complex fuzzy set :
Learn the definition of complex fuzzy set CFS and the operations of complex fuzzy set CFS. This article explains