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Convexity of Picture Fuzzy Multisets

This paper introduces the concept of convexity for picture fuzzy multisets and presents several of their fundamental properties following an analysis of the underlying structure.

Original authors: Taiwo O. Sangodapo

Published 2026-03-25

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Taiwo O. Sangodapo

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

The Big Picture: Why Do We Need This?

Imagine you are trying to describe a group of people at a party, but you want to be very precise about their feelings toward a specific topic (like "The new music playlist").

Old School (Fuzzy Sets): You could say, "Most people like it." (Yes/No, but with a degree of "Yes").
Better (Intuitionistic Fuzzy Sets): You could say, "Some like it, some hate it." (Yes/No/Undecided).
Even Better (Picture Fuzzy Sets): You can now say, "Some Love it, some are Neutral, some Hate it, and some Refuse to listen." This captures the full spectrum of human opinion.

Now, imagine that instead of just one person giving an opinion, you have a crowd where the same person might give multiple opinions at different times, or you are counting votes from a group where the number of people matters. This is a Multiset.

This paper is about taking that complex "crowd opinion" system and figuring out how to draw a smooth, unbroken shape around it. In math, this shape is called a Convex Set.

The Core Concept: What is "Convexity"?

The Jello Analogy:
Imagine a blob of Jello on a plate.

If you pick any two points inside the Jello and draw a straight line between them, the entire line stays inside the Jello. That is convex.
If the Jello has a hole in the middle (like a donut), and you draw a line from one side of the hole to the other, the line goes through the empty air. That is NOT convex.

In the world of fuzzy logic, "convexity" means that if you have two valid opinions (or states of being), any "mix" of those two opinions must also be a valid state.

What is a "Picture Fuzzy Multiset" (PFMS)?

Let's break down the title:

Picture Fuzzy: Instead of just "Yes" or "No," we have four colors: Positive (Green), Neutral (Yellow), Negative (Red), and Refusal (Grey).
Multiset: Unlike a normal list where you can't have duplicates, a multiset is like a bag of marbles. You can have three red marbles, two blue ones, and five green ones. The count matters.

The Real-World Scenario:
Imagine a voting booth for a new park design.

Person A votes: 100% "Love it" (Positive).
Person B votes: 100% "Hate it" (Negative).
The Multiset: We have a bag containing 50 "Love" votes and 50 "Hate" votes.

The paper asks: If we mix these votes together, does the result still make sense within our rules?

The Main Discovery: Defining the "Shape" of the Bag

The author, Taiwo O. Sangodapo, introduces a new rulebook for these "bags of opinions."

1. The Mixing Rule (Convex Combination)

If you take a vote from Person A (Green) and a vote from Person B (Red) and mix them 50/50, you get a "Grey" vote (Neutral).
The paper proves that if your original bag of votes is "convex" (well-organized), then any mixture of votes you create from that bag will also stay inside the "well-organized" zone.

Analogy: If you have a smooth, round hill of sand (convex), and you take a handful from the left and a handful from the right and mix them, the new pile of sand will still sit nicely on the hill. It won't magically jump off the hill.

2. The "Cut" Rule (Slicing the Cake)

The paper uses a clever trick called a "Cut Set."
Imagine your complex bag of votes is a multi-layered cake.

Layer 1: Only the "Strongly Positive" votes.
Layer 2: The "Positive" and "Neutral" votes.
Layer 3: Everything.

The paper proves a simple but powerful rule: If every single layer of the cake is a smooth, round hill (convex), then the whole cake is a smooth, round hill.
This makes it much easier for mathematicians to check if a complex system is "convex." They just need to check the layers!

3. The "Hull" (The Rubber Band)

The paper also defines the "Picture Convex Hull."
Imagine you have a bunch of scattered marbles (votes) on a table. If you stretch a rubber band around all of them, the shape inside the rubber band is the "Hull."
The paper shows that for these complex picture fuzzy multisets, the "Hull" is simply every possible mix you can make of the votes inside.

Why Does This Matter? (The "So What?")

You might ask, "Who cares about mixing fuzzy votes?"

This is crucial for Artificial Intelligence (AI) and Decision Making.

Scenario: A self-driving car needs to decide whether to stop or go. It gets data from 100 sensors. Some say "Stop" (Red), some say "Go" (Green), some are unsure (Yellow).
The Problem: If the AI's logic isn't "convex," it might make a weird, illogical decision when mixing these sensor inputs. It might think a 50/50 mix of "Stop" and "Go" is actually "Jump off a cliff" (which is outside the logical shape).
The Solution: By proving that these "Picture Fuzzy Multisets" are convex, the author ensures that AI systems can mix complex, multi-layered human opinions or sensor data smoothly and safely. It guarantees that the "middle ground" is always a valid, safe place to be.

Summary in One Sentence

This paper builds a mathematical safety net that ensures when we mix complex, multi-colored opinions from a crowd (multisets), the resulting "average" opinion always stays logical and within the boundaries of reality, just like mixing colors of paint always gives you another valid color, never a hole in the universe.

1. Problem Statement

The paper addresses a gap in the mathematical theory of fuzzy sets and computational intelligence. While the concepts of Fuzzy Sets (FS), Intuitionistic Fuzzy Sets (IFS), and Picture Fuzzy Sets (PFS) have been extensively studied, and the concepts of Fuzzy Multisets (FMS) and Picture Fuzzy Multisets (PFMS) have been established as generalizations, the specific property of convexity had not yet been rigorously defined or analyzed for Picture Fuzzy Multisets.

