The recent hybrid cosine-Jaccard measure for neutrosophic sets proposed to handle uncertainty, indeterminacy, and inconsistency in the complex data more effectively. The three main properties: boundedness, symmetry, and reflexivity have been established; therefore, algebraic characteristics should be explored in depth for the establishment of advanced computational frameworks. This paper is the extension of the theoretical foundation by introducing monotonicity and a distributive-like property over weighted combinations. Mathematical proof is provided for each property, followed by numerical examples to demonstrate their validity and computational consistency. This research establishes a theoretical framework for a hybrid cosine-Jaccard measure for neutrosophic sets, facilitating its facilitating reliable implementation in intricate algebraic operations related to neutrosophic multi-criteria decision-making, clustering algorithms, and information retrieval systems.
In complex real-world decision-making scenarios, there are many uncertainties and inaccuracies exist in the complexity of a data set where it is necessary to have advanced mathematical frameworks that can capture the robustness of information processing
In this paper, we investigate two main extension of algebraic properties in hybrid cosine-Jaccard similarity measure for neutrosophic set: monotonicity and linearity in weights. This theoretical framework is important so that it can prove the consistency. Monotonicity is an important property in the hybrid cosine-Jaccard measure for neutrosophic sets. This is to reinforce the three main property: symmetry, boundedness and reflexivity to ensure the reliability of the measure and logical consistency when comparing sets. In the neutrosophic set, it is vital to highlight the membership functions as truth (T), indeterminacy (I), and falsity (F) memberships; monotonicity guarantees that the similarity score behaves predictably. For instance, it is said that two neutrosophic sets become more alike when their truth memberships increase simultaneously, their indeterminacies align more closely, and their falsities decrease in a correlated manner. Therefore, the hybrid similarity measure must also increase or, at a minimum, but not decrease. This monoticity property is crucial because a similarity measure that violates monotonicity can yield error results; for instance, two sets might be objectively more alike than another pair but receive a lower similarity score, rendering the measure untrustworthy for practical decision-making. The hybrid cosine-Jaccard measure strengthens of both cosine and Jaccard so that can handle both the vector-like and set-like aspects of neutrosophic information. Enforcing monotonicity ensures that the robustness in hybrid approach. This property is applicable in real-life application such as pattern recognition, clustering, and multi-criteria decision-making, where the correctness of the similarity ordering directly impacts the quality and validity of the final results. Without monotonicity, the measure lacks a foundational guarantee of its logical soundness.
Besides, the linearity property in hybrid cosine-Jaccard measure for neutrosophic is best investigated where it examines the accuracy and interpretability of similarity computation in decision-making complexity that involves uncertainties
The paper is organized into different sections. Section 2 provides the preliminaries related to neutrosophic sets and similarity measures. Section 3 presents the extension of algebraic properties in hybrid cosine-Jaccard measure for neutrosophic set with proof. Later, Section 4 the illustrates numerical examples. Finally, the conclusion and future research are provide.
This section provides some preliminaries of the core concept and necessary definitions for neutrosophic sets and similarity measures.
Definition 1 (
A = { ⟨x, (TA(x), IA(x), FA(x))⟩ ∶ x ∈ N }, where N is the universe of discourse, x is a generic element in N, T(x), I(x), and F(x) are functions that define the degree of membership (truth), indeterminacy, and non-membership (falsity) of the element x to the set A. Definition 2 (
Σx∈X[TA(x)TB(x)+ IA(x)IB(x)+ FA(x)FB(x)]
SC(A, B) = . (1)
√Σ [T (x)2+ I (x)2+ F (x)2]· √Σ [T (x)2+ I (x)2+ F (x)2]
x∈X A A A x∈X B B B
Definition 3 (
Σx∈X[min(TA(x),TB(x)) + min(IA(x),IB(x)) + min(FA(x),FB(x))]
SJ(A, B) = . (2)
Σx∈X[max(TA(x),TB(x)) + max(IA(x),IB(x)) + max(FA(x),FB(x))]
Definition 4 (
SHCJ(A, B) = α · Scosine(A, B) + β · SJaccard(A, B), (3) where Scosine(A, B) is the cosine similarity measure between A and B, and SJaccard(A, B) is the Jaccard similarity measure between A and B; α and β are weights such that α + β = 1 and α, β ∈ [0,1]. These weights determine the relative importance of the cosine and Jaccard components according to the decision maker.
