On some similarity measures and entropy on quadripartitioned single valued neutrosophic sets

Abstract

A notion of Quadripartitioned Single Valued Neutrosophic Sets (QSVNS) is introduced and a theoretical study on various set-theoretic operations on them has been carried out. The definitions of distance, similarity measure and entropy have been proposed. Finally an application of the proposed similarity measure in a problem pertaining to pattern recognition has been shown.

Keywords

Single valued neutrosophic sets quadripartitioned single valued neutrosophic sets similarity measure entropy

1 Introduction

The study of logic stretches from the fundamental classical 2-valued or Boolean logic to the study of the most general multi-valued logic. In case of classical logic, the values attributed to truth T and falsity F are 1 and 0 respectively. Later, the development of fuzzy logic was proposed as a generalization of the Boolean Logic [16] where T and F could assume any values from [0, 1]. Although the theory of fuzzy sets, which was proposed by L. A. Zadeh [19] in 1965, revolutionized the approach of dealing with uncertainties, yet it had its own limitations. Hence, in due course of time, several other improvizations of the fuzzy theory came into existance. Some of these include the theory of L-Fuzzy sets by Goguen [7], the theory of rough sets by Pawlak [12], the theory of intuitionistic fuzzy sets by K. T. Atanassov [1, 2] etc. Unlike the theory of fuzzy sets which associates a certain degree of membership, μ ε [0, 1] to each element of the universe of discourse, intuitionistic fuzzy sets associate a degree of non-membership ν ε [0, 1] as well, to each element where 0 ≤ μ + ν ≤ 1. However, the notion of indeterminacy, generally referred to as the hesitation margin, π, defined as, π = 1 - μ - ν in case of intuitionistic fuzzy sets was somewhat specific and completely dependent on the values of membership and non-membership of an element. This particular shortcoming of the theory of intuitionistic fuzzy sets was compensated by the introduction of the theory of neutrosophic sets by Florentin Samarandache in 1995 [13 –15]. Neutrosophic sets were proposed as a generalization of the intuitionistic fuzzy sets and neutrosophic logic sprouted from the branch of philosophy ’neutrosophy’ which means the study of neutralities. In case of neutrosophic sets, indeterminacy is taken care of separately and each element x is characterized by a truth-membership function T_A (x), an indeterminacy membership function I_A (x) and a falsity-membership function F_A (x), each of which belongs to the non-standard unit interval] 0^-, 1⁺ [.

Although the neutrosophic indeterminacy is independent of the truth and falsity-membership values and is more general than the hesitation margin of intuitionistic fuzzy sets yet, it is not very clear whether the indeterminacy associated to a particular element refers to the hesitation regarding its belongingness or non-belongingness. Expounding it more clearly, it might be stated that if for a particular event x, a person associates an indeterminacy membership I_A (x), it becomes difficult to comprehend whether the degree of uncertainty of the person regarding the occurance of the event is I_A (x) or the degree of uncertainty of the person regarding the non-occurance of that event is I_A (x). Thus, while some authors prefer to model the behaviour of indeterminacy in a way similar to that of the truth-membership, others may prefer to model its behaviour in a way similar to that of the falsity-membership. Quite naturally, this often leads to diverse approaches in dealing with uncertainty while executing various operations over neutrosophic sets as can be seen from the works of [17, 18].

At this juncture, it became necessary to look for means to find a solution to this conflict of interests. In this regard, Belnap’s four valued logic [3], which involves the study of truth T, falsity F, unknown U and contradictiton C proves to be a more general approach. Based on this, Smarandache proposed the notion of Four Numerical-valued neutrosophic logic [16] where the indeterminacy is split into two parts namely, ‘unknown’ viz. neither true nor false and ‘contradiction’ viz. both true and false, thereby providing a solution to the difficulties encountered in dealing with usual neutrosophic indeterminacy. This four-valued neutrosophic logic being of special interest to us, a notion of Quadripartitioned Single Valued Neutrosophic Sets (QSVNS, in short) is introduced in this paper whereby some of their properties have been studied and an application to an example of a pattern recognition problem has been shown.

The organization of the paper is as follows: Section 1 provides a brief introduction; Section 2 is dedicated to recalling some preliminary results; Section 3 introduces the concept of a quadripartitioned neutrosophic set and deals with some basic set-theoretic operations over quadripartitioned neutrosophic sets; Section 4 introduces the definition of similarity and distance measure; Section 5 deals with the concept of entropy over QSVNS; Section 6 consists of a comparative study of the proposed similarity measures in the context of classification of patterns; Section 7 concludes the paper.

2 Preliminaries

In this section we discuss some preliminary results that would prove to be useful in the following sections.

2.1 An overview of four valued logic

Belnap [3], with a view to device a practical tool for inference, introduced the concept of a four valued logic. In his work, corresponding to a certain information he considered four possibilities namely T: just True F: just false None: neither True nor False and, Both: both True and False. He symbolized these four truth values as 4 ={ T, F, Both, None } such that the possible values satisfied the conditions as shown in Table 1.

Also, for a mapping s from any atomic information into 4, the semantics was induced as $\begin{matrix} s (A & B) & = & s (A) & s (B) \\ s (A \lor B) & = & s (A) \lor s (B) \\ s (\sim A) & = & \sim s (A) \end{matrix}$

And, for any formulae A, B, C the following results hold:

A ∨ B ⇔ B ∨ A; A & B ⇔ B & A.

A ∨ (B ∨ C) ⇔ (A ∨ B) ∨ C; A & (B & C) ⇔ (A & B) & C.

A & (B ∨ C) ⇔ (A & B) ∨ (A & C); A ∨ (B & C) ⇔ (A ∨ B) & (A ∨ C).

