3}≥ a_{3}). It follows that a_{1}> max(x_{1}, y_{1}) = w_{1}and a_{3}≤ min(x_{3}, y_{3}) = w_{3}. Thus, w ≤_{3}a. - Case 2:
- (a
_{1}= x_{1}, x_{3}> a_{3}) and (a_{1}= y_{1}, y_{3}> a_{3}). It follows that a_{1}= max(x_{1}, y_{1}) = w_{1}, a_{3}< min(x_{3}, y_{3}) = w_{3}, thus w ≤_{3}a. - Case 3:
- (a
_{1}= x_{1}, a_{3}= x_{3}, x_{2}≤ a_{2}) and (a_{1}= y_{1}, a_{3}= y_{3}, y_{2}≤ a_{2}). It follows that a_{1}= max(x_{1}, y_{1}) = w_{1}, a_{3}= min(x_{3}, y_{3}) = w_{3}and a_{2}≥ max(x_{2}, y_{2}) ≥ 0, thus w ≤_{3}a. - Case 4:
- (a
_{1}= x_{1}, x_{3}> a_{3}) and (y_{1}< a_{1}, y_{3}≥ a_{3}). It follows that a_{1}≥ max(x_{1}, y_{1}) = w_{1}and a_{3}≤ min(x_{3}, y_{3}) = w_{3}. If (a_{1}> max(x_{1}, y_{1}) = w_{1}, a_{3}≤ min(x_{3}, y_{3}) = w_{3}) or ((a_{1}= max(x_{1}, y_{1}) = w_{1}, a_{3}< min(x_{3}, y_{3}) = w_{3}), then w ≤_{3}a; If a_{1}= max(x_{1}, y_{1}) = w_{1}and a_{3}= min(x_{3}, y_{3}) = w_{3}, according the hidden condition a_{2}≥ 0, we can get w ≤_{3}a. - Case 5:
- (x
_{1}< a_{1}, x_{3}≥ a_{3}) and (a_{1}= y_{1}, y_{3}> a_{3}). It follows that a_{1}≥ max(x_{1}, y_{1}) = w_{1}and a_{3}≤ min(x_{3}, y_{3}) = w_{3}, similar to Case 4, so we can get w ≤_{3}a. - Case 6:
- (x
_{1}< a_{1}, x_{3}≥ a_{3}) and (a_{1}= y_{1}, a_{3}= y_{3}, y_{2}≤ a_{2}). It follows that x_{1}< y_{1}and x_{3}≥ y_{3}, so x ≤_{3}y, which is a contradiction with hypothesis $x|{|}_{{\le}_{3}}y$. - Case 7:
- (a
_{1}= x_{1}, x_{3}> a_{3}) and (a_{1}= y_{1}, a_{3}= y_{3}, y_{2}≤ a_{2}). It follows that y_{1}= x_{1}and y_{3}< x_{3}, so x ≤_{3}y, which is a contradiction with hypothesis $x|{|}_{{\le}_{3}}y$. - Case 8:
- (a
_{1}= x_{1}, a_{3}= x_{3}, x_{2}≤ a_{2}) and (y_{1}< a_{1}, y_{3}≥ a_{3}). It follows that y_{1}< x_{1}and y_{3}≥ x_{3}, so y ≤_{3}x, which is a contradiction with hypothesis $x|{|}_{{\le}_{3}}y$. - Case 9:
- (a
_{1}= x_{1}, a_{3}= x_{3}, x_{2}≤ a_{2}) and (a_{1}= y_{1}, y_{3}> a_{3}). It follows that y_{1}= x_{1}and y_{3}> x_{3}, so y ≤_{3}x, which is a contradiction with hypothesis $x|{|}_{{\le}_{3}}y$.

_{3}a. That is, w = (max(x

_{1}, y

_{1}), 0, min(x

_{3}, y

_{3})) is the smallest upper bound of x, y.