Let A and B be the neutrosophic sets defined on a universal set X. The hybrid Cosine-Jaccard similarity measure SHCJ(A, B) between A and B can be decomposed into three independent components corresponding to the truth (T), indeterminacy (I), and falsity (F) membership. By integrating the weighted sum of Cosine-Jaccard similarity for each neutrosophic component can be represented as follows:
T I F
SHCJ(A, B) = wT · SHCL(A, B) + wI · SHCJ(A, B) + wF · SHCJ(A, B), (4) where wT, wI, wF ∈ [0,1] such that the weight, wT + wI + wF = 1. If the weights are equal, Eq. (4) is simplified as follow:
T I F
SHCJ(A, B) = ( ) [SHCJ(A, B) + SHCJ(A, B) + SHCJ(A, B)].
In this section, we show the extension properties for hybrid cosine-Jaccard similarity measure for neutrosophic set: monotonicity and linearity in weights. These properties are important to reinforce the three main properties
Let A, B and C be three single-valued neutrosophic sets defined by a finite universal set X = {x1, x2, … , xn}. Suppose that for every element x ∈ X, the components of B are "closer" to those of A than the components of C are. Formally, we assume the following conditions hold for all x ∈ X:
I. Truth-Monotonicity: ∣ TA(x) − TB(x) ∣≤∣ TA(x) − TC(x) ∣.
II. Indeterminacy-Monotonicity: ∣ IA(x) − IB(x) ∣≤∣ IA(x) − IC(x).
III. Falsity-Monotonicity: ∣ FA(x) − FB(x) ∣≤∣ FA(x) − FC(x).
Furthermore, we assume that the membership values are non-negative and bounded, where T, I, F ∈ [0,1]. Then, the Hybrid Cosine-Jaccard similarity measure is monotonic, meaning the similarity between A and B is greater than or equal to the similarity between A and C:
SHCJ(A, B) ≥ SHCJ(A, C), where SHCJ is defined with fixed weights α, β ∈ [0,1]such that α + β = 1.
Proof: Firstly, from the cosine similarity for neutrosophic set is defined as follows:
Σᵢ₌₁ⁿ(TA TB + IA I B+ FA FB)
SC(A, B) = 2 2 2 2 2 2 .
√Σᵢ₌₁ⁿ(TA + IA + FA) · √Σᵢ₌₁ⁿ(TB + I B+ FB)
Let us consider the difference in a single component, say Truth, for an element x. The given condition ∣ TA − TB ∣≤∣ TA − TC ∣ implies that (TA − TB) ≤ (TA − TC) .
Consider the squared Euclidean distance between the entire neutrosophic vectors A and B:
2 =1 2 2 2
dE(A, B) = Σᵢ ⁿ[(TA − TB) + (IA − IB) + (FA − FB) ].
2 =1 2 2 2
dE(A, C) = Σᵢ ⁿ[(TA − TC) + (IA − IC) + (FA − FC) ].
From the given Conditions (1)-(3), (TA − TB)² + (IA − IB)² + (FA − FB)² ≤ (TA − TC)² + (IA − IC)² + (FA − FC)².
Summing over all n elements in X dE(A, B) ≤ dE(A, C).
By improving the relationship between Euclidean distance and Cosine Similarity, it is known that for any two vectors u and v:
2 2 2
dE(u, v) = ‖u‖ + ‖v‖ − 2‖u‖‖v‖cos(θ).
where cos (θ) is the cosine similarity.
Applying this to the cosine similarity for neutrosophic sets
2 2 2
dE(A, B) = ‖A‖ + ‖B‖ − 2‖A‖‖B‖SC(A, B).
2 2 2
dE(A, C) = ‖A‖ + ‖C‖ − 2‖A‖‖C‖SC(A, C).
We know dE(A, B) ≤ dE(A, C).
Thus,
2 2 2 2
‖A‖ + ‖B‖ − 2‖A‖‖B‖SC(A, B) ≤ ‖A‖ + ‖C‖ − 2‖A‖‖C‖SC(A, C).
SC(A, B) ≤ SC(A, C).
Thus, under the given distance conditions and the assumption of comparable magnitudes for B and C, the Cosine Similarity for neutrosophic set is monotonic. □ Secondly, the Jaccard similarity for neutrosophic set is defined as followed:
Σᵢ₌₁ⁿ[min(TA,TB) + min(IA,IB) + min(FA,FB)]
SJ(A, B) = .
Σᵢ₌₁ⁿ[max(TA,TB) + max(IA,IB) + max(FA,FB)]
In the case for minimum, min (TA, TB) ≥ min (TA, TC). For instance, I. If TA > TC, then TB ≥ TC, so min (TA, TB) ≥ min (TA, TC) = TC.
II. If TA < TC, then TB ≤ TC, so min (TA, TB) ≥ min (TA, TC) = TA. Therefore, symmetric argument holds. In the case for maximum, max (TA, TB) ≤ max (TA, TC). For example, I. If TA > TC, then TB ≤ TA, so max (TA, TB) ≤ max (TA, TC) = TA.
