**3. Quadripartitioned Single Valued Bipolar Neutrosophic Sets**

In this section, we introduce the concept of quadripartitioned single valued bipolar neutrosophic sets (QSVBNS).

**Definition 6.** *A quadripartitioned single valued bipolar neutrosophic set (QSVBNS) A in X defined as an object of the form A* = h*x*, *T P A* (*x*), *C P A* (*x*), *U P A* (*x*), *F P A* (*x*), *T N A* (*x*), *C N A* (*x*), *U N A* (*x*), *F N A* (*x*)i : *x* ∈ *X, where, T P A* , *C P A* , *U P A* , *F P A* : *X* → [0, 1] *and T N A* , *C N A* , *U N A* , *F N A* : *X* → [−1, 0]*. The positive membership degrees T P A* (*x*), *C P A* (*x*), *U P A* (*x*), *F P A* (*x*) *denote respectively the truth-membership, a contradiction-membership, an ignorance membership, and falsity membership of x* ∈ *X corresponding to a QSVBNS A. The negative membership degrees T N A* (*x*), *C N A* (*x*), *U N A* (*x*), *F N A* (*x*) *denote respectively the truth-membership, a contradictionmembership, an ignorance membership, and falsity membership of x* ∈ *X to some explicit counter-property corresponding to a QSVBNS A.*

*With respect to* (*T P A* (*x*), *F P A* (*x*)) *and* (*C P A* (*x*), *U P A* (*x*)) *and +ve and -ve membership grade, there is a sense of symmetry in the structure of QSVBNS.*

**Example 1.** *Suppose an environment organization desires to know peoples opinion on the following statement: "The fashion industry has helped economic growth but it also has a bad impact on the environment due to a large amount of carbon emissions".*

*To help the cause, a group of four experts, say X* = {*x*1, *x*2, *x*3, *x*4}*, has been asked to give their opinion. The statement can be divided into two parts as:*


*It may so happen that the opinion has the following outcomes: "a degree of agreement with statement* (*a*) *and disagreement with statement* (*b*)*", "a degree of agreement and disagreement with both the statements* (*a*) *and* (*b*)*", "a degree of neither agreement nor disagreement regarding both the statements", and "a degree of disagreement with statement* (*a*) *and agreement with statement* (*b*)*". According to the views of the four experts, the outcome represented in terms of QSVBNS as follows:*

$$\langle 0.9, 0.4, 0.3, 0.1, -0.1, -0.4, -0.3, -0.9 \rangle / \ge\_1 + \langle 0.3, 0.8, 0.2, 0.3, -0.4, -0.4, -0.9, -0.5 \rangle / \ge\_2 + \langle 0.2, 0.5, 0.5, 0.5, 0.9, -0.8, -0.1, -0.4, -0.2 \rangle / \ge\_4$$

*the above QSVBNS reflects that the expert x*<sup>1</sup> *agrees to the fact that the fashion industry has helped economic growth, whereas the expert x*<sup>2</sup> *believes that fashion industry might have helped economic growth but it also has affected the environment a bit. On the other side the expert x*<sup>3</sup> *is ignorant regarding the truth of both the statements and the expert x*<sup>4</sup> *opines that fashion industry does not have much impact on the world economy but he believes that it causes damage to the environment.*

**Remark 1.** *The relationship between QSVBNS and other extensions of fuzzy sets are diagrammatically depicted in the following figure (Figure 1).*

**Figure 1.** Relationship of quadripartitioned single valued bipolar neutrosophic sets (QSVBNS) with other extensions of fuzzy sets. .

**Definition 7.** *A QSVBNS, A over a universe X is said to be an absolute QSVBNS, denoted by* **X***, if for each x* ∈ *X*, *T P A* (*x*) = 1, *C P A* (*x*) = 1, *U P A* (*x*) = 0, *F P A* (*x*) = 0, *T N A* (*x*) = −1, *C N A* (*x*) = −1, *U N A* (*x*) = 0, *F N A* (*x*) = 0*.*

**Definition 8.** *A QSVBNS, A over a universe X is said to be a null QSVBNS, denoted by* **Θ***, if for each x* ∈ *X the membership values are respectively T P A* (*x*) = 0, *C P A* (*x*) = 0, *U P A* (*x*) = 1, *F P A* (*x*) = 1, *T N A* (*x*) = 0, *C N A* (*x*) = 0, *U N A* (*x*) = −1, *F N A* (*x*) = −1*.*

Alongside this, we expound some set theoretic operations on quadripartitioned single valued bipolar neutrosophic sets over a universe *X* and analyze some of their properties.