(B ∨ C) & A ⇔ (B & A) ∨ (C & A); (B & C) ∨ A ⇔ (B ∨ A) & (C ∨ A).

∼∼ A ⇔ A

∼ (A & B) ⇔ ∼ A ∨ ∼ B; ∼ (A ∨ B) ⇔ ∼ A & ∼ B.

In [16], Smarandache recast Belnap’s concept of four valued logic as “Four-numerical valued neutrosophic logic” where the indeterminacy I is split as U = unknown and C = contradiction. T, F, C, U are subsets of [0, 1] instead of symbols.

2.2 Some results regarding neutrosophic sets

Definition 2.1. [17] Let X be a space of points with a generic element in X denoted by x. A single valued neutrosophic set A in X is an object of the form $A = \sum_{i = 1}^{n} 〈T_{A} (x_{i}), F_{A} (x_{i}), I_{A} (x_{i})〉 / x_{i}, x_{i} ɛ X$ when the universe of discourse is discrete.

It is represented as $A = \int_{X} 〈T_{A} (x), I_{A} (x), F_{A} (x)〉 / x, x ɛ X$ when the universe of discourse is continuous.

T_A, I_A, F_A respectively denote the truth-membership, indeterminacy membership and falsity-membership functions such that for each point x in X, $T_{A} (x), I_{A} (x), F_{A} (x) ε [0, 1] .$

The various operations are defined as, Containment: A ⊆ B iff T_A (x_i) ≤ T_B (x_i), I_A (x_i) ≤ I_B (x_i), F_A (x_i) ≥ F_B (x_i) for x_i ∈ X Complement: $c (A) = \sum_{i = 1}^{n} < F_{A} (x_{i}), 1 - I_{A} (x_{i}), T_{A} (x_{i}) > / x_{i}$ for x_i ∈ X Union: $A \cup B = \sum_{i = 1}^{n} < T_{A} (x_{i}) \lor T_{B} (x_{i}), I_{A} (x_{i}) \lor I_{B} (x_{i}), F_{A} (x_{i}) \land F_{B} (x_{i}) > / x_{i}$ for x_i ∈ X Intersection: $A \cap B = \sum_{i = 1}^{n} < T_{A} (x_{i}) \land T_{B} (x_{i}), I_{A} (x_{i}) \land I_{B} (x_{i}), F_{A} (x_{i}) \lor F_{B} (x_{i}) > / x_{i}$ for x_i ε X

Remark 2.2. [18] As stated before, an alternative approach where the behaviour of the indeterminacy is assumed to be similar to that of the falsity membership, exists and is widely in use. In such cases the operations are defined as, Containment: A ⊆ B iff T_A (x_i) ≤ T_B (x_i), I_A (x_i) ≥ I_B (x_i), F_A (x_i) ≥ F_B (x_i) for x_i ∈ X Union: $A \cup B = \sum_{i = 1}^{n} < T_{A} (x_{i}) \lor T_{B} (x_{i}), I_{A} (x_{i}) \land I_{B} (x_{i}), F_{A} (x_{i}) \land F_{B} (x_{i}) > / x_{i}$ for x_i ∈ X Intersection: $A \cap B = \sum_{i = 1}^{n} < T_{A} (x_{i}) \land T_{B} (x_{i}), I_{A} (x_{i}) \lor I_{B} (x_{i}), F_{A} (x_{i}) \lor F_{B} (x_{i}) > / x_{i}$ for x_i ε X.

Proposition 2.3. [17] SVNS satisfy the following properties under set-theoretic operations:

A ∪ B = B ∪ A; A ∩ B = B ∩ A.

A ∪ (B ∪ C) = (A ∪ B) ∪ C; A ∩ (B ∩ C) = (A ∩ B) ∩ C.

A ∪ A = A; A ∩ A = A.

A ∪ (A ∩ B) = A; A ∩ (A ∪ B) = A.

c (c (A)) = A.

De-Morgan’s laws hold viz. $\begin{matrix} c (A \cup B) = c (A) \cap c (B); \\ c (A \cap B) = c (A) \cup c (B) . \end{matrix}$

3 Quadripartitioned single valued neutrosophic sets

Based on Smarandache’s “Four Numerical-Valued neutrosophic logic” and “n-valued refined neutrosophic set” [16], we propose the concept of a “Quartipartitioned single valued neutrosophic set” (QSVNS). The term “quadripartitioned” means something that is divided into four characteristic features. Thus a quadripartitioned single valued neutrosophic set is an improvization of Wang’s single valued neutrosophic set in the sense that in case of QSVNS, the indeterminacy is split into two parts signifying contradiction and ignorance respectively. We now define a QSVNS as follows:

Definition 3.1. Let X be a non-empty set. A quadripartitioned neutrosophic set (QSVNS) A, over X characterizes each element x in X by a truth-membership function T_A, a contradiction-membership function C_A, an ignorance-membership function U_A and a falsity membership function F_A such that for each x ε X, T_A, C_A, U_A, F_A ε [0, 1] and 0 ≤ T_A (x) + C_A (x) + U_A (x) + F_A (x) ≤4 When X is discrete, A is represented as, $A = \sum_{i = 1}^{n} 〈T_{A} (x_{i}), C_{A} (x_{i}), U_{A} (x_{i}), F_{A} (x_{i})〉 / x_{i}, x_{i} ɛ X .$ However, when the universe of discourse is continuous, A is represented as, $A = \int_{X} 〈T_{A} (x), C_{A} (x), U_{A} (x), F_{A} (x)〉 / x, x ɛ X .$

Example 3.2. Consider the statement “A: Technology has subdued the art of letter writing”.

