What is a Cubic Set?
A Cubic Set is a highly advanced extension of fuzzy sets that combines two different types of fuzzy information into one structure. It allows us to handle uncertainty by looking at both a range (interval) and a specific value (fuzzy point) simultaneously.
1. An Interval-Valued Fuzzy Set (IVFS): Represents the “uncertain range” of membership.
2. A Standard Fuzzy Set (FS): Represents a “specific degree” of membership.
Where A(x) is the interval [μ_lower, μ_upper] and λ(x) is a standard fuzzy value between 0 and 1.
Internal vs. External Cubic Sets
Cubic sets are classified based on where the specific fuzzy value (λ) sits in relation to the interval (A).
1. Internal Cubic Set (ICS)
An ICS occurs when the fuzzy value falls inside the defined interval.
2. External Cubic Set (ECS)
An ECS occurs when the fuzzy value falls outside the defined interval.
Basic Operations on Cubic Sets
When performing operations on Cubic Sets, we must calculate the interval part and the fuzzy point part separately.
1. P-Union (Union)
The union of two cubic sets takes the “Maximum” of the intervals and the “Maximum” of the fuzzy points.
2. P-Intersection (Intersection)
The intersection takes the “Minimum” of the intervals and the “Minimum” of the fuzzy points.
Comparison: Why Use Cubic Sets?
| Feature | Standard Fuzzy Set | Cubic Set |
|---|---|---|
| Membership | Single value (0.7) | Interval + Value ([0.6, 0.8], 0.75) |
| Data Depth | Low (Static) | High (Combined perspectives) |
| Decision Making | Simple preference | Complex risk assessment |
• Interval: “The fever is likely between 101°F and 103°F.”
• Fuzzy Value: “But my specific feeling of urgency is 0.9.”
The Cubic Set captures both the statistical range and the expert’s specific intuition.
Designed by: Dr. M.U. Mirza
Mathematical Researcher & Educator