A few basic definitions are given here below as found in [5] .

Let $X$ be a nonempty set. A pair $\mu =\left({\mu }^{+},{\mu }^{-}\right)$ is called a bipolar-valued fuzzy set (or bipolar fuzzy set) in $X$ if ${\mu }^{+}:X\to \left[0,1\right]$ and ${\mu }^{-}:X\to \left[-1,0\right]$ are mappings.

In particular, the bipolar fuzzy empty set (resp. the bipolar fuzzy whole set), denoted by $0_{bp}=\left(0_{bp}^{+},0_{bp}^{-}\right)\left[\text{resp}\text{.}{\text{1}}_{bp}=\left(1_{bp}^{+},1_{bp}^{-}\right)\right]$, is a bipolar fuzzy set in $X$ defined by

$\forall x\in X,{\text{0}}_{bp}^{+}\left(x\right)=0=0_{bp}^{-}\left(x\right)\left[\text{resp}{\text{. 1}}_{bp}^{+}\left(x\right)=1{\text{and 1}}_{bp}^{-}\left(x\right)=-1\right]$

The collection of all bipolar fuzzy sets of the set $X$ will be denoted by $BPF\left(X\right)$.

A fundamental question in the realm of fuzzy sets involves the use of relative comparisons to calculate the similarities and distances between fuzzy sets [6] [7] . Similarity involves recognizing patterns and making associations which enable one to classify objects and concepts, particularly in the domain of classification and clustering. In these applications, an unknown object is assigned to a particular category if its similarity measure to objects within that category is higher than its similarity to objects in other categories [8] .

In essence, the similarity of two fuzzy sets is 1 if they are identical i.e., if they both contain the same values with the same degree of membership, and 0 if they have nothing in common, i.e., they do not contain any of the same values. These two properties are referred to as reflexivity and overlapping respectively.

Definition (Similarity Measure)

Intuitively, the similarity of two fuzzy sets is 1 if they are identical i.e., if they both contain the same values with the same degree of membership, and 0 if they have nothing in common, i.e., they do not contain any of the same values. These two properties are referred to as reflexivity and overlapping respectively.

A function $S:A\times B\to \left[0,1\right]$ is called a similarity measure if

1) Reflexivity: $S\left(A,A\right)=1$, meaning an element always has full similarity to itself.

2) Symmetry: $S\left(A,B\right)=S\left(B,A\right)$ $\forall A,B\in X$. This property ensures that similarity is bidirectional.

3) Overlapping: $S\left(A,B\right)\ge \max \left(\frac{\left|A\cap B\right|}{\left|A\right|},\frac{\left|A\cap B\right|}{\left|B\right|}\right)$. This property ensures that shared characteristics contribute strongly to the similarity score.

4) Transitivity: if $S\left(A,B\right)\ge \alpha$ and $S\left(B,C\right)\ge \alpha$ then $S\left(A,C\right)\ge \alpha$. This property is important for clustering and classification, ensuring that if $A$ is similar to $B$ and $B$ is similar to $C$, then $A$ should be similar to $C$.

It is not necessary for a similarity measure to have all these properties [9] . There are situations when symmetry does not need to be satisfied and in other situations, it has been argued whether transitivity is necessary or even useful.

The Jaccard similarity/index compares the similarity between two fuzzy sets by calculating the size of their intersection divided by the size of their union. According to [10] , the fuzzy Jaccard similarity between two fuzzy sets $A$ and $B$ in a universal set $X$ is defined as

$J\left(A,B\right)=\frac{{\displaystyle \underset{x\in X}{\sum }\min \left({\mu }_A\left(x\right),{\mu }_B\left(x\right)\right)}}{{\displaystyle \underset{x\in X}{\sum }\max \left({\mu }_A\left(x\right),{\mu }_B\left(x\right)\right)}}$

Instead of simply counting shared elements, fuzzy Jaccard similarity is computed using the ratio of the sum of the minimum membership values (fuzzy intersection) to the sum of the maximum membership values (fuzzy union).