Suppose, this statement is posed in front of a group of four people, say, X ={ x₁, x₂, x₃, x₄ } (which constitute the universe under consideration) and they are requested to express their opinion regarding this statement. Now it may so happen that the opinion of the people may vary among the following possible options: “a degree of agreement with the statement”, “a degree of both agreement as well as disagreement regarding the statement”, “a degree of neither agreement nor disagreement regarding the statement” and “a degree of disagreement with respect to the statement”. According to the response of the people, the available information can be represented in terms of a QSVNS as follows: $\begin{matrix} A & = & 〈 0.7, 0.5, 0.2, 0.1 〉 / x_{1} + 〈 0.2, 0.6, 0.1, 0.5 〉 / x_{2} \\ + 〈 0.5, 0.2, 0.7, 0.1 〉 / x_{3} + 〈 0.4, 0.2, 0.5, 0.8 〉 / x_{4} \end{matrix}$

From the above QSVNS, it is seen that the person x₁ is to a great extent, in agreement with the statement whereas, x₄ mostly disagrees with the statement while x₂ opines that the statement is both true as well as false and x₃ is mainly in ignorance regarding the truth of the statement.

Remark 3.3. A QSVNS A, can be decomposed to yield two SVNS, say, A_t and A_f where the respective membership functions of both these sets are defined as $\begin{matrix} T_{A_{t}} (x) & = & T_{A} (x) = T_{A_{f}} (x) \\ I_{A_{t}} (x) & = & C_{A} (x); I_{A_{f}} (x) = U_{A} (x) \\ F_{A_{t}} (x) & = & F_{A} (x) = F_{A_{f}} (x), for all x ε X . \end{matrix}$

In this respect it needs to be stated that while performing set-theoretic operations over these SVNS, the behaviour of I_{A
_t} is treated similar to that of T_{A
_t} while the behaviour of I_{A
_f} is modelled in a way similar to that of F_{A
_f}.

Definition 3.4. A QSVNS is said to be an absolute QSVNS, denoted by A, iff its membership values are respectively defined as T_A (x) =1, C_A (x) =1, U_A (x) =0 and F_A (x) =0.

Definition 3.5. A QSVNS is said to be a null QSVNS, denoted by varTheta, iff its membership values are respectively defined as T_varTheta (x) =0, C_varTheta (x) =0, U_varTheta (x) =1 and F_varTheta (x) =1

We now propose some set-theoretic operations on quadripartitioned neutrosophic sets over a common universe X and study some of their basic properties as follows:

Definition 3.6. Consider two QSVNS A and B, over X. A is said to be contained in B, denoted by A ⊆ B iff T_A (x) ≤ T_B (x) , C_A (x) ≤ C_B (x) , U_A (x) ≥ U_B (x) and F_A (x) ≥ F_B (x).

Definition 3.7. The complement of a QSVNS A is denoted by A^c and is defined as

, x_i ε X

i.e. T_{A
^c} (x_i) = F_A (x_i), C_{A
^c} (x_i) = U_A (x_i), U_{A
^c} (x_i) = C_A (x_i) and F_{A
^c} (x_i) = T_A (x_i), x_i ε X.

Definition 3.8. The union of two QSVNS A and B is denoted by A ∪ B and is defined as

$A \cup B = \sum_{i = 1}^{n} < T_{A} (x_{i}) \lor T_{B} (x_{i}), C_{A} (x_{i}) \lor C_{B} (x_{i}), U_{A} (x_{i}) \land U_{B} (x_{i}), F_{A} (x_{i}) \land F_{B} (x_{i}) > / X$

Definition 3.9. The intersection of two QSVNS A and B is denoted by A ∩ B and is defined as

A ∩ B = < T_A (x_i) ∧ T_B (x_i) , C_A (x_i) ∧ C_B (x_i) , U_A (x_i) ∨ U_B (x_i) , F_A (x_i) ∨ F_B (x_i) >/X

Example 3.10. Consider two QSVNS defined over X, given by $\begin{matrix} A & = & 〈 0.7, 0.5, 0.2, 0.1 〉 / x_{1} + 〈 0.2, 0.6, 0.1, 0.5 〉 / x_{2} \\ + 〈 0.5, 0.2, 0.7, 0.1 〉 / x_{3} + 〈 0.4, 0.2, 0.5, 0.8 〉 / x_{4} \\ B & = & 〈 0.0, 0.25, 0.7, 0.6 〉 / x_{1} + 〈 0.5, 0.5, 0.3, 0.1 〉 / x_{2} \\ + 〈 0.9, 0.2, 0.0, 0.0 〉 / x_{3} + 〈 0.01, 0.1, 0.2, 1.0 〉 / x_{4} \end{matrix}$ Then we have $\begin{matrix} A^{c} & = & 〈 0.1, 0.2, 0.5, 0.7 〉 / x_{1} + 〈 0.5, 0.1, 0.6, 0.2 〉 / x_{2} \\ + 〈 0.1, 0.7, 0.2, 0.5 〉 / x_{3} + 〈 0.8, 0.5, 0.2, 0.4 〉 / x_{4} \end{matrix}$

$\begin{matrix} A \cup B \\ = 〈 0.7, 0.5, 0.2, 0.1 〉 / x_{1} + 〈 0.5, 0.6, 0.1, 0.1 〉 / x_{2} \\ + 〈 0.9, 0.2, 0.0, 0.0 〉 / x_{3} + 〈 0.4, 0.2, 0.2, 0.8 〉 / x_{4} \\ A \cap B \\ = 〈 0.0, 0.25, 0.7, 0.6 〉 / x_{1} + 〈 0.2, 0.5, 0.3, 0.5 〉 / x_{2} \\ + 〈 0.5, 0.2, 0.7, 0.1 〉 / x_{3} + 〈 0.01, 0.1, 0.5, 1.0 〉 / x_{4} \end{matrix}$

Proposition 3.11.Quadripartitioned single valued neutrosophic sets satisfy the following properties under the aforementioned set-theoretic operations:

A ∪ B = B ∪ A ; A ∩ B = B ∩ A

A ∪ (B ∪ C) = (A ∪ B) ∪ C ; A ∩ (B ∩ C) = (A ∩ B) ∩ C

A ∪ (A ∩ B) = A ; A ∩ (A ∪ B) = A

(i) (A^c) ^c = A

A^c = Θ

Θ^c = A

De-Morgan’s laws hold viz.

