# Another Note on Paraconsistent Neutrosophic Sets

## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

**Definition**

**1.**

- sup T = t
_{sup}inf T = t_{inf}, - sup I = i
_{sup}, inf I = i_{inf}; - sup F =f
_{sup}, inf F = f_{inf}and - n
_{sup}= t_{sup}+ i_{sup}+ f_{sup}, n_{inf}= t_{inf}+ i_{inf}+ f_{inf}.

**Definition**

**2.**

**Definition**

**3.**

_{N}and 1

_{N}are defined as follows:

_{N}may be defined as:

- (0
_{1}) - 0
_{N}= x (0, 0, 1) - (0
_{2}) - 0
_{N}= x (0, 1, 1) - (0
_{3}) - 0
_{N}= x (0, 1, 0) - (0
_{4}) - 0
_{N}= x (0, 0, 0)

_{N}may be defined as:

- (1
_{1}) - 1
_{N}= x (1, 0, 0) - (1
_{2}) - 1
_{N}= x (1, 0, 1) - (1
_{3}) - 1
_{N}= x (1, 1, 0) - (1
_{4}) - 1
_{N}= x (1, 1, 1)

**Definition**

**4.**

_{A}, I

_{A}, F

_{A}), B = x (T

_{B}, I

_{B}, F

_{B}) be NSs. Then:

- (I
_{1}) - A∩B = x (T
_{A}⋅T_{B}, I_{A}⋅I_{B}, F_{A}⋅F_{B}) - (I
_{2}) - A∩B = x (T
_{A}∧T_{B}, I_{A}∧I_{B}, F_{A}∨F_{B}) - (I
_{3}) - A∩B = x (T
_{A}∧T_{B}, I_{A}∨I_{B}, F_{A}∨F_{B})

- (U
_{1}) - A∪B = x (T
_{A}∨T_{B}, I_{A}∨I_{B}, F_{A}∧F_{B}) - (U
_{2}) - A∪B = x (T
_{A}∨T_{B}, I_{A}∧I_{B}, F_{A}∧F_{B})

**Definition**

**5.**

_{j}∣j ∈ J} be an arbitrary family of NSs in X, then:

- (1)
- ∩A
_{j}may be defined as:- (i)
- ∩A
_{j}= x (∧, ∧, ∨) - (ii)
- ∩A
_{j}=x (∧, ∨, ∨)

- (2)
- ∪A
_{j}may be defined as:- (i)
- ∪A
_{j}= x (∨, ∨, ∧) - (ii)
- ∪A
_{j}= x (∨, ∧, ∧)

**Definition**

**6.**

- (1)
- 0
_{N}and 1_{N}ϵ τ_{τ}; - (2)
- G
_{1}∩G_{2}∈ τ, for any G_{1};, G_{2}∈ τ; - (3)
- ∪G
_{j}∈ τ or any subfamily {G_{j}}_{j ∈}_{J}of τ.

**X,τ**) is called a neutrosophic topological space.

## 3. Results

**Proposition**

**1.**

**Proof.**

- (1)
- It is necessary to omit a definition of ∩, because we will need ∩ of paraconsistent NSs to be paraconsistent. Indeed, let
**A**= x (1/2, 1/2, 1/2) and B = x (1/2, 1/3, 1/3) (both are paraconsistent NSs), but 1/4 + 1/6 + 1/6 is not > 1. Then, the case with product ((I_{1}), in Definition 4) must be deleted for paraconsistent NSs. - (2)
- The definitions of 0
_{N}and 1_{N}also have problems for paraconsistent NSs:- (a)
- Only (0
_{2}) and (1_{2}), (1_{3}), (1₄) are paraconsistent; - (b)
- If we want all NSs: 0
_{N}∪0_{N}, 0_{N}∪1_{N}, 1_{N}∪1_{N}, 0_{N}∩0_{N}, and 0_{N}∩1_{N}to be paraconsistent NSs, it is necessary to delete 1_{2}in Definition 3, because with this definition,

_{N}∩1

_{N}is equal either to x (0, 0, 1) which is not paraconsistent, or to x (0,1,1) = 0

_{N}.

_{N}∪0

_{N}= x (0, 1, 1) = 0

_{N},

_{N}∪1

_{N}is equal either to x (1,0,1), or to x (1,1,0), or x (1,1,1), i.e equal to 1

_{N},

_{N}∪1

_{N}is equal either to x (1,0,1), or to x (1,1,0), or x (1,1,1), i.e equal to 1

_{N},

_{N}∩0

_{N}= x (0,1,1) = 0

_{N},

_{N}∩1

_{N}is equal either to x (1,0,1), or to x (1,1,0), or x (1,1,1), i.e equal to 1

_{N}.

**Definition**

**7.**

- (1)
- 0
_{N}= x (0,1,1), and 1_{N}= x (1,1,0) or x (1,1,1), are in τ; - (2)
- G
_{1}∩G_{2}∈ τ for any G_{1}, G_{2}∈ τ (where ∩ is defined by (I_{2}) or (I_{3})); - (3)
- ∪G
_{j}∈ τ for any subfamily {G_{j}}_{j ∈}_{J}of τ (where ∪ is defined by Definition 5).

**Remark.**

## 4. Discussion

**Proposition**

**2.**

- (a)
- 0
_{N}= x (0,1,1), and 1_{N}= x (1,1,0) or x (1,1,1) - (b)
- ∩ defined by (I
_{2}) or (I_{3}) - (c)
- ∪ defined by Definition 5 is a bounded lattice.

**Proof.**

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**MDPI and ACS Style**

Lupiáñez, F.G. Another Note on Paraconsistent Neutrosophic Sets. *Symmetry* **2017**, *9*, 140.
https://doi.org/10.3390/sym9080140

**AMA Style**

Lupiáñez FG. Another Note on Paraconsistent Neutrosophic Sets. *Symmetry*. 2017; 9(8):140.
https://doi.org/10.3390/sym9080140

**Chicago/Turabian Style**

Lupiáñez, Francisco Gallego. 2017. "Another Note on Paraconsistent Neutrosophic Sets" *Symmetry* 9, no. 8: 140.
https://doi.org/10.3390/sym9080140