Pythagorean fuzzy set PFS
Gain insights into the definition of pythagorean fuzzy set PFS and standard operations of pythagorean fuzzy set. This guide explains the PFS membership degree, PFS non-membership degree, and includes a pythagorean fuzzy set with example to show its expanded search space
What is a Pythagorean Fuzzy Set (PFS)?
Introduced by Ronald R. Yager in 2013, Pythagorean Fuzzy Sets are a powerful extension of Intuitionistic Fuzzy Sets (IFS). They are designed to handle situations where the sum of membership and non-membership degrees exceeds 1, but the sum of their squares does not.
Mathematical Definition
A Pythagorean Fuzzy Set P in a universe U is defined by a membership function μP(x) and a non-membership function νP(x).
Unlike IFS, where μ + ν ≤ 1, in PFS the rule is:
0 ≤ (μP(x))² + (νP(x))² ≤ 1
In standard IFS: 0.8 + 0.5 = 1.3 (Invalid).
In PFS: (0.8)² + (0.5)² = 0.64 + 0.25 = 0.89 (Valid).
Operations of Pythagorean Fuzzy Sets
Let A and B be two Pythagorean Fuzzy Sets. The fundamental operations are:
- Union (A ∪ B): Takes the maximum of membership and minimum of non-membership.
max(μA, μB), min(νA, νB) - Intersection (A ∩ B): Takes the minimum of membership and maximum of non-membership.
min(μA, μB), max(νA, νB) - Complement (Ac): Swaps the membership and non-membership values.
Ac = (νA, μA)
The Degree of Indeterminacy (π)
In PFS, the hesitancy or indeterminacy degree is calculated using the square root:
πP(x) = √(1 – (μP(x)² + νP(x)²))Pythagorean Fuzzy Numbers (PFN)
A Pythagorean Fuzzy Number is typically represented as a pair p = (μ, ν). These numbers are used extensively in multi-criteria decision-making (MCDM) models.
Arithmetic Operations
For two PFNs, p1 = (μ1, ν1) and p2 = (μ2, ν2):
| Operation | Formula |
|---|---|
| Addition (⊕) | ( √(μ₁² + μ₂² – μ₁²μ₂²), ν₁ν₂ ) |
| Multiplication (⊗) | ( μ₁μ₂, √(ν₁² + ν₂² – ν₁²ν₂²) ) |
Let p1 = (0.6, 0.4) and p2 = (0.5, 0.3)
Calculation: μ = √(0.6² + 0.5² – 0.6²*0.5²) = √(0.36 + 0.25 – 0.09) = √0.52 ≈ 0.72
Calculation: ν = 0.4 * 0.3 = 0.12
Result: (0.72, 0.12)
Designed by: Dr. M.U. Mirza
Mathematical Researcher & Educator