(A ∪ B) ^c = A^c ∩ B^c; (A ∩ B) ^c = A^c ∪ B

(i)A ∪ A = A

A ∩ A = A

A ∪ Θ = A

A ∩ Θ = Θ

Proof. The proofs are straight-forward.

4 Various similarity measures on quadripartitioned neutrosophic sets

Definition 4.1. Let QSVNS (X) denote the set of all quadripartitioned neutrosophic sets, over the non-empty universe of discourse X. Then a mapping s : QSVNS (X) × QSVNS (X) → [0, 1] is said to be a similarity measure iff for A, B ε QSVNS (X) it satisfies the following properties viz.

s (A, B) = s (B, A)

0 ≤ s (A, B) <1 and s (A, B) =1 iff A = B

for any A, B, C ε QSVNS (X), such that, A ⊂ B ⊂ C, s (A, C) ≤ s (A, B) ∧ s (B, C).

A recent study shows that several measures of similarity exist in the literature which do not satisfy the triangle inequality (S3). Some example of such similarity measures are,

Weighted similarity measure for SVNS based on matching function [11]: $\begin{matrix} s^{w} (A, B) \\ = \frac{\sum_{i = 1}^{n} ω_{i} {(T_{A} (x_{i}) . T_{B} (x_{i}) + I_{A} (x_{i}) . I_{B} (x_{i}) + F_{A} (x_{i}) . F_{B} (x_{i}))}^{2}}{\sum_{i = 1}^{n} ω_{i} {(T_{A}^{2} (x_{i}) + I_{A}^{2} (x_{i}) + F_{A}^{2} (x_{i})) . (T_{B}^{2} (x_{i}) + I_{B}^{2} (x_{i}) + F_{B}^{2} (x_{i}))}} \end{matrix}$ where ω_i ε [0, 1] is the weight associated to each x_i ε X.

Cosine similarity measure for interval valued neutrosophic sets [4]:

$C_{N} (A, B) = \frac{1}{n} \sum_{i = 1}^{n} [(▵ T_{A} (x_{i}) . ▵ T_{B} (x_{i}) + ▵ I_{A} (x_{i}) . ▵ I_{B} (x_{i}) + ▵ F_{A} (x_{i}) . ▵ F_{B} (x_{i})) / (▵ T_{A}^{2} (x_{i}) + ▵ I_{A}^{2} (x_{i}) + ▵ F_{A}^{2} (x_{i}))^{\frac{1}{2}} .$

$(▵ T_{B}^{2} (x_{i}) + ▵ I_{B}^{2} (x_{i}) + ▵ F_{B}^{2} (x_{i}))^{\frac{1}{2}}]$

Dice similarity measure for single valued neutrosophic multisets [18]:

$S_{D} (A, B) = \frac{1}{n} \sum_{j = 1}^{n} \sum_{i = 1}^{l_{j}} [2 {(T_{A}^{i} (x_{j}) . T_{B}^{i} (x_{j}) + I_{A}^{i} (x_{j})) . (I_{B}^{i} (x_{j}) + F_{A}^{i} (x_{j}) . F_{B}^{i} (x_{j}))} / {(T_{A}^{i} (x_{j})^{2} + I_{A}^{i} (x_{j})^{2} + F_{A}^{i} (x_{j})^{2}) + (T_{B}^{i} (x_{j})^{2} + I_{B}^{i} (x_{j})^{2} + F_{B}^{i} (x_{j})^{2})}$ ]

where l_j = L (x_j : A, B) = max {L (x_j : A) , L (x_j : B)} is the maximum length of an element, j = 1, 2, . . . , n.

These measures have found extensive applicability in various spheres pertaining to decision making problems and yet they do not satisfy (S3). We thus introduce the definition of a different kind of similarity measure, which we term as quasi-similarity, a term which was first mentioned in [6] as follows:

Definition 4.2. A mapping s : QSVNS (X) × QSVNS (X) → [0, 1] is said to be a quasi-similarity measure if it satisfies (S1)and (S2).

4.1 Distance based similarity measure

Before proceeding to define the distance based similarity measure, the notion of distance between QSVNS is introduced first.

Definition 4.3. A mapping d : QSVNS (X) × QSVNS (X) → R⁺, where R⁺ is the set of all positivereal numbers, is said to be a distance measure iff for A, B, C ε QSVNS (X), it satisfies the following properties

d (A, B) ≥0 and equality holds iff A = B.

d (A, B) = d (B, A)

d (A, B) ≤ d (A, B) + d (A, C)

Let A, B ε QSVNS (X) then for all x_i ε X, we define the following distance measures.