_{3}y = inf(x, y), x ∨

_{3}y = sup(x, y), and (D*, ≤

_{3}) is a lattice. □

## 4. New Operations and Properties of Totally Dependent-Neutrosophic Sets (Picture Fuzzy Sets)

_{3}and the rank of totally dependent-neutrosophic sets determined by score function and accuracy function (see Definition 6).

_{3}B if and only if (μ

_{A}(x), η

_{A}(x), ν

_{A}(x)) ≤

_{3}(μ

_{B}(x), η

_{B}(x), ν

_{B}(x)), ∀x∈X.

**Proposition**

**4.**

- (1)
- A ⊆
_{3}A; - (2)
- (A ⊆
_{3}B, B ⊆_{3}A) ⇒ A = B; - (3)
- (A ⊆
_{3}B, B ⊆_{3}C) ⇒ A ⊆_{3}C.

**Definition**

**9.**

- (1)
- (A ∪
_{3}B)(x) = $\{\begin{array}{ll}\left({\mu}_{A}\left(x\right),{\eta}_{A}\left(x\right),{\nu}_{A}\left(x\right)\right),& \mathrm{if}\u200a\left({\mu}_{B}\left(x\right),{\eta}_{B}\left(x\right),{\nu}_{B}\left(x\right)\right){\le}_{3}\left({\mu}_{A}\left(x\right),{\eta}_{A}\left(x\right),{\nu}_{A}\left(x\right)\right)\\ \left({\mu}_{B}\left(x\right),{\eta}_{B}\left(x\right),{\nu}_{B}\left(x\right)\right)\u200a,& \mathrm{if}\left({\mu}_{A}\left(x\right),{\eta}_{A}\left(x\right),{\nu}_{A}\left(x\right)\right)\u200a{\le}_{3}\u200a\left({\mu}_{B}\left(x\right),{\eta}_{B}\left(x\right),{\nu}_{B}\left(x\right)\right)\\ \left(\mathrm{max}\left({\mu}_{A}\left(x\right),{\mu}_{B}\left(x\right)\right),\hspace{0.17em}0,\hspace{0.17em}\mathrm{min}\left({\nu}_{A}\left(x\right),{\nu}_{B}\left(x\right)\right)\right),& \mathrm{otherwise}\end{array}$ - (2)
- (A ∩
_{3}B)(x) = $\{\begin{array}{l}\left({\mu}_{A}\left(x\right),{\eta}_{A}\left(x\right),{\nu}_{A}\left(x\right)\right),\u200a\u200a\u200a\u200a\u200a\u200a\u200a\u200a\u200a\hspace{0.17em}\u200a\u200a\u200a\u200a\u200a\u200a\u200a\u200a\u200a\hspace{0.17em}\mathrm{if}\left({\mu}_{A}\left(x\right),{\eta}_{A}\left(x\right),{\nu}_{A}\left(x\right)\right)\u200a{\le}_{3}\u200a\left({\mu}_{B}\left(x\right),{\eta}_{B}\left(x\right),{\nu}_{B}\left(x\right)\right)\\ \left({\mu}_{B}\left(x\right),{\eta}_{B}\left(x\right),{\nu}_{B}\left(x\right)\right)\u200a,\u200a\u200a\u200a\u200a\u200a\u200a\u200a\u200a\hspace{0.17em}\u200a\u200a\u200a\u200a\u200a\u200a\u200a\u200a\hspace{0.17em}\mathrm{if}\u200a\left({\mu}_{B}\left(x\right),{\eta}_{B}\left(x\right),{\nu}_{B}\left(x\right)\right){\le}_{3}\left({\mu}_{A}\left(x\right),{\eta}_{A}\left(x\right),{\nu}_{A}\left(x\right)\right)\\ (\mathrm{min}\left({\mu}_{A}\left(x\right),{\mu}_{B}\left(x\right)\right),\hspace{0.17em}1-\mathrm{min}\left({\mu}_{A}\left(x\right),{\mu}_{B}\left(x\right)\right)-\mathrm{max}({\nu}_{A}\left(x\right),{\nu}_{B}\left(x\right)),\mathrm{max}({\nu}_{A}\left(x\right),{\nu}_{B}\left(x\right))),\u200a\u200a\u200a\mathrm{otherwise}\end{array}$ - (3)
- ${A}^{{c}_{3}}$ = $\u200a\{\left(x,{\nu}_{A}\left(x\right),1-{\mu}_{A}\left(x\right)-{\eta}_{A}\left(x\right)-{\nu}_{A}\left(x\right),{\mu}_{A}\left(x\right)\right)|x\in X)\}$.