Entropy, a fundamental concept in information theory, measures the uncertainty or randomness of a system. Shannon entropy, introduced in 1948, is the most well-known measure of information uncertainty and forms the basis of modern information theory [2] . However, Shannon entropy has limitations when dealing with higher-order moments of probability distributions, long-tailed distributions, or multi-fractal systems. In such scenarios, a more generalized entropy measure is required.

To address these limitations, Alfréd Rényi introduced the Rényi entropy in 1961 as a one-parameter generalization of Shannon entropy [11] . Rényi entropy provides flexibility in quantifying information by introducing an order parameter, α, which allows fine-tuning sensitivity to different probabilities within a distribution. This feature makes it highly useful in fields like machine learning, cryptography, statistical mechanics, quantum computing, and image processing.

The Rényi entropy of order $\alpha$ for a probability distribution $P=\left\{p_1,p_2,\cdots ,p_n\right\}$ is defined as $H_{\alpha }\left(P\right)=\frac{1}{1-\alpha }\ln {\displaystyle \underset{i=1}{\overset{n}{\sum }}{p}_i^{\alpha }}$ where $\alpha >0,\alpha \ne 1$ is the order of the entropy, controlling the weighting of probabilities.

Note

1) When $\alpha \to 1$ Renyi entropy converges to Shannon entropy

$H_1\left(P\right)={\displaystyle \underset{i=1}{\overset{n}{\sum }}{p}_i\ln p_i}$

2) When $\alpha \to \infty$ Renyi entropy focusses only on the most probable event.

The motivation behind Rényi entropy arises from its ability to:

1) Extend Shannon Entropy: While Shannon entropy assumes an equal weighting of probabilities, Rényi entropy allows adjustable sensitivity to rare or dominant events based on the value of αalphaα.

2) Enhance Information Divergence Measures: The Rényi divergence, a generalization of Kullback-Leibler (KL) divergence, provides a more robust way to measure differences between probability distributions [12] .

3) Improve Uncertainty Quantification in Fuzzy and Quantum Systems: Many real-world problems involve uncertainty beyond probabilistic randomness—for instance, fuzzy logic and quantum mechanics, where entropy measures must be adapted accordingly [13] .

4) Enable Robust Feature Selection and Learning Algorithms: In machine learning and deep learning, Rényi entropy helps in feature selection, adversarial learning, and information-theoretic optimization [12] .

5) Enhance Image Processing Methods: Rényi entropy is particularly effective in image segmentation and edge detection, where different levels of entropy sensitivity are required to enhance contrast and extract meaningful structures [14] .

Given these advantages, Rényi entropy has become a fundamental tool in information science, artificial intelligence, and complex system analysis.

In the context of fuzzy sets, the concept of the entropy of a fuzzy set was first introduced by [6] as a means of quantifying the degree of fuzziness within a fuzzy set. This laid the foundation for measuring the uncertainty or ambiguity present in fuzzy systems. The formal axiomatization of fuzzy entropy was later developed by [15] , who drew parallels between fuzzy entropy and Shannon probability entropy, interpreting it as a measure of the amount of information contained in a fuzzy set.

Further advancements were made by [16] who proposed that the entropy of a fuzzy set could be calculated based on its distance from the nearest crisp (non-fuzzy) set, providing a geometric perspective on fuzziness measurement. Yager [17] [18] offered an alternative interpretation, defining entropy as the distance between a fuzzy set and its complement. According to [16] , any meaningful measure of fuzziness should reflect the lack of distinction between a fuzzy set and its negation, leading to the proposal of a metric for quantifying the distance between a fuzzy set and its complement.

Formally, in [8] the entropy $H\left(A\right)$ of a fuzzy set $A$ defined on a universal set $X$ with membership function is expressed as:

$H\left(A\right)=-{\displaystyle \underset{x\in X}{\sum }\left[{\mu }_A\left(x\right)\log {\mu }_A\left(x\right)+\left(1-{\mu }_A\left(x\right)\right)\log \left(1-{\mu }_A\left(x\right)\right)\right]}$

where ${\mu }_A\left(x\right)$ represents the membership degree of element $x$ in the fuzzy set $A$.