Definition 4.4. The Hamming distance between A and B is defined as $\begin{matrix} h (A, B) \\ = \sum_{i = 1}^{n} (| T_{A} (x_{i}) - T_{B} (x_{i}) | + | C_{A} (x_{i}) - C_{B} (x_{i}) | \\ + | U_{A} (x_{i}) - U_{B} (x_{i}) | + | F_{A} (x_{i}) - F_{B} (x_{i}) |) . \end{matrix}$

Definition 4.5. The Normalized Hamming distance between A and B is defined as $h_{N} (A, B) = \frac{1}{4 n} h (A, B) .$

Definition 4.6. The Euclidean distance between A and B is defined as $\begin{matrix} e (A, B) \\ = \sum_{i = 1}^{n} ({| T_{A} (x_{i}) - T_{B} (x_{i}) |}^{2} + {| C_{A} (x_{i}) - C_{B} (x_{i}) |}^{2} \\ + {| U_{A} (x_{i}) - U_{B} (x_{i}) |}^{2} + {| F_{A} (x_{i}) - F_{B} (x_{i}) |}^{2})^{\frac{1}{2}} \end{matrix}$

Definition 4.7. The Normalized Euclidean distance between A and B is defined as $e_{N} (A, B) = \frac{1}{2 \sqrt{n}} e (A, B)$ .

The notion of distance measure plays a significant role since distance measures are not restricted to the study of distances between sets only. As can be seen from the works of [5 , 11], the distance measure between two sets was implemented to deduce a distance-based similarity measure for the sets concerned. Also, induced ordered weighted aggregation distance (IOWAD) operators introduced by J. Merigo and M. Casanovas, are aggregation operators that extend the usual OWA operators by using distance measures and a re-ordering of the arguments depending on the order-inducing variables concerned. These operators are powerful tools for decision making using distance measures as can be seen from noteworthy works like [8 –10]. However, in the present paper we restrict ourselves to the study of distance-based similarity measures only.

Definition 4.8. The distance based similarity measure between A, B ε QSVNS (X) is defined as $S_{d} (A, B) = \frac{1}{1 + d (A, B)}$ where the distance between A and B can be evaluated using any of the afore-stated methods.

Example 4.9. Consider the QSVNS of Example 3.5, then we have, h (A, B) =5.14, h_N (A, B) =0.32, e (A, B) =1.52 and e_N (A, B) =0.38. Also, the distance based similarity measure corresponding to the Hamming distance, Normalized Hamming distance, Euclidean distance and Normalized Euclidean distance are found to be 0.163, 0.758, 0.397 and 0.725 respectively.

4.2 Similarity measures based on membership values

Suppose A, B ε QSVNS (X). At first some functions are defined which would be useful in defining the similarity measure.

For each x_i ε X, i = 1, 2, . . . , n and for j = 1, 2, 3, 4 define the functions $h_{j}^{A, B} : X \to [0, 1]$ respectively as, $\begin{matrix} h_{1}^{A, B} (x_{i}) & = & | T_{B} (x_{i}) - T_{A} (x_{i}) | \\ h_{2}^{A, B} (x_{i}) & = & | F_{A} (x_{i}) - F_{B} (x_{i}) | \\ h_{3}^{A, B} (x_{i}) & = & \frac{1}{3} (h_{1}^{A, B} (x_{i}) + h_{2}^{A, B} (x_{i}) \\ + | C_{B} (x_{i}) - C_{A} (x_{i}) |) \\ h_{4}^{A, B} (x_{i}) & = & | U_{A} (x_{i}) - U_{B} (x_{i}) | \end{matrix}$

The functions defined above measure the difference between the various membership values corresponding to the two sets A and B w.r.t. each x_i.Define a mapping, $S^{'} (A, B) = 1 - \frac{1}{4 n} \sum_{i = 1}^{n} \sum_{j = 1}^{4} h_{j}^{A, B} (x_{i})$

Theorem 4.10. is a measure of similarity between the two quadripartitioned single valued neutrosophic sets A and B over X.

Proof. The proofs are straight-forward.

When an information is represented in terms of a QSVNS, the uncertainty associated with the information are characterized by four membership functions describing the aspects ‘true’, ‘both true and false’, ‘neither true nor false’ and ‘false’. Naturally, it would be a better attempt if most of the available information could be put into best use while defining the measure of similarity between two QSVNS. Thus, we improvize the definition of the proposed similarity measure as follows:

Suppose, $\begin{matrix} τ_{1}^{A, B} (x_{i}) \\ = \frac{1}{2} | (T_{A} (x_{i}) . F_{A} (x_{i}) - C_{A} (x_{i})) \\ - (T_{B} (x_{i}) . F_{B} (x_{i}) - C_{B} (x_{i})) | \\ = \frac{1}{2} | (T_{A} (x_{i}) . F_{A} (x_{i}) - T_{B} (x_{i}) . F_{B} (x_{i})) \\ + (C_{B} (x_{i}) - C_{A} (x_{i})) | \\ τ_{2}^{A, B} (x_{i}) = \frac{1}{3} (h_{1}^{A, B} (x_{i}) + h_{2}^{A, B} (x_{i}) \\ + | U_{B} (x_{i}) - U_{A} (x_{i}) |) \end{matrix}$ Finally define, $S^{″} (A, B) = 1 - [\frac{1}{n} \sum_{i = 1}^{n} \frac{1}{4} (h_{1}^{A, B} (x_{i}) + h_{2}^{A, B} (x_{i}) {+ τ_{1}^{A, B} (x_{i}) + τ_{2}^{A, B} (x_{i}))^{p}]}^{\frac{1}{p}}$ where p is any positive integer, and is defined to be the “order of similarity”.

Theorem 4.11. is a similarity measure.

Proof.

(i) It is easy to prove that .