**Proposition**

**5.**

- (1)
- A ∪
_{3}B = {(μ_{A}(x), η_{A}(x), ν_{A}(x))∨_{3}(μ_{B}(x), η_{B}(x), ν_{B}(x))**|**x ∈ X}; - (2)
- A ∩
_{3}B = {(μ_{A}(x), η_{A}(x), ν_{A}(x))∧_{3}(μ_{B}(x), η_{B}(x), ν_{B}(x))**|**x ∈ X}.

**Proposition**

**6.**

- (1)
- A ∩
_{3}A = A, A ∪_{3}A = A; - (2)
- A ∩
_{3}B = B ∩_{3}A, A ∪_{3}B = B ∪_{3}A; - (3)
- (A ∩
_{3}B) ∩_{3}C = A ∩_{3}(B ∩_{3}C), (A ∪_{3}B) ∪_{3}C = A ∪_{3}(B ∪_{3}C); - (4)
- A ∩
_{3}(B ∪_{3}A) = A, A ∪_{3}( B ∩_{3}A) = A; - (5)
- A ⊆
_{3}B ⇔ A ∪_{3}B = B; A ⊆_{3}B ⇔ A ∩_{3}B = A.

**Proposition**

**7.**

**Proposition**

**8.**

- (1)
- ${\left({A\cap}_{3}B\right)}^{{c}_{3}}={A}^{{c}_{3}}{\cup}_{3}{B}^{{c}_{3}}$;
- (2)
- ${\left({A\cup}_{3}B\right)}^{{c}_{3}}={A}^{{c}_{3}}{\cap}_{3}{B}^{{c}_{3}}$.

**Proof.**

^{c3}= {(x, 𝜈

_{A}(x), 1 − 𝜇

_{A}(x) − 𝜂

_{A}(x) − 𝜈

_{A}(x), μ

_{A}(x))

**|**x ∈ X},

^{c3}= {(x, 𝜈

_{B}(x), 1 − 𝜇

_{B}(x) − 𝜂

_{B}(x) − 𝜈

_{B}(x), μ

_{B}(x))

**|**x ∈ X}.

_{3}A, then:

- Case 1:
- μ
_{B}(x) < μ_{A}(x) and ν_{B}(x) ≥ ν_{A}(x). It follows that ${A}^{{c}_{3}}$ ⊆_{3}${B}^{{c}_{3}}$. Thus ${\left({A\cap}_{3}B\right)}^{{c}_{3}}={A}^{{c}_{3}}{\cup}_{3}{B}^{{c}_{3}}$. - Case 2:
- μ
_{B}(x) = μ_{A}(x) and ν_{B}(x) > ν_{A}(x). It follows that ${A}^{{c}_{3}}$ ⊆_{3}${B}^{{c}_{3}}$. Thus ${\left(A{\cap}_{3}B\right)}^{{c}_{3}}={A}^{{c}_{3}}{\cup}_{3}{B}^{{c}_{3}}$. - Case 3:
- μ
_{B}(x) < μ_{A}(x), ν_{B}(x) = ν_{A}(x) and 𝜂_{B}(x) ≤ 𝜂_{A}(x). Then 1 − 𝜇_{A}(x) − 𝜂_{A}(x) − 𝜈_{A}(x) ≤ 1 − 𝜇_{B}(x) − 𝜂_{B}(x) − 𝜈_{B}(x). Thus ${A}^{{c}_{3}}$ ⊆_{3}${B}^{{c}_{3}}$, and ${\left(A{\cap}_{3}B\right)}^{{c}_{3}}={B}^{{c}_{3}}={A}^{{c}_{3}}{\cup}_{3}{B}^{{c}_{3}}$.