Illustrative example

Define $d_p\left(A,A^c\right)={\left[{\displaystyle \underset{i=1}{\overset{n}{\sum }}\left|{\mu }_A\left(x_i\right)-{\mu }_{A^c}{\left(x_i\right)}^p\right|}\right]}^{\frac{1}{p}},p=1,2,3,\cdots$ Then the measure of fuzziness of $A$ can be defined as

$f_p\left(A\right)=1-\frac{d_p\left(A,A^c\right)}{\Vert \sup \left(A\right)\Vert }$

For $p=1$, $d_p\left(A,A^c\right)$ yields the Hamming metric.

$d_p\left(A,A^c\right)={\displaystyle \underset{i=1}{\overset{n}{\sum }}\left|{\mu }_A\left(x_i\right)-{\mu }_{A^c}\left(x_i\right)\right|}$

Since ${\mu }_{A^c}\left(x\right)=1-{\mu }_A\left(x\right)$, this becomes $d_p\left(A,A^c\right)={\displaystyle \underset{i=1}{\overset{n}{\sum }}\left|2{\mu }_A\left(x_i\right)-1\right|}$.

For $p=2$ we have the Euclidean metric

$d_2\left(A,A^c\right)={\left[{\displaystyle \underset{i=1}{\overset{n}{\sum }}\left|{\mu }_A\left(x_i\right)-{\mu }_{A^c}{\left(x_i\right)}^2\right|}\right]}^{\frac{1}{2}}$ and

since ${\mu }_{A^c}\left(x\right)=1-{\mu }_A\left(x\right)$, then this becomes

$d_2\left(A,A^c\right)={\left[{\displaystyle \underset{i=1}{\overset{n}{\sum }}{\left(2{\mu }_A\left(x_i\right)-1\right)}^2}\right]}^{\frac{1}{2}}$

Min-max normalization is a widely used technique in image processing, machine learning, and data preprocessing for transforming data into a predefined range, typically [0, 1] or [−1, 1]. This method is particularly effective in scenarios where pixel intensity values or feature distributions need to be scaled for consistency across different datasets.

This sub-section reviews min-max normalization from the perspective of fuzzy set theory, highlighting its applications, limitations, and recent advancements.

In digital topology, a grayscale image can be represented as a digital image where pixel intensities are defined on a discrete grid, and adjacency relations determine connectivity According to [19] , a grayscale image is a function $I:X\to \left[0,L\right]$ where

1) $X\subseteq {\mathbb{Z}}^n$ is a finite subset of the integer lattice representing the image domain.

2) $L$ is the maximum intensity level.

3) $I\left(x\right)$ represents the grayscale intensity value at pixel $x$, where 0 corresponds to black, L corresponds to white, and intermediate values represent shades of gray [20] .

In Graph-based approach, a grayscale image can be modeled as an undirected graph $G=\left(V,E\right)$ where the vertices $V$ represent pixels in the image, $E$ define the adjacency relations.

Each vertex will possess a grayscale intensity value $I\left(v\right)\in \left[0,255\right]$ with 0 corresponding to a low intensity pixel (black) and 255 corresponding to the highest intensity pixel (white). All intermediate values represent shades of gray [20] .

Illustrative example

A $3\times 3$ grayscale image

$\begin{array}{ccc}\left(50\right)& \left(120\right)& \left(200\right)\\ \left(80\right)& \left(150\right)& \left(220\right)\\ \left(90\right)& \left(180\right)& \left(250\right)\end{array}$

will have a graphical representation with 4-adjacency relation as shown below

Min-max normalization is widely applied in fuzzy-based image processing, where it ensures that data is transformed into a standardized range suitable for fuzzy logic operations. By normalizing image intensities, feature values, or clustering inputs, it enhances the effectiveness of fuzzy techniques in image segmentation, similarity measurement, and contrast enhancement. Two of the areas where this formulation finds application are:

1) Fuzzy clustering techniques, such as Fuzzy C-Means (FCM), rely heavily on min-max normalization to standardize input features before assigning fuzzy memberships to clusters. Without proper normalization, feature scales may distort clustering results, leading to misclassification [21] [22] .