II. If TA < TC, then TB ≤ TC, so max (TA, TB) ≤ max (TA, TC) = TC.
Thus, a symmetric argument holds.
The same logic applies to the Indeterminacy and Falsity components.
For every element x ∈ X, the numerator can be written as min(TA, TB) + min(IA, IB) + min(FA, FB) ≥ min(TA, TC) + min(IA, IC) + min(FA, FC), while the denominator can be written as max(TA, TB) + max(IA, IB) + max(FA, FB) ≤ max(TA, TC) + max(IA, IC) + max(FA, FC).
SJ(A, B) ≥ SJ(A, C).
The Jaccard Similarity for neutrosophic set is monotonic under the given conditions. □ The hybrid measure is a convex combination of the two monotonic measures:
SHCJ(A, B) = α · SC(A, B) + β · SJ(A, B).
SHCJ(A, C) = α · SC(A, C) + β · SJ(A, C).
From the monotonicity of cosine similarity and Jaccard similarity for neutrosophic set:
SC(A, B) ≥ SC(A, C), and SJ(A, B) ≥ SJ(A, C).
Since the weights α and β are non-negative, α · SC(A, B) ≥ α · SC(A, C).
β · SJ(A, B) ≥ β · SJ(A, C).
By adding these two inequalities together yields the final result:
α · SC(A, B) + β · SJ(A, B) ≥ α · SC(A, C) + β · SJ(A, C).
∴ SHCJ(A, B) ≥ SHCJ(A, C). ∎
The hybrid cosine-Jaccard similarity measure SHCJ(A, B) is linear with respect to the weights α, β, wT, wI, and wF. For any two sets of the weights (α1, β1) and (α2, β2), and for any scalar λ ∈ [0,1] can be written as followed:
(λα₁+(1−λ)α₂,λβ₁+(1−λ)β₂) (α₁,β₁) (α₂,β₂)
SHCJ(A, B) = λ · SHCJ(A, B) + (1 − λ) · SHCJ(A, B) .
Therefore, linearity property holds for the component weights wT, wI, wF.
Proof: Let (α1, β1) and (α2, β2) be two pairs of weights such that α1 + β1 = 1 and α2 + β2 = 1. Let λ ∈ [0,1]. We define a new pair of weights as α′ = λα₁ + (1 − λ)α₂, β′ = λβ₁ + (1 − λ)β₂.
α′ + β′ = λ(α₁ + β₁) + (1 − λ)(α₂ + β₂) = λ · 1 + (1 − λ) · 1 = 1.
Thus, (α′, β′) is a valid weighting pair.
Now, consider the hybrid measure under these weights:
(α′,β′) ′ ′
SHCJ(A, B) = α · SC(A, B) + β · SJ(A, B).
= (λα1 + (1 − λ)α2) ⋅ SC(A, B) + (λβ1 + (1 − λ)β2) ⋅ SJ(A, B).
= λ(α1SC(A, B) + β1SJ(A, B)) + (1 − λ)(α2SC(A, B) + β2SJ(A, B)).
(α1,β1) (α2,β2)
= λ ⋅ SHCJ(A, B) + (1 − λ) ⋅ SHCJ(A, B) .
This completes the proof for the linearity in the hybrid weights α and β. Thus, the component weights wT, wI, wF can be derived in the similar way.
1 1 1 2 2 2
Next, let (wT, wI , wF) and (wT, wI , wF) be two sets of component weights, where the total weight is 1. For ∈[0,1], we define the weights as
′ 1 2 ′ 1 2 ′ 1 2
wT = λwT + (1 − λ)wT, wI = λwI + (1 − λ)wI , wF = λwF + (1 − λ)wF.
′ ′ ′
(wT,wI,wF) ′ T ′ I ′ F
SHCJ(A, B) = wT ⋅ SHCJ(A, B) + wI ⋅ SHCJ(A, B) + wF ⋅ SHCJ(A, B).
1 T 1 I 1 F 2 T 2 I 2 F
= λ(wTSHCJ + wI SHCJ + wFSHCJ) + (1 − λ)(wTSHCJ + wI SHCJ + wFSHCJ).
1 1 1 2 2 2
(wT,wI ,wF) (wT,wI ,wF)
= λ ⋅ SHCJ(A, B) + (1 − λ) ⋅ SHCJ(A, B) .
Therefore, the measure is also linear in the component of weights. ∎ This linear in weight theorem allows decision makers to adjust weights such as giving the importance to truth in medical diagnosis and have predictable results. It also helps to combine decision maker’s opinions.
In this section, we demonstrate two numerical examples to show the monotonicity and linearity in weights theorem from the previous section.