_{3}B, then ${\left(A{\cap}_{3}B\right)}^{{c}_{3}}={A}^{{c}_{3}}{\cup}_{3}{B}^{{c}_{3}}$.

_{3}A nor A ⊆

_{3}B, then:

_{3}B = {(x, min(μ

_{A}(x), μ

_{B}(x)), 1 − min(μ

_{A}(x), μ

_{B}(x)) − max(𝜈

_{A}(x), ν

_{B}(x)), max(𝜈

_{A}(x), ν

_{B}(x)))

**|**x ∈ X},

_{3}B = {(x, max(μ

_{A}(x), μ

_{B}(x)), 0, min(𝜈

_{A}(x), ν

_{B}(x)))

**|**x ∈ X}.

_{3}B)

^{c3}= {(x, max(𝜈

_{A}(x), ν

_{B}(x)), 0, min(μ

_{A}(x), μ

_{B}(x)))|x ∈ X},

^{c3}∪

_{3}B

^{c3}= {(x, max(𝜈

_{A}(x), ν

_{B}(x)), 0, min(μ

_{A}(x), μ

_{B}(x)))

**|**x ∈ X}.

**Theorem**

**1.**

_{TDNS}= {(x, 0, 0, 1) | x∈X}, 1

_{TDNS}={(x, 1, 0, 0) | x∈X}.

_{3}, ∩

_{3},

^{c}

_{3}, 0

_{TDNS}, 1

_{TDNS}) is a GM-algebra (i.e., generalized De Morgan algebra).

**Proof.**

_{3}, ∩

_{3},

^{c}

_{3}, 0

_{TDNS}, 1

_{TDNS}) is a GM-algebra.

_{3}, ∩

_{3},

^{c}

_{3}, 0

_{TDNS}, 1

_{TDNS}), that is, it is not a De Morgan algebra. □

**Example**

**3.**

_{3}B) ∪

_{3}C = {(a, 0.2, 0.2, 0.4), (b, 1, 0, 0)}; (A ∪

_{3}C) ∩

_{3}(B ∪

_{3}C) = {(a, 0.2, 0.4, 0.4), (b, 1, 0, 0)};

_{3}B) ∩

_{3}C = {(a, 0.2, 0.2, 0.4), (b, 0, 0, 1)}; (A ∩

_{3}C) ∪

_{3}(B ∩

_{3}C) = {(a, 0.2, 0, 0.4), (b, 0, 0, 1)}.

_{3}B) ∪

_{3}C ≠ (A ∪

_{3}C) ∩

_{3}(B ∪

_{3}C), (A ∪

_{3}B) ∩

_{3}C ≠ (A ∩

_{3}C) ∪

_{3}(B ∩

_{3}C).

**Proposition**

**9.**

_{α}, η

_{α}, ν

_{α}, ρ

_{α}), β = (μ

_{β}, η

_{β}, ν

_{β}, ρ

_{β}) are two totally dependent-neutrosophic numbers, and (μ

_{α}, η

_{α}, ν

_{α}) ≤

_{3}(μ

_{β}, η

_{β}, ν

_{β}), then α ≺ β or α ∼ β.

**Proof.**

_{α}, η

_{α}, ν

_{α}) ≤

_{3}(μ

_{β}, η

_{β}, ν

_{β}). By the definition of type-3 order relation ≤

_{3}, we have

- Case 1:
- (μ
_{α}< μ_{β}, ν_{α}≥ ν_{β}) or (μ_{α}= μ_{β}, ν_{α}> ν_{β}). It follows that S(α) = μ_{α}− ν_{α}< μ_{β}− ν_{β}= S(β). Thus, α ≺ β. - Case 2:
- (μ
_{α}= μ_{β}, ν_{α}= ν_{β}and η_{α}< η_{β}. It follows that S(α) = S(β), H(α) < H(β). Thus, α ≺ β. - Case 3:
- μ
_{α}= μ_{β}, ν_{α}= ν_{β}and η_{α}= η_{β}. It follows that S(α) = S(β), H(α) = H(β). Thus, α ∼ β.