2) In fuzzy-based image retrieval, similarity measures compute distances between images in a feature space. Min-max normalization ensures that feature values fall within a comparable range, enhancing the accuracy of fuzzy histogram similarity measures and fuzzy correlation techniques [23] .

Despite its effectiveness in fuzzy-based image processing, min-max normalization has several limitations that can affect its performance in certain applications. One major drawback is its sensitivity to outliers, as extreme values in the dataset disproportionately influence the scaling range, leading to distorted fuzzy membership values [24] . This can be particularly problematic in medical imaging or low-contrast image analysis, where small variations in intensity carry significant meaning. Additionally, min-max normalization assumes a linear transformation, which may not always be suitable for complex or non-linearly distributed data, such as images with varying lighting conditions or textures [23] . The technique also depends on the fixed minimum and maximum values within the dataset, which can make it ineffective for dynamically changing environments, such as real-time image processing or adaptive fuzzy systems [25] . Moreover, by strictly mapping values between 0 and 1, min-max normalization can lead to loss of contrast and information, particularly when applied to images with already narrow intensity distributions [26] .

As a result, alternative normalization methods, such as sigmoid-based transformations which uses a logistic function to transform data, preserving fine-grained differences in middle-range values [27] or adaptive fuzzy normalization, which dynamically adjusts min-max scaling based on data distributions [28] are often preferred for applications requiring greater robustness and flexibility.

In the context of fuzzy image representation, each pixel intensity $I\left(p_i\right)$ in an image is transformed into a single-valued membership function. To extend this idea into the bipolar fuzzy context, each pixel will be assigned two membership values,

$B\left(p_i\right)=\left({\mu }^{+}\left(p_i\right),{\mu }^{-}\left(p_i\right)\right)$

where ${\mu }^{+}\left(p_i\right)$ and ${\mu }^{-}\left(p_i\right)$ represent the degree of relevance to the desired feature and undesired features respectively.

For grayscale images, the bipolar fuzzy memberships functions may be defined as

${\mu }^{+}\left(p_i\right)=\frac{I\left(p_i\right)-\min \left(I\right)}{\max \left(I\right)-\min \left(I\right)}$

where:

This proposed transformation can be shown by taking $I\left(p_i\right)=200$. In this case

${\mu }^{+}\left(p_i\right)=\frac{200-0}{255-0}=0.784$

(a pixel with higher intensity will have a higher positive membership).

The negative membership function ${\mu }^{-}\left(p_i\right)$ determines how strongly a pixel belongs to the background, noise, or undesired features. By definition this will be given by

${\mu }^{-}\left(p_i\right)=-\left(1-{\mu }^{+}\left(p_i\right)\right)\in \left[-1,0\right]$

This formulation ensures that brighter pixels have higher ${\mu }^{+}$- values and lower ${\mu }^{-}$ values and vice versa.

In example above,

if ${\mu }^{+}\left(p_i\right)=0.784$ then

${\mu }^{-}\left(p_i\right)=-\left(1-0.784\right)=-0.216$

This corresponds perfectly with the fact that since the positive membership measures the presence of a desired feature, the negative membership naturally represents the presence of the counter property of that feature. Therefore, a pixel of higher intensity will have a lower negative membership, representing higher presence of the low intensity characteristics.