(ii) We have, T_A (x_i) , C_A (x_i) , U_A (x_i) , F_A (x_i) ε [0, 1]. Thus, $h_{1}^{A, B} (x_{i})$ attains its maximum value if either one of T_A (x_i) or T_B (x_i) is equal to 1 while the other is 0 and in that case the maximum value is 1. Similarly, $h_{1}^{A, B} (x_{i})$ attains a minimum value 0 if T_A (x_i) and T_B (x_i) are equal. So, it follows that $0 \leq h_{1}^{A, B} (x_{i}) \leq 1$ . Similarly it can be shown that $h_{2}^{A, B} (x_{i})$ , lies within [0, 1] for all x_i ε X. Similar proofs apply for $τ_{1}^{A, B} (x_{i})$ and $τ_{2}^{A, B} (x_{i})$ . So, $0 \leq h_{1}^{A, B} (x_{i}) + h_{2}^{A, B} (x_{i}) + τ_{1}^{A, B} (x_{i}) + τ_{2}^{A, B} (x_{i}) \leq 4$ ⇒ $0 \leq \frac{1}{n} \sum_{i = 1}^{n} [\frac{1}{4} (h_{1}^{A, B} (x_{i}) + h_{2}^{A, B} (x_{i}) + τ_{1}^{A, B} (x_{i}) + τ_{2}^{A, B} (x_{i}))]^{p} \leq 1$

which implies .

Now iff $h_{1}^{A, B} (x_{i}) + h_{2}^{A, B} (x_{i}) + τ_{1}^{A, B} (x_{i}) + τ_{2}^{A, B} (x_{i}) = 0$ for each $x_{i} ε X \Leftrightarrow h_{1}^{A, B} (x_{i}) = 0, h_{2}^{A, B} (x_{i}) = 0, τ_{1}^{A, B} (x_{i}) = 0 = τ_{2}^{A, B} (x_{i})$ ⇔T_A (x_i) = T_B (x_i) , C_A (x_i) = C_B (x_i) , U_A (x_i) = U_B (x_i) , F_A (x_i) = F_B (x_i)

i.e. iff A = B.

(iii) Suppose P ⊂ Q ⊂ R. then, we have, T_P (x_i) ≤ T_Q (x_i) ≤ T_R (x_i) , C_P (x_i) ≤ C_Q (x_i) ≤ C_R (x_i) , U_P (x_i) ≥ U_Q (x_i) ≥ U_R (x_i) and F_P (x_i) ≥ F_Q (x_i) ≥ F_R (x_i) for all x_i ε X.

Consider $h_{1}^{P, Q} (x_{i})$ and $h_{1}^{P, R} (x_{i})$ . Since T_Q (x_i) _leqTR (x_i), it follows that, |T_R (x_i) - T_P (x_i) | ≥ |T_Q (x_i) - T_P (x_i) | $\Rightarrow h_{1}^{P, R} (x_{i}) \geq h_{1}^{P, Q} (x_{i})$ .

Similarly it can be shown that $h_{2}^{P, R} (x_{i}) \geq h_{2}^{P, Q} (x_{i})$ , for all x_i ε X.

Next, consider $τ_{1}^{P, Q} (x_{i})$ and $τ_{1}^{P, R} (x_{i})$ .

Since T_P (x_i) ≤ T_Q (x_i) ≤ T_R (x_i) , F_P (x_i) ≥ F_Q (x_i) ≥ F_R (x_i), it follows that, 0 ≤ T_P (x_i) . F_P (x_i) - T_Q (x_i) . F_Q (x_i) ≤ T_P (x_i) . F_P (x_i) - T_R (x_i) . F_R (x_i). Again, C_P (x_i) ≤ C_Q (x_i) ≤ C_R (x_i), we have 0 ≤ C_Q (x_i) - C_P (x_i) ≤C_R (x_i) - C_P (x_i). Thus, | (T_P (x_i) . F_P (x_i) - T_Q (x_i) . F_Q (x_i)) + (C_Q (x_i) - C_P (x_i)) | ≤| (T_P (x_i) . F_P (x_i) - T_R (x_i) . F_R (x_i)) + (C_R (x_i) - C_P (x_i)) | $\Rightarrow τ_{1}^{P, Q} (x_{i}) \leq τ_{1}^{P, R} (x_{i})$

Similar proof can be constructed for $τ_{2}^{P, Q} (x_{i})$ respectively for each x_i. Thus, one can safely say that, $\begin{matrix} \frac{1}{4} (h_{1}^{P, R} (x_{i}) + h_{2}^{P, R} (x_{i}) + τ_{1}^{P, R} (x_{i}) + τ_{2}^{P, R} (x_{i})) \\ \geq \frac{1}{4} (h_{1}^{P, Q} (x_{i}) + h_{2}^{P, Q} (x_{i}) + τ_{1}^{P, Q} (x_{i}) + τ_{2}^{P, Q} (x_{i})) \end{matrix}$ which means $\begin{matrix} {{(\frac{1}{n} \sum_{i = 1}^{n} [\frac{1}{4} (h_{1}^{P, R} (x_{i}) + h_{2}^{P, R} (x_{i}) + τ_{1}^{P, R} (x_{i}) + τ_{2}^{P, R} (x_{i}))]}^{p})}^{\frac{1}{p}} \\ \geq {(\frac{1}{n} \sum_{i = 1}^{n} {[\frac{1}{4} (h_{1}^{P, Q} (x_{i}) + h_{2}^{P, Q} (x_{i}) + τ_{1}^{P, Q} (x_{i}) + τ_{2}^{P, Q} (x_{i}))]}^{p})}^{\frac{1}{p}}, \end{matrix}$

for any positive integer p.

Thus, it automatically follows that, .

The proof of follows in a similar manner. Hence, it can be concluded that which completes the proof.

Definition 4.12. The weighted similarity measure can be defined in a similar way as, $\begin{matrix} S_{w}^{″} (A, B) \\ = 1 - [\frac{1}{n} \sum_{i = 1}^{n} (\frac{1}{3} w_{i} | h_{1}^{A, B} (x_{i}) + h_{2}^{A, B} (x_{i}) \\ {{+ τ_{1}^{A, B} (x_{i}) + τ_{2}^{A, B} (x_{i}) |)}^{p}]}^{\frac{1}{p}} \end{matrix}$ where w_i are the weights associated to the elements x_i of the universe, i = 1, 2, . . . , n such that 0 ≤ w_i ≤ 1 and $\sum_{i = 1}^{n} w_{i} = 1$ .