**Example**

**4.**

## 5. New Operations and Properties of Totally Dependent-Neutrosophic Soft Sets

**Remark**

**5.**

**Definition**

**10**

**[28].**The type-2 complement of a totally dependent-neutrosophic soft set (F, A) over U is denoted as (F, A) ${}^{{c}_{2}}$ and is defined by (F, A) ${}^{{c}_{2}}$ = (F${}^{{c}_{2}}$, A), where F${}^{{c}_{2}}$: A → SNS(U) is a mapping given by:

**Definition**

**11**

**[28].**The type-2 intersection of two totally dependent-neutrosophic soft sets (F, A) and (G, B) over a common universe U is a totally dependent-neutrosophic soft set (H, C), where C = A ∪ B and for all e ∈ C,

_{F(e)}(x), μ

_{G(e)}(x)), min(η

_{F(e)}(x), η

_{G(e)}(x)), max(ν

_{F(e)}(x), ν

_{G(e)}(x)|x ∈ U}. This relation is denoted by (F, A) ∩

_{2}(G, B) = (H, C).

**Definition**

**12**

**[28].**The type-2 union of two totally dependent-neutrosophic soft sets (F, A) and (G, B) over a common universe U is a totally dependent-neutrosophic soft set (H, C), where C = A ∪ B and for all e ∈ C,

_{F(e)}(x), μ

_{G(e)}(x)), min(η

_{F(e)}(x), η

_{G(e)}(x)), min(ν

_{F(e)}(x), ν

_{G(e)}(x)|x ∈ U}. This relation is denoted by (F, A) ∪

_{2}(G, B) = (H, C).

**Definition**

**13.**

_{3}(G, B), if:

- (1)
- A ⊆ B;
- (2)
- ∀e ∈ A, F(e) ⊆
_{3}G(e), that is, ∀x ∈ U, (μ_{F(e)}(x) < μ_{G(e)}(x), ν_{F(e)}(x) ≥ ν_{G(e)}(x)), or (μ_{F(e)}(x) = μ_{G(e)}(x), ν_{F(e)}(x) > ν_{G(e)}(x)), or (μ_{F(e)}(x) = μ_{G(e)}(x), ν_{F(e)}(x) = ν_{G(e)}(x) and η_{F(e)}(x) ≤ η_{G(e)}(x)).

**Example**

**5.**

_{1}, x

_{2}, x

_{3}, x

_{4}} and E = {e

_{1}, e

_{2}, e

_{3}, e

_{4}, e

_{5}}. Suppose that (F, A) and (G, B) are two SNSSs over U, A = {e

_{1}, e

_{2}}, B = {e

_{1}, e

_{2}, e

_{5}} and

**Definition**

**14.**

_{3}(G, B) and (G, B) ⊆

_{3}(F, A).

**Definition**

**15.**

**Definition**

**16.**

_{3}(G, B) = (H, C), where C = A ∪ B, and ∀e ∈ C,

**Example**

**6.**

_{1}, x

_{2}, x

_{3}, x

_{4}}, E = {e

_{1}, e

_{2}, e

_{3}, e

_{4}, e

_{5}}, A = {e

_{1}, e

_{2}}, B = {e

_{1}, e

_{3}, e

_{5}}, and

_{3}(G, B) = (H, C), where C = A ∪ B = {e

_{1}, e

_{2}, e

_{3}, e

_{5}} and

_{3}(G, B) = (H, C) =

**Definition**

**17.**

_{3}(G, B) = (H, C), where C = A ∩ B, and ∀e ∈ C, H(e) = F(e) ∩

_{3}F(e).