To compare two bipolar fuzzy images $B_1$ and $B_2$ we extend classical fuzzy similarity measures to account for both positive and negative memberships. The general form of a bipolar fuzzy similarity measure may be expressed as:

$S_B\left(B_1,B_2\right)=\alpha S^{+}\left(F_1,F_2\right)-\beta S^{-}\left(F_1,F_2\right)$

where:

In Section 4.2.3, a characterization of Bipolar Jaccard similarity for Bipolar fuzzy sets was proposed. This was defined for two bipolar fuzzy sets $A$ and $B$ may as

$s\left(A,B\right)={\displaystyle \underset{i=1}{\overset{n}{\sum }}\left(\frac{A^{+}\left(x_i\right)\wedge B^{+}\left(x_i\right),A^{-}\left(x_i\right)\wedge B^{-}\left(x_i\right)}{A^{+}\left(x_i\right)\vee B^{+}\left(x_i\right),A^{-}\left(x_i\right)\vee B^{-}\left(x_i\right)}\right)}$

This characterization is equivalent to

$s\left(A,B\right)=\frac{{\displaystyle \sum \min \left(A^{+}\left(x_i\right),B^{+}\left(x_i\right)\right)-{\displaystyle \sum \min \left(A^{-}\left(x_i\right),B^{-}\left(x_i\right)\right)}}}{{\displaystyle \sum \max \left(A^{+}\left(x_i\right),B^{+}\left(x_i\right)\right)+{\displaystyle \sum \max \left(A^{-}\left(x_i\right),B^{-}\left(x_i\right)\right)}}}$

In this formulation, the numerator captures the overlap in positive memberships while taking care of dissimilarities in negative memberships, while the denominator normalizes the measure to ensure that $s\left(A,B\right)\in \left[-1,1\right]$.

In this section, we present a systematic step-by-step approach for computing the bipolar Jaccard similarity between two grayscale images for experimental validation. The utility of this approach can be seen in domains where intensity differences carry critical information and therefore the bipolar approach is potentially more effective in smoothening out minor variations thereby making similarity computation more robust against unwanted variations in pixel intensity.

This step-by-step approach is presented as a means of validating the effectiveness of the proposed Bipolar Fuzzy Jaccard Similarity, demonstrating its ability to enhance similarity computations.

Suppose we have to $4\times 4$ grayscale images as below

Image 1

$A=\begin{array}{cccc}50& 80& 120& 200\\ 30& 90& 130& 210\\ 60& 100& 140& 220\\ 70& 110& 150& 230\end{array}$

Image 2

$B=\begin{array}{cccc}55& 85& 125& 205\\ 35& 95& 135& 215\\ 65& 105& 145& 225\\ 75& 115& 155& 235\end{array}$

Step 1: Compute the minimum and maximum pixel intensities

Find the minimum and maximum pixel values across both images in order to normalize the images

$\begin{array}{l}\min \left(A,B\right)=\min \left(30,35\right)=30\\ \max \left(A,B\right)=\max \left(230,235\right)=235\end{array}$

Step 2: Compute positive membership function ${\mu }^{+}$ using min-max normalization

For every pixel $p_i$, we can then by min-max normalization

${\mu }^{+}\left(p_i\right)=\frac{I\left(p_i\right)-\min \left(I\right)}{\max \left(I\right)-\min \left(I\right)}$

${\mu }^{+}\left(p_i\right)=\frac{A\left(p_i\right)-30}{235-30}$

In $A$ we have

${\mu }_A^{+}=\left[\begin{array}{cccc}\frac{50-30}{205}& \frac{80-30}{205}& \frac{120-30}{205}& \frac{200-30}{205}\\ \frac{30-30}{205}& \frac{90-30}{205}& \frac{130-30}{205}& \frac{210-30}{205}\\ \frac{60-30}{205}& \frac{100-30}{205}& \frac{140-30}{205}& \frac{220-30}{205}\\ \frac{70-30}{205}& \frac{110-30}{205}& \frac{150-30}{205}& \frac{230-30}{205}\end{array}\right]$

${\mu }_A^{+}=\left[\begin{array}{cccc}0.0976& 0.3439& 0.4390& 0.8293\\ 0& 0.2978& 0.4878& 0.8780\\ 0.1463& 0.3415& 0.5366& 0.9268\\ 0.1951& 0.3902& 0.5854& 0.9756\end{array}\right]$