**Definition 9.** *Let A and B be two QSVBNS over X. Then A is said to be included in B, denoted by A* ⊆ *B, if for each x* ∈ *X*, *T P A* (*x*) ≤ *T P B* (*x*), *C P A* (*x*) ≤ *C P B* (*x*), *U P A* (*x*) ≥ *U P B* (*x*), *F P A* (*x*) ≥ *F P B* (*x*) *and T N A* (*x*) ≥ *T N B* (*x*), *C N A* (*x*) ≥ *C N B* (*x*), *U N A* (*x*) ≤ *U N B* (*x*), *F N A* (*x*) ≤ *F N B* (*x*)*.*

**Definition 10.** *Two QSVBNSs A and B are said to be equal if for each x* ∈ *X*, *T P A* (*x*) = *T P B* (*x*), *C P A* (*x*) = *C P B* (*x*), *U P A* (*x*) = *U P B* (*x*), *F P A* (*x*) = *F P B* (*x*) *and T N A* (*x*) = *T N B* (*x*), *C N A* (*x*) = *C N B* (*x*), *U N A* (*x*) = *U N B* (*x*), *F N A* (*x*) = *F N B* (*x*)*.*

**Definition 11.** *The complement of a QSVBNS A, denoted by A c , is defined as, A <sup>c</sup>* <sup>=</sup> <sup>h</sup>*x*, *<sup>F</sup> P A* (*x*), *U P A* (*x*), *C P A* (*x*), *T P A* (*x*), *F N A* (*x*), *U N A* (*x*), *C N A* (*x*), *T N A* (*x*)i : *x* ∈ *X, where, T P <sup>A</sup><sup>c</sup>* (*x*) = *F P A* (*x*), *C P <sup>A</sup><sup>c</sup>* (*x*) = *U P A* (*x*), *U P <sup>A</sup><sup>c</sup>* (*x*) = *C P A* (*x*), *F P <sup>A</sup><sup>c</sup>* (*x*) = *T P A* (*x*) *and T N <sup>A</sup><sup>c</sup>* (*x*) = *F N A* (*x*), *C N <sup>A</sup><sup>c</sup>* (*x*) = *U N A* (*x*), *U N <sup>A</sup><sup>c</sup>* (*x*) = *C N A* (*x*), *F N <sup>A</sup><sup>c</sup>* (*x*) = *T N A* (*x*), *x* ∈ *X.*

**Definition 12.** *The union of two QSVBNS A and B, denoted by A* ∪ *B is defined as, A* ∪ *B* = h*x*, *T P A* (*x*) ∨ *T P B* (*x*), *C P A* (*x*) ∨ *C P B* (*x*), *U P A* (*x*) ∧ *U P B* (*x*), *F P A* (*x*) ∧ *F P B* (*x*), *T N A* (*x*) ∧ *T N B* (*x*), *C N A* (*x*) ∧ *C N B* (*x*), *U N A* (*x*) ∨ *U N B* (*x*), *F N A* (*x*) ∨ *F N B* (*x*)i : *x* ∈ *X.*

**Definition 13.** *The intersection of two QSVBNS, A and B, denoted by A* ∩ *B is defined as, A* ∩ *B* = h*x*, *T P A* (*x*) ∧ *T P B* (*x*), *C P A* (*x*) ∧ *C P B* (*x*), *U P A* (*x*) ∨ *U P B* (*x*), *F P A* (*x*) ∨ *F P B* (*x*), *T N A* (*x*) ∨ *T N B* (*x*), *C N A* (*x*) ∨ *C N B* (*x*), *U N A* (*x*) ∧ *U N B* (*x*), *F N A* (*x*) ∧ *F N B* (*x*)i : *x* ∈ *X.*

**Example 2.** *For two QSVBNS A and B over X given by A* = h0.8, 0.6, 0.4, 0.1, −0.2, −0.3, −0.5, −0.7i/*x*<sup>1</sup> + h0.6, 0.5, 0.2, 0.3, −0.5, −0.4, −0.7, −0.8i/*x*<sup>2</sup> + h0.2, 0.5, 0.6, 0.7, −0.6, −0.1, −0.5, −0.7i/*x*<sup>3</sup> *and B* = h0.6, 0.5, 0.4, 0.3, −0.4, −0.7, −0.5, −0.6i/*x*<sup>1</sup> + h0.4, 0.5, 0.7, 0.5, −0.6, −0.4, −0.3, −0.4i/*x*<sup>2</sup> + h0.3, 0.7, 0.4, 0.2, −0.2, −0.1, −0.4, −0.8i/*x*<sup>3</sup>


**Theorem 1.** *Under the aforesaid set-theoretic operation, the quadripartitioned single valued bipolar neutrosophic sets satisfy the following properties:*

	- *A* ∪ **Θ** = *A and A* ∩ **X** = *A.*
	- (*A* ∪ *B*) ∪ *C* = *A* ∪ (*B* ∪ *C*) *and* (*A* ∩ *B*) ∩ *C* = *A* ∩ (*B* ∩ *C*)*.*
	- *A* ∪ (*B* ∩ *C*) = (*A* ∪ *B*) ∩ (*A* ∪ *C*) *and A* ∩ (*B* ∪ *C*) = (*A* ∩ *B*) ∪ (*A* ∩ *C*)*.*
	- (*A* ∪ *B*) *<sup>c</sup>* = *A <sup>c</sup>* <sup>∩</sup> *<sup>B</sup> c and* (*A* ∩ *B*) *<sup>c</sup>* = *A <sup>c</sup>* <sup>∪</sup> *<sup>B</sup> c .*

**Proof.** Proofs are plain-dealing.