**Example**

**7.**

_{3}(G, B) = (H, C), where C = A ∩ B = {e

_{1}} and

**Proposition**

**10.**

- (1)
- ${\left({\left(F,A\right)}^{{c}_{3}}\right)}^{{c}_{3}}=(F,A)$;
- (2)
- (F, A) ∪
_{3}(F, A) = (F, A), (F, A) ∩_{3}(F, A) = (F, A); - (3)
- (F, A) ∪
_{3}(G, B) = (G, B) ∪_{3}(F, A), (F, A) ∩_{3}(G, B) = (G, B) ∩_{3}(F, A); - (4)
- ((F, A) ∪
_{3}(G, B)) ∪_{3}(H, C) = (F, A) ∪_{3}((G, B) ∪_{3}(H, C)); - (5)
- ((F, A) ∩
_{3}(G, B)) ∩_{3}(H, C) = (F, A) ∩_{3}((G, B) ∩_{3}(H, C)), when A ∩ B ∩ C ≠ ∅.

**Proof.**

**Proposition**

**11.**

- (1)
- ((F, A) ∪
_{3}(G, A))^{c3}= (F, A)^{c3}∩_{3}(G, A)^{c3}; - (2)
- ((F, A) ∩
_{3}(G, A))^{c3}= (F, A)^{c3}∪_{3}(G, A)^{c3}.

**Proof.**

- (1)
- Assume that (F, A) ∪
_{3}(G, A) = (H, A) and (F, A)^{c}^{3}∩_{3}(G, A)^{c}^{3}= (I, A). Then:∀e ∈ A, H(e) = F(e) ∪_{3}G(e) (by Definition 16);∀e ∈ A, I(e)= F^{c}^{3}(e) ∩_{3}G^{c}^{3}(e) = (F(e))^{c}^{3}∩_{3}(G(e))^{c}^{3}= (F(e) ∪_{3}G(e))^{c}^{3}

(by Definitions 15, 17 and Proposition 8).Thus ∀e ∈ A, H^{c}^{3}(e) = (H(e))^{c}^{3}= (F(e) ∪_{3}G(e))^{c}^{3}= I(e). Since ((F, A) ∪_{3}(G, A))^{c}^{3}= (H^{c}^{3}, A), it follows that ((F, A) ∪_{3}(G, A))^{c}^{3}= (H^{c}^{3}, A) = (I, A) = (F, A)^{c}^{3}∩_{3}(G, A)^{c}^{3}. - (2)
- By (1), and using Proposition 10(1) we can get ((F, A) ∩
_{3}(G, A))^{c}^{3}= (F, A)^{c}^{3}∪_{3}(G, A)^{c}^{3}. □

**Proposition**

**12.**

- (1)
- (F, A) ∩
_{3}((F, A) ∪_{3}(G, A)) = (F, A); - (2)
- (F, A) ∪
_{3}((F, A) ∩_{3}(G, A)) = (F, A); - (3)
- (F, A) ⊆
_{3}(G, A) ⇔ (F, A) ∪_{3}(G, A) = (G, A); - (4)
- (F, A) ⊆
_{3}(G, A) ⇔ (F, A) ∩_{3}(G, A) = (F, A).

**Proof.**

- (1)
- Assume that (F, A) ∪
_{3}(G, A) = (H, A) and (F, A) ∩_{3}((F, A) ∪_{3}(G, A)) = (I, A). Then:∀e ∈ A, H(e) = F(e) ∪_{3}G(e) (by Definition 16);∀e ∈ A, I(e) = F(e) ∩_{3}H(e) = F(e) ∩_{3}(F(e) ∪_{3}G(e)) = F(e)

(by Definition 17 and Proposition 6(3))._{3}((F, A) ∪_{3}(G, A)) = (F, A). - (2)
- The proof is similar to (1).
- (3)
- From Proposition 6(4) and Definition 16, we get that (F, A) ⊆
_{3}(G, A) ⇔ (F, A) ∪_{3}(G, A) = (G, A). - (4)
- From Proposition 6(4) and Definition 17, we get that (F, A) ⊆
_{3}(G, A) ⇔ (F, A) ∩_{3}(G, A) = (F, A). □

**Theorem**

**2.**

_{3}, ∩

_{3},

^{c3}, (0

_{TDNSS}, A), (1

_{TDNSS}, A)) is a GM-algebra.

