**2. Preliminaries**

**Definition 1** ([7])**.** *Let X be a universal set. A single valued neutrosophic set A over X is defined as, A* = {h*x*,h*TA*(*x*), *IA*(*x*), *FA*(*x*)ii : *x* ∈ *X*}*, where, TA*(*x*), *IA*(*x*), *FA*(*x*) *are respectively called truth-membership function, indeterminacy-membership function and falsity-membership function. These are defined by T<sup>A</sup>* : *X* → [0, 1], *I<sup>A</sup>* : *X* → [0, 1], *F<sup>A</sup>* : *X* → [0, 1] *respectively with the property that* 0 ≤ *TA*(*x*) + *IA*(*x*) + *FA*(*x*) ≤ 3*.*

**Definition 2** ([17])**.** *Let X be a universal set. A quadripartitioned neutrosophic set (QSVNS) A, over X is defined as, A* = {h*x*,h*TA*(*x*), *CA*(*x*), *UA*(*x*), *FA*(*x*)ii : *x* ∈ *X*}*, where TA*(*x*), *CA*(*x*), *UA*(*x*), *FA*(*x*) *are respectively called truth-membership function, contradiction-membership function, ignorance-membership function, and falsity-membership function. These are defined by T<sup>A</sup>* : *X* → [0, 1], *C<sup>A</sup>* : *X* → [0, 1], *U<sup>A</sup>* : *X* → [0, 1], *F<sup>A</sup>* : *X* → [0, 1] *respectively with the property that* 0 ≤ *TA*(*x*) + *CA*(*x*) + *UA*(*x*) + *FA*(*x*) ≤ 4*. When X is discrete, A is represented as, A* = *n* ∑ *k*=*i* h*TA*(*xi*), *CA*(*xi*), *UA*(*xi*), *FA*(*xi*),i/*x<sup>i</sup>* , *x<sup>i</sup>* ∈ *X. When X is continuous, A is represented as,* <sup>Z</sup> *X* h*TA*(*x*), *CA*(*x*), *UA*(*x*), *FA*(*x*)i/*x*, *x* ∈ *X.*

**Definition 3** ([26])**.** *A bipolar neutrosophic set (BNS) A in X is defined to be an object of the form A* = {h*x*, *T* <sup>+</sup>(*x*), *I* <sup>+</sup>(*x*), *F* <sup>+</sup>(*x*), *T* −(*x*), *I* −(*x*), *F* <sup>−</sup>(*x*)i : *x* ∈ *X*}*, where, T* <sup>+</sup>, *I* <sup>+</sup>, *F* <sup>+</sup> *are functions from X to* [0, 1] *and T* −, *I* −, *F* <sup>−</sup> *are functions from X to* [−1, 0]*. The +ve membership degrees T* <sup>+</sup>(*x*), *I* <sup>+</sup>(*x*), *F* <sup>+</sup>(*x*) *denote respectively the truth-membership, indeterminate-membership, and falsity-membership of x* ∈ *X corresponding to a bipolar neutrosophic set A and the -ve membership degrees T* −(*x*), *I* −(*x*), *F* −(*x*) *denote respectively the truth-membership, indeterminate-membership, and falsity-membership x* ∈ *X to some implicit counter-property corresponding to a bipolar neutrosophic set A.*

**Definition 4** ([17])**.** *Let A and B be two QSVNS. Then*

*(1) A* ⊆ *B if TA*(*x*) ≤ *TB*(*x*), *CA*(*x*) ≤ *CB*(*x*), *UA*(*x*) ≥ *UB*(*x*), *FA*(*x*) ≥ *FB*(*x*) *for all x* ∈ *X. n*

*i*=1

*(2) The complement of A is denoted by A c and is defined as A <sup>c</sup>* = ∑ *i*=1 h*FA*(*xi*), *UA*(*xi*), *CA*(*xi*), *TA*(*xi*),i/*x<sup>i</sup>* , *x<sup>i</sup>* ∈ *X, where TA<sup>c</sup>* (*xi*) = *FA*(*xi*), *CA<sup>c</sup>* (*xi*) = *UA*(*xi*) *and UA<sup>c</sup>* (*xi*) = *CA*(*xi*), *FA<sup>c</sup>* (*xi*) = *TA*(*xi*)*. (3) A* ∪ *B is defined as A* ∪ *B* = *n* ∑ h*TA*(*xi*) ∨ *TB*(*xi*), *CA*(*xi*) ∨ *CB*(*xi*), *UA*(*xi*) ∧ *UB*(*xi*), *FA*(*xi*) ∧

$$\begin{aligned} \mathsf{F\_B(\mathbf{x\_i})} / \langle X. \\ \mathsf{A} \cap B \text{ is defined as } A \cap B &= \sum\_{i=1}^n \langle \mathsf{T\_A(\mathbf{x\_i})} \wedge \mathsf{T\_B(\mathbf{x\_i})}, \mathsf{C\_A(\mathbf{x\_i})} \wedge \mathsf{C\_B(\mathbf{x\_i})}, \mathsf{U\_A(\mathbf{x\_i})} \vee \mathsf{U\_B(\mathbf{x\_i})}, \mathsf{F\_A(\mathbf{x\_i})} \vee \mathsf{C\_A(\mathbf{x\_i})}, \\ \mathsf{F\_B(\mathbf{x\_i})} / \langle X. \end{aligned}$$

**Definition 5** ([26])**.** *Let A*<sup>1</sup> = h*x*, *T* + 1 (*x*), *I* + 1 (*x*), *F* + 1 (*x*), *T* − 1 (*x*), *I* − 1 (*x*), *F* − 1 (*x*)i, *A*<sup>2</sup> = h*x*, *T* + 2 (*x*), *I* + 2 (*x*), *F* + 2 (*x*), *T* − 2 (*x*), *I* − 2 (*x*), *F* − 2 (*x*)i *be two bipolar neutrosophic sets (BNS) over the universe of discourse X. Then,*