Example 1 (monotonicity). By using the example of the previous paper
Let A and B be neutrosophic sets over U = {u}:
A = (TA, IA, FA) = (0.5, 0.3, 0.2).
B = (TB, IB, FB) = (0.6, 0.2, 0.1).
C = (T , I , F ) = (0.4, 0.5, 0.3).
Besides, we also consider the sum weight of truth (T), indeterminacy (I), and falsity (F) is equal to 1 and α = β = 0.5.
Step 1. Calculating the cosine similarity and Jaccard similarity Cosine Similarity:
[(0.5)(0.6)+(0.3)(0.2)+(0.2)(0.1)] 0.38
SC(A, B) = = ≈ 0.9628.
[√0.52+0.32+0.22·√0.62+0.22+0.12] 0.6164×0.6403 [(0.5)(0.4)+(0.3)(0.5)+(0.2)(0.3)] 0.41
SC(A, C) = = ≈ 0.9407.
[√0.52+0.32+0.22·√0.42+0.52+0.32] (0.6164×0.7071)
Jaccard Similarity:
[min(0.5,0.6)+min(0.3,0.2)+min(0.2,0.1)] (0.5+0.2+0.1)
SJ(A, B) = = ≈ 0.7273.
[max(0.5,0.6)+max(0.3,0.2)+max(0.2,0.1)] (0.6+0.3+0.2) [min(0.5,0.4)+min(0.3,0.5)+min(0.2,0.3)] (0.4+0.3+0.2)
SJ(A, C) = = ≈ 0.6923.
[max(0.5,0.4)+max(0.3,0.5)+max(0.2,0.3)] (0.5+0.5+0.3)
Step 2. Using the formula of hybrid cosine-Jaccard similarity measures By applying the hybrid Cosine-Jaccard Similarity measures:
SHCJ(A, B) = (0.5 × 0.9628) + (0.5 × 0.7273) ≈ 0.84505.
SHCJ(B, A) = (0.5 × 0.7273) + (0.5 × 0.9628) ≈ 0.84505.
SHCJ(A, C) = (0.5 × 0.940) + (0.5 × 0.6923) ≈ 0.8162.
Step 3. Check the monotonicity SC(A, B) ≈ 0.9628 > SC(A, C) ≈ 0.940.
SJ(A, B) ≈ 0.7273 > SJ(A, C) ≈ 0.6923.
SHCJ(A, B) ≈ 0.8451 > SHCJ(A, C) ≈ 0.8162.
Example 2 (Linearity in Weights). Given the two neutrosophic sets are as followed:
A = (0.6, 0.3, 0.2).
B = (0.5, 0.4, 0.3).
Besides, we also assume the component similarities have been computed as followed:
T T T
SHCJ(A, B) = 0.9, SHCJ(A, B) = 0.6, SHCJ(A, B) = 0.7.
Step 1. Calculating the weight.
If the weight given are w1 = (wT = 0.5, wI = 0.3, wF = 0.2).
w2 = (wT = 0.2, wI = 0.5, wF = 0.3).
SHCJ(A, B) = 0.5 · 0.9 + 0.3 · 0.6 + 0.2 · 0.7 = 0.77.
SHCJ(A, B) = 0.2 · 0.9 + 0.5 · 0.6 + 0.3 · 0.7 = 0.69.
let λ = 0.4. The estimating weights are w = 0.4 · w1 + 0.6 · w2 = (0.32, 0.42, 0.26).
The measure with w′ is:
SHCJ(A, B) = 0.32 · 0.9 + 0.42 · 0.6 + 0.26 · 0.7 = 0.722.
Step 2. Verifying the linearity
w1 w2
λ · S + (1 − λ) · S = 0.4 · 0.77 + 0.6 · 0.69 = 0.722.
This research investigates the extensive algebraic properties: monotonicity and linearity in weights in hybrid cosine-Jaccard measure for neutrosophic set enhances the consistency with the three main properties: symmetry, boundedness and reflexivity and also increase the quality of decision-making processes. This theoretical framework can also help to optimize practical applications which give better accuracy and dependable evaluations in ambiguous contexts. Monotonicity theorem helps to enhance the establishment of three main properties: boundedness, symmetry, and reflexivity. Besides, the linearity in weight theorems also allows weights adjustment based on different real-life application. Thus, the future research should aim for creating more advanced algorithms that utilize these characteristics for improved performance across diverse applications. Future research on the hybrid cosine-Jaccard measure for neutrosophic sets can also be extended in improving these algebraic characteristics and investigating the results in practical applications. Using more complex models to test whether it can handle more accurately during decision-making processes.
This research is supported by Fundamental Research Grant (FRGS/1/2020/STG06/USM/02/1) by Ministry of Higher Education Malaysia.
The authors declare no conflict of interest.