Similarly for $B$

${\mu }_B^{+}=\left[\begin{array}{cccc}0.1220& 0.2683& 0.4634& 0.8537\\ 0.0244& 0.3171& 0.5122& 0.9024\\ 0.1707& 0.3659& 0.5610& 0.9512\\ 0.2195& 0.4146& 0.6098& 1\end{array}\right]$

Step 3: Compute negative membership function ${\mu }^{-}$

The negative membership function ${\mu }^{-}\left(p_i\right)$ is defined by ${\mu }^{-}\left(p_i\right)=-\left(1-{\mu }^{+}\left(p_i\right)\right)\in \left[-1,0\right]$.

In $A$ we have

${\mu }_A^{-}=\left[\begin{array}{cccc}-0.9024& -0.7561& -0.5610& -0.1707\\ -1& -0.7073& -0.5122& -0.1220\\ -0.8537& -0.6585& -0.4634& -0.0732\\ -0.8049& -0.6098& -0.4146& -0.0244\end{array}\right]$

For $B$

${\mu }_B^{-}=\left[\begin{array}{cccc}-0.8780& -0.7317& -0.5366& -0.1463\\ -0.9756& -0.6829& -0.4878& -0.0976\\ -0.8293& -0.6341& -0.4390& -0.0488\\ -0.7805& -0.5854& -0.3902& 0\end{array}\right]$

Step 4: Compute bipolar Jaccard similarity

The bipolar Jaccard similarity is given by

$s\left(A,B\right)=\frac{{\displaystyle \sum \min \left(A^{+}\left(x_i\right),B^{+}\left(x_i\right)\right)-{\displaystyle \sum \min \left(A^{-}\left(x_i\right),B^{-}\left(x_i\right)\right)}}}{{\displaystyle \sum \max \left(A^{+}\left(x_i\right),B^{+}\left(x_i\right)\right)+{\displaystyle \sum \max \left(A^{-}\left(x_i\right),B^{-}\left(x_i\right)\right)}}}$

It can be verified that

$s\left(A,B\right)=\frac{-0.5584}{16.9884}=-0.077$

The Bipolar Jaccard Similarity value of −0.077 suggests that the images have slightly more dissimilarities than similarities. This means that while some pixels have similar intensity values, there are still noticeable variations. The negative membership values indicate a higher degree of dissimilarity than similarity (if the similarity score were closer to 1, it would suggest that the images are nearly identical, whereas a score approaching −1 would indicate significant differences between them).

Instead of using linear min-max normalization, we define sigmoid-based fuzzy membership functions by

$\begin{array}{l}{\mu }^{+}\left(p\right)=\frac{1}{1+{\text{e}}^{-k\left(I\left(p\right)-T\right)}}\\ {\mu }^{-}\left(p\right)=-\left(1-{\mu }^{+}\left(p\right)\right)\end{array}$

where:

Suppose we have to $4\times 4$ grayscale images as below

Image 1

$A=\begin{array}{cccc}50& 80& 120& 200\\ 30& 90& 130& 210\\ 60& 100& 140& 220\\ 70& 110& 150& 230\end{array}$

Image 2

$B=\begin{array}{cccc}55& 85& 125& 205\\ 35& 95& 135& 215\\ 65& 105& 145& 225\\ 75& 115& 155& 235\end{array}$

Then computing this in Python using the Bipolar fuzzy Jaccard formula,

$s\left(A,B\right)=\frac{{\displaystyle \sum \min \left(A^{+}\left(x_i\right),B^{+}\left(x_i\right)\right)-{\displaystyle \sum \min \left(A^{-}\left(x_i\right),B^{-}\left(x_i\right)\right)}}}{{\displaystyle \sum \max \left(A^{+}\left(x_i\right),B^{+}\left(x_i\right)\right)+{\displaystyle \sum \max \left(A^{-}\left(x_i\right),B^{-}\left(x_i\right)\right)}}}$

we obtain a Bipolar Fuzzy Jaccard Similarity −0.1120.