Existing literature had established convexity for:

Classical Fuzzy Sets (Zadeh, 1965).
Intuitionistic Fuzzy Sets (Huang, 2012; Das & Mukhlalsah, 2018).
Picture Fuzzy Sets (Sangodapo, 2022).

However, PFMSs are a more complex structure that allows for multiple occurrences of membership values (count functions) for positive, neutral, and negative degrees. The problem is to extend the definition of convexity to this multi-valued, multi-occurrence structure and investigate its fundamental properties.

2. Methodology

The author employs a deductive mathematical approach, building upon established definitions from fuzzy set theory and extending them to the multiset domain. The methodology involves:

Literature Review: Tracing the evolution from Zadeh's fuzzy sets to Atanassov's IFS, Cuong's PFS, and Cao et al.'s PFMS.
Definition Extension: Adapting the definition of convexity from Picture Fuzzy Sets (PFS) to Picture Fuzzy Multisets (PFMS). This involves modifying the membership functions to account for the "count" nature of multisets (sequences of membership values rather than single values).
Formal Proofs: Using mathematical induction and set-theoretic logic to prove theorems regarding:
- The equivalence between the convexity of a PFMS and the convexity of its $(r, s, t)$ -cuts.
- The behavior of convexity under set operations (intersection).
- The characterization of convexity via finite convex combinations.
- The construction of the "Picture Convex Hull."

3. Key Contributions

The paper makes the following specific theoretical contributions:

A. Definition of Convex PFMS

The author defines a Picture Fuzzy Multiset $D$ as convex if, for any two points $x, y$ in the universe and a scalar $\lambda \in [0, 1]$ , the membership sequences satisfy specific inequalities. Specifically, for the $k$ -th element in the membership sequence:

Positive Membership ( $\sigma$ ): $\sigma^k_D[(1-\lambda)x + \lambda y] \geq \sigma^k_D(x) \wedge \sigma^k_D(y)$
Neutral Membership ( $\tau$ ): $\tau^k_D[(1-\lambda)x + \lambda y] \geq \tau^k_D(x) \wedge \tau^k_D(y)$
Negative Membership ( $\eta$ ): $\eta^k_D[(1-\lambda)x + \lambda y] \leq \eta^k_D(x) \vee \eta^k_D(y)$

B. Generalization to Finite Points

The definition is extended from two points to $n$ points using a Picture Convex Combination (PCC). A PFMS is convex if the membership values of any convex combination of points within the set are greater than or equal to (for positive/neutral) or less than or equal to (for negative) the minimum/maximum of the membership values of the constituent points.

C. Picture Convex Hull (Pch)

The paper defines the Picture Convex Hull of a PFMS $D$ , denoted as $Pch(D)$, as the set of all possible convex combinations of points in $D$ . It establishes that $Pch(D)$ is the smallest convex PFMS containing $D$ .

D. Equivalence with Cut Sets

A significant theoretical result is the proof that a PFMS is convex if and only if its $(r, s, t)$ -cuts (crisp sets derived by thresholding membership values) are convex. This bridges the gap between fuzzy/multiset convexity and classical convex geometry.

4. Key Results and Theorems

The paper establishes several propositions and theorems:

Proposition 3.1 (Cut Set Equivalence): A PFMS $D$ is convex if and only if its $(r, s, t)$ -cut $C_{r,s,t}(D)$ is a convex crisp set for all valid thresholds $r, s, t$ .
Proposition 3.2 (Intersection Property): The intersection of two convex PFMSs is also a convex PFMS.
Corollary 3.1 (Arbitrary Intersection): The intersection of an arbitrary family of convex PFMSs is convex.
Proposition 3.3 (Finite Combination Characterization): A PFMS is convex if and only if it satisfies the convexity condition for all finite families of points and their convex combinations.
Theorem 3.1 (Convex Hull Property): For a convex PFMS $D$ , the Picture Convex Hull $Pch(D)$ consists exactly of all convex combinations of the points of $D$ .

5. Significance and Implications

Theoretical Advancement: This work fills a critical gap in the hierarchy of fuzzy set generalizations. By defining convexity for PFMSs, it unifies the concepts of "neutrality" (from PFS) and "multiplicity" (from Multisets) within the geometric framework of convexity.
Computational Intelligence Applications: PFMSs are designed to handle complex decision-making scenarios involving positive, neutral, and negative opinions (e.g., voting systems, medical diagnosis, social media sentiment). The introduction of convexity allows for:
- Optimization: Applying convex optimization techniques to problems modeled with PFMSs.
- Clustering: Enhancing clustering algorithms (like the Hierarchical Picture Clustering mentioned in the introduction) by ensuring cluster boundaries are convex.
- Decision Support: Providing a rigorous mathematical basis for aggregating multi-criteria decision-making data where multiple instances of the same opinion exist.
Foundation for Future Research: The paper sets the stage for further extensions, such as affine PFMSs, fuzzy convex programming with PFMS constraints, and applications in group theory (which the author notes has been explored in other contexts).

In summary, Sangodapo successfully extends the geometric concept of convexity to the complex domain of Picture Fuzzy Multisets, providing the necessary definitions, proofs, and structural properties to enable future mathematical and computational applications.