**Proof.**

_{3}(1

_{TDNS}

_{S}, A) and (0

_{TDNS}

_{S}, A) ⊆

_{3}(F, A), ∀(F, A) ∈ TDNSS(U, A).

_{3}, ∩

_{3}, (0

_{TDNS}

_{S}, A), (1

_{TDNS}

_{S}, A)) is a bounded lattice. Therefore, by Propositions 10(1), 12 and Definition 7, we get that (TDNSS(U, A), ∪

_{3}, ∩

_{3},

^{c}

^{3}, (0

_{TDNS}

_{S}, A), (1

_{TDNS}

_{S}, A)) is a GM-algebra. □

_{3}and ∩

_{3}) in TDNSS(U, A) is not satified.

**Example**

**8.**

_{1}, x

_{2}, x

_{3}, x

_{4}}, E = {e

_{1}, e

_{2}, e

_{3}, e

_{4}}, and A = {e

_{1}, e

_{2}}. Suppose that (F, A), (G, A), and (H, A) are totally dependent-neutrosophic soft sets over U, and

_{3}(G, A)) ∩

_{3}(H, A) ≠ ((F, A) ∩

_{3}(H, A)) ∪

_{3}((G, A)) ∩

_{3}(H, A));

_{3}(G, A)) ∪

_{3}(H, A) ≠ ((F, A) ∪

_{3}(H, A)) ∩

_{3}((G, A)) ∪

_{3}(H, A)).

## 6. Conclusions

_{3}is a partial ordering relation and D* makes up a lattice with respect to type-3 intersection and type-3 union operations. Then, we give some new operations of totally dependent-neutrosophic (picture fuzzy) sets and totally dependent-neutrosophic (picture fuzzy) soft sets, and their properties are presented. At the same time, we point out all of the totally dependent-neutrosophic (picture fuzzy) sets on X make up generalized De Morgan algebra with respect to type-3 intersection, type-3 union, and type-3 complement operations. Moreover, we prove that for appointed parameter sets, all of the totally dependent-neutrosophic (picture fuzzy) soft sets over U can also generate a generalized De Morgan algebra based on type-3 algebraic operations.

_{2}B if (∀x∈X, μ

_{A}(x) ≤ μ

_{B}(x), η

_{A}(x) ≤ η

_{B}(x), ν

_{A}(x) ≥ ν

_{B}(x)), which means that the first two membership functions (μ, η) have the same effect, but the three membership functions in the original definition of neutrosophic sets are completely independent, which is incongruous. For the type-1 inclusion relation, there is a similar problem. From Definition 8, we know that the type-3 inclusion relation has overcome this defect.

## Author Contributions

## Acknowledgments

## Conflicts of Interest

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**MDPI and ACS Style**

Zhang, X.; Bo, C.; Smarandache, F.; Park, C.
New Operations of Totally Dependent-Neutrosophic Sets and Totally Dependent-Neutrosophic Soft Sets. *Symmetry* **2018**, *10*, 187.
https://doi.org/10.3390/sym10060187

**AMA Style**

Zhang X, Bo C, Smarandache F, Park C.
New Operations of Totally Dependent-Neutrosophic Sets and Totally Dependent-Neutrosophic Soft Sets. *Symmetry*. 2018; 10(6):187.
https://doi.org/10.3390/sym10060187

**Chicago/Turabian Style**

Zhang, Xiaohong, Chunxin Bo, Florentin Smarandache, and Choonkil Park.
2018. "New Operations of Totally Dependent-Neutrosophic Sets and Totally Dependent-Neutrosophic Soft Sets" *Symmetry* 10, no. 6: 187.
https://doi.org/10.3390/sym10060187