It can be easily shown that $S_{w}^{″} (A, B)$ is a measure of similarity.

4.3 Similarity measure based on correlation coefficient

Definition 4.13. The correlation coefficient between two QSVNS A and B is defined as, $\begin{matrix} β (A, B) \\ = [\sum_{i = 1}^{n} (T_{A} (x_{i}) . T_{B} (x_{i}) + C_{A} (x_{i}) . C_{B} (x_{i}) \\ + U_{A} (x_{i}) . U_{B} (x_{i}) + F_{A} (x_{i}) . F_{B} (x_{i}))] / \\ [(\sum_{i = 1}^{n} (T_{A}^{2} (x_{i}) + C_{A}^{2} (x_{i}) + U_{A}^{2} (x_{i}) + F_{A}^{2} (x_{i}))) . \\ {(\sum_{i = 1}^{n} (T_{B}^{2} (x_{i}) + C_{B}^{2} (x_{i}) + U_{B}^{2} (x_{i}) + F_{B}^{2} (x_{i})))]}^{\frac{1}{2}} \end{matrix}$

Remark 4.14.β (A, B) is a quasi-similarity between the sets A and B.

4.4 Quadripartitioned similarity measure

Definition 4.15. For any two QSVNS A and B over a universe X ={ x₁, x₂, . . . , x_n }, define the quadripartitioned similarity measure S₄ : QSVNS (X) × QSVNS (X) → [0, 1] ⁴ between two QSVNS A and B in the sense of Broumi and Smarandache [5] as $S_{4} (A, B) = 〈S_{T} (A, B), S_{C} (A, B), S_{U} (A, B), S_{F} (A, B)〉$ where $\begin{matrix} S_{T} (A, B) & = & \frac{\sum_{i = 1}^{n} \min (T_{A} (x_{i}), T_{B} (x_{i}))}{\sum_{i = 1}^{n} \max (T_{A} (x_{i}), T_{B} (x_{i}))} \\ S_{C} (A, B) & = & \frac{\sum_{i = 1}^{n} \min (C_{A} (x_{i}), C_{B} (x_{i}))}{\sum_{i = 1}^{n} \max (C_{A} (x_{i}), C_{B} (x_{i}))} \\ S_{U} (A, B) & = & \frac{\sum_{i = 1}^{n} \min (U_{A} (x_{i}), U_{B} (x_{i}))}{\sum_{i = 1}^{n} \max (U_{A} (x_{i}), U_{B} (x_{i}))} \\ S_{F} (A, B) & = & \frac{\sum_{i = 1}^{n} \min (F_{A} (x_{i}), F_{B} (x_{i}))}{\sum_{i = 1}^{n} \max (F_{A} (x_{i}), F_{B} (x_{i}))} \end{matrix}$

A quadripartitioned similarity measure is in fact a quadruple comprising four different similarity measures in terms of the four membership values of a QSVNS. At times, for the sake of convenience S₄ (A, B) is also represented in the form of a matrix: $S_{4} (A, B) = 〈 \begin{matrix} S_{T} (A, B) & S_{F} (A, B) \\ S_{C} (A, B) & S_{U} (A, B) \end{matrix} 〉$ Such a similarity measure takes into account, separately, the degrees of similarity between the various membership values. It is easy to show that S₄ (A, B) satisfies all the axiomatic properties of a similarity measure. We only give an outline of the proof of the triangle inequality as follows:

When A ⊂ B ⊂ C, for each x_i ε X,

min (T_A (x_i) , T_B (x_i)) = T_A (x_i) max (T_A (x_i) , T_B (x_i)) = T_B (x_i) min (T_A (x_i) , T_C (x_i)) = T_A (x_i) and max (T_A (x_i) , T_C (x_i)) = T_C (x_i). Thus, in such a case, $S_{T} (A, B) = \frac{\sum_{i = 1}^{n} T_{A} (x_{i})}{\sum_{i = 1}^{n} T_{B} (x_{i})}$ .

Similarly, $S_{T} (A, C) = \frac{\sum_{i = 1}^{n} T_{A} (x_{i})}{\sum_{i = 1}^{n} T_{C} (x_{i})}$ .

Since, T_B (x_i) ≤ T_C (x_i), it follows that S_T (A, C) ≤ S_T (A, B).

Similar proofs follow for S_C (A, B), S_U (A, B) and S_F (A, B). Thus, all the components of S₄ (A, B) individually satisfy the properties (S1) - (S3).

If, further, we introduce the notations $\bar{0} = 〈 0, 0, 0, 0 〉$ , $\bar{1} = 〈 1, 1, 1, 1 〉$ and an ordering on [0, 1] ⁴ of the form, iff a_i ≤ b_i, for a_i, b_i ε [0, 1], i = 1, 2, 3, 4; then for any A, B ε QSVNS (X)

$\bar{0} ⪯ S_{4} (A, B) ⪯ \bar{1}$

S₄ (A, B) = S₄ (B, A)

for A, B, C ε QSVNS (X) such that A ⊆ B ⊆ C, we have S₄ (A, C) ⪯ S₄ (A, B) ∧ S₄ (B, C).

5 Entropy measure for QSVNS

Definition 5.1. Let X be a non-empty universe of discourse. A mapping e : QSVNS (X) → [0, 1] is said to be an entropy on QSVNS (X) if e satisfies the following properties:

e (A) =0 iff $A ε P (X)$

e (A) =1 for A ε QSVNS (X) if T_A (x) = C_A (x) = U_A (x) = F_A (x) =0.5 for all x ε X.

e (A) ≥ e (B) iff T_A (x) + F_A (x) ≤ T_B (x) + F_B (x) and |C_A (x) - U_A (x) | ≤ |C_B (x) - U_B (x) |, for all x ε X.

e (A) = e (A^c), for all A ε QSVNS (X).