See Appendix xx for the Full Python Implementation of the Bipolar Fuzzy Jaccard Similarity Algorithm.

The BJS with sigmoid transformation is lower (−0.112) than with the min-max transformation (−0.077) implying the two functions assigns bipolar fuzzy values differently. This difference could be as a result of:

1) Min-max transformation preserves relative pixel differences, making it more sensitive to slight variations.

2) Sigmoid-based transformation compresses extreme values, emphasizing mid-range intensities, which affects how differences in brightness levels contribute to similarity.

3) The penalty for mismatches (negative membership function) behaves differently in sigmoid-based transformation, leading to a more negative similarity score in some cases.

Key differences between the two approaches

To extend Rényi entropy to bipolar fuzzy sets, we define a new entropy function that considers both the positive and negative membership degrees of elements in a BFS.

Given a bipolar fuzzy set $A$, the bipolar fuzzy probability distribution may be defined as

$p_i^{+}=\frac{{\mu }^{+}\left(x_i\right)}{{\displaystyle \underset{j=1}{\overset{n}{\sum }}{\mu }^{+}\left(x_j\right)}}$

where $p_i^{+}$ represents the positive probability contribution.

$p_i^{+}=\frac{\left|{\mu }^{-}\left(x_i\right)\right|}{{\displaystyle \underset{j=1}{\overset{n}{\sum }}\left|{\mu }^{-}\left(x_j\right)\right|}}$

where $p_i^{-}$ represents the negative probability contribution.

Using these probability distributions, we can define the bipolar Rényi entropy (BRE) by

$H_{\alpha }^B\left(A\right)=\frac{1}{1-\alpha }\ln \left({\displaystyle \underset{i=1}{\overset{n}{\sum }}{\left(p_i^{+}\right)}^{\alpha }}+{\displaystyle \underset{i=1}{\overset{n}{\sum }}{\left(p_i^{-}\right)}^{\alpha }}\right)$

This formulation ensures that the definition encapsulates both the supporting and opposing membership degrees.

1) If ${\mu }^{-}\left(x_i\right)\forall i$ then $H_{\alpha }^B\left(A\right)$ reduces to the classical Rényi entropy. The proposed BRE for bipolar fuzzy sets generalizes the standard Rényi entropy.

2) The parameter $\alpha$ enables dynamic control over the entropy computation, with higher $\alpha$ emphasizing dominant elements.

To rigorously examine the proposed BRE, we analyze its mathematical properties, ensuring its validity as a generalization of classical Rényi entropy.

Non-negativity

A fundamental property of entropy measures is that it must be non-negative i.e., $H_{\alpha }^B\left(A\right)\ge 0$.

Proof

$\forall i,0\le p_i^{+}\le 1$ and $0\le p_i^{-}\le 1$. Since $\alpha >0$ then $0\le {\left(p_i^{+}\right)}^{\alpha }\le 1$ and $0\le {\left(p_i^{-}\right)}^{\alpha }\le 1$ $\forall i$.

Therefore $H_{\alpha }^B\left(A\right)=\frac{1}{1-\alpha }\ln \left({\displaystyle \underset{i=1}{\overset{n}{\sum }}{\left(p_i^{+}\right)}^{\alpha }}+{\displaystyle \underset{i=1}{\overset{n}{\sum }}{\left(p_i^{-}\right)}^{\alpha }}\right)\ge 0$.

Monotonicity

BRE, like classical entropy measures, is expected to be monotonically decreasing in $\alpha$ meaning that as the probability contribution increases with increase in $\alpha$, the entropy associated with it decreases, signifying a reduction in uncertainty.

