interval valued fuzzy set :
Learn the definition of interval valued fuzzy set and the algebraic operations of interval valued fuzzy set. We look at IVFS membership intervals and provide an interval valued fuzzy set with example to show how ranges represent uncertainty.
What is an Interval-Valued Fuzzy Set (IVFS)?
An Interval-Valued Fuzzy Set is a generalization of the standard fuzzy set. Instead of assigning a single precise membership value (like 0.7), it assigns a range or interval (like [0.6, 0.8]).
MA(x) = [μAL(x), μAU(x)]
• μL: Lower Bound (Minimum degree of membership).
• μU: Upper Bound (Maximum degree of membership).
• Constraint: 0 ≤ μL ≤ μU ≤ 1.
Example Scenario
If you ask a doctor how “High” a patient’s fever is, the doctor might say: “The membership of this temperature in the ‘High Fever’ set is between 0.7 and 0.9.” This is a perfect application of IVFS.
Set Operations on IVFS
Let A = [aL, aU] and B = [bL, bU] be two interval membership values.
1. Union (OR)
The union is calculated by taking the maximum of the lower bounds and the maximum of the upper bounds.
Calculation:
Lower Bound: max(0.3, 0.4) = 0.4
Upper Bound: max(0.6, 0.8) = 0.8
Result: [0.4, 0.8]
2. Intersection (AND)
The intersection takes the minimum of both bounds.
Calculation:
Lower Bound: min(0.3, 0.4) = 0.3
Upper Bound: min(0.6, 0.8) = 0.6
Result: [0.3, 0.6]
Interval-Valued Fuzzy Numbers (IVFN) Arithmetic
When working with numerical calculations, IVFNs use standard interval arithmetic or specific generalized formulas.
| Operation | Formula for A=[aL, aU] and B=[bL, bU] |
|---|---|
| Addition (⊕) | [ aL + bL – aLbL , aU + bU – aUbU ] |
| Multiplication (⊗) | [ aLbL , aUbU ] |
| Complement | [ 1 – aU , 1 – aL ] |
Arithmetic Step-by-Step Examples
Let A = [0.2, 0.5] and B = [0.3, 0.6].
Upper: 0.5 + 0.6 – (0.5 × 0.6) = 1.1 – 0.30 = 0.80
Result: [0.44, 0.80]
Upper: 1 – 0.2 = 0.8
Result: [0.5, 0.8]
Designed by: Dr. M.U. Mirza
Mathematical Researcher & Educator