Proposition 5.2. $e_{m} (A) = 1 - \frac{1}{n} \sum_{i = 1}^{n} (T_{A} (x_{i}) + F_{A} (x_{i})) . | C_{A} (x_{i}) - U_{A} (x_{i}) |$ , for all x_i ε X is an entropy measure.

Proof. Proofs are straight-forward.

Remark 5.3.e_m is referred to as the entropy based on membership degrees.

Example 5.4. Consider the QSVNS A of Example 3.5 then we have

e_m (A) =0.688. Also, e_m (A^c) =0.688 = e_m (A).

6 A comparative study of the proposed similarity measures in the context of an example pertaining to pattern recognition

Suppose x₁, x₂, x₃ respectively denote the saturation, sharpness and contrast of three similar hued patterns A, B and C which are represented in terms of three QSVNS as $\begin{matrix} A & = & 〈 0.5, 0.4, 0.2, 0.01 〉 / x_{1} + 〈 0.2, 0.1, 0.3, 0.5 〉 / x_{2} \\ + 〈 0.45, 0.2, 0.1, 0.3 〉 / x_{3} \\ B & = & 〈 0.4, 0.4, 0.01, 0.0 〉 / x_{1} + 〈 0.5, 0.3, 0.4, 0.4 〉 / x_{2} \\ + 〈 0.5, 0.4, 0.3, 0.1 〉 / x_{3} \\ C & = & 〈 0.56, 0.8, 0.0, 0.0 〉 / x_{1} + 〈 0.6, 0.5, 0.3, 0.2 〉 / x_{2} \\ + 〈 0.4, 0.4, 0.5, 0.6 〉 / x_{3} \end{matrix}$

Further suppose that there are two unidentified patterns P₁ and P₂ given by $\begin{matrix} P_{1} & = & 〈 0.45, 0.4, 0.11, 0.05 〉 / x_{1} \\ + 〈 0.35, 0.2, 0.35, 0.45 〉 / x_{2} \\ + 〈 0.45, 0.3, 0.2, 0.2 〉 / x_{3} \\ P_{2} & = & 〈 0.6, 0.7, 0.01, 0.0 〉 / x_{1} \\ + 〈 0.5, 0.5, 0.4, 0.2 〉 / x_{2} \\ + 〈 0.3, 0.2, 0.5, 0.6 〉 / x_{3} \end{matrix}$

In order to determine which unidentified pattern belongs to which one of the specified patterns at hand, the similarity measures between the given patterns A and B and the unknown patterns P₁ and P₂ are calculated. Finally, the unidentified pattern bearing the highest similarity to the given set of patterns is concluded to belong to that particular set of patterns. In this respect, it needs to be stated that the similarity measure has been calculated taking 3 values of the order of similarity p viz. p = 1, p = 2 and p = 3. The obtained results are represented in a tabular form (ref. Table 2).

Discussion: From the given set of data, it is seen that, the calculated values of the similarity measures , and S₄ between P₁ and A, B, C indicate that the pattern P₁ belongs to the class of patterns, of which A is a member. However, the quasi similarity measure β throws close values for β (P₁, A)and β (P₂, B) which creates confusion in classifying the pattern P₁. Thus, , and S₄ are better measures of similarity compared to β. Moreover, from the similarity measure S₄ (P₁, A), it is seen that membership values T, C, U and F of the patterns P₁ and A are similar upto 84.6%, 77.8%, 68% and 77.6% respectively. Similarly, all the measures , S″, β and S₄ show that the pattern P₂ belongs to the class of patterns C.

7 Conclusion

In this paper, Belnap’s four valued logic has been used as a framework for proposing a set-theoretic structure which involves partitioned indeterminacies. Although apparently it might so appear that the indeterminacy membership functions C and U are inter-dependent, often, in reality, while dealing with linguistic approaches, it is quite natural that the values corresponding to the functions are actually independent and the mutual dependence of these functional values boils down to a particular case under speculation.As for example, concerning a particular sample of information, a particular person may be utterly clueless as to whether the piece of information is true, false, both true and false or neither of them. In such cases his judgement may instinctively, yet quite involuntarily hover between ’both true and false’ and ’neither true nor false’, being totally unaware of the fact that these two truth values are somewhat logically complementary. For another instance, given two patterns under consideration, it might so happen that at a particular moment the graphical representations of the two patterns are same while they differ in terms of the constituent hues. Thus, at such stances the information that the patterns are similar are both true and false and simultaneously neither true nor false. Putting into considerations such situations such as these, it is evident that a structure like QSVNS prove to be useful and at times essential in representing and tackling the available information. At present some basic set-theoretic operations and similarity measures have been stated.

Future works may involve the study of OWA and IOWAD operators in the context of quadripartitioned single valued neutrosophic sets while dealing with actual problems and implementing them in decision making problems.

Footnotes

Acknowledgments

The authors express their gratitude to the anonymous reviewers whose recommendations aided in improving the quality of the manuscript.

The authors also express their thankfulness to the Associate Editor of this journal for his suggestions and for extending his help in preparing the article.

The research of the first author is supported by University JRF (Junior Research Fellowship).

The research of the second author is supported by UGC (ERO) Minor Research Project (Project no. F.PSW-19/12-13).

The research of the third author is partially supported by the Special Assistance Programme (SAP) of UGC, New Delhi, India [Grant no. F 510/3/DRS-III/(SAP-I)].

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