Proof

Consider the function

$S_{\alpha }={\displaystyle \underset{i=1}{\overset{n}{\sum }}{\left(p_i^{+}\right)}^{\alpha }}+{\displaystyle \underset{i=1}{\overset{n}{\sum }}{\left(p_i^{-}\right)}^{\alpha }}$

Taking the derivative with respect to $\alpha$.

$\frac{\text{d}}{\text{d}\alpha }{S}_{\alpha }={\displaystyle \underset{i=1}{\overset{n}{\sum }}{\left(p_i^{+}\right)}^{\alpha }\ln }\left(p_i^{+}\right)+{\displaystyle \underset{i=1}{\overset{n}{\sum }}{\left(p_i^{-}\right)}^{\alpha }\ln \left(p_i^{-}\right)}$

Since $0\le p_i^{+}\le 1$ and $0\le p_i^{-}\le 1$ $\forall i$ then their logarithms will be negative

$\frac{\text{d}}{\text{d}\alpha }{S}_{\alpha }<0$

which implies $S_{\alpha }$ is monotonically decreasing in $\alpha$. Therefore,

$H_{\alpha }^B\left(A\right)=\frac{1}{1-\alpha }\ln \left({\displaystyle \underset{i=1}{\overset{n}{\sum }}{\left(p_i^{+}\right)}^{\alpha }}+{\displaystyle \underset{i=1}{\overset{n}{\sum }}{\left(p_i^{-}\right)}^{\alpha }}\right)$ is monotonically decreasing

Regarding the monotonicity of entropy, it is important to note that maximum entropy occurs when the probability distribution is uniform, and the entropy decreases as the probability distribution becomes more skewed towards a few specific events. Therefore, intuitively, with more information about a system, the uncertainty about its state decreases, leading to a decrease in entropy.

Concavity

Entropy is generally expected to be concave. To establish that BRE is concave, we need to compute its second derivative and show that it is non-positive i.e., $\frac{{\text{d}}^2}{\text{d}{\alpha }^2}\left(H_{\alpha }^B\left(A\right)\right)\le 0$.

Proof

Differentiating $S_{\alpha }={\displaystyle \underset{i=1}{\overset{n}{\sum }}{\left(p_i^{+}\right)}^{\alpha }}+{\displaystyle \underset{i=1}{\overset{n}{\sum }}{\left(p_i^{-}\right)}^{\alpha }}$ a second time

$\frac{\text{d}}{\text{d}\alpha }\left(\frac{\text{d}}{\text{d}\alpha }{S}_{\alpha }\right)=\frac{\text{d}}{\text{d}\alpha }\left({\displaystyle \underset{i=1}{\overset{n}{\sum }}{\left(p_i^{+}\right)}^{\alpha }\ln }\left(p_i^{+}\right)+{\displaystyle \underset{i=1}{\overset{n}{\sum }}{\left(p_i^{-}\right)}^{\alpha }\ln \left(p_i^{-}\right)}\right)$

$\begin{array}{l}={\displaystyle \underset{i=1}{\overset{n}{\sum }}{\left(p_i^{+}\right)}^{\alpha }{\left(\ln p_i^{+}\right)}^2}+{\displaystyle \underset{i=1}{\overset{n}{\sum }}{\left(p_i^{-}\right)}^{\alpha }}{\left(\ln p_i^{-}\right)}^2-\frac{{\displaystyle \underset{i=1}{\overset{n}{\sum }}{\left(p_i^{+}\right)}^{\alpha }\left(\ln p_i^{+}\right)}+{\displaystyle \underset{i=1}{\overset{n}{\sum }}{\left(p_i^{-}\right)}^{\alpha }}{\left(\ln p_i^{-}\right)}^2}{{\displaystyle \underset{i=1}{\overset{n}{\sum }}{\left(p_i^{+}\right)}^{\alpha }}+{\displaystyle \underset{i=1}{\overset{n}{\sum }}{\left(p_i^{-}\right)}^{\alpha }}}\\ \le \text{0, since}0\le p_i^{+}\le 1\text{and}0\le p_i^{-}\le 1\forall i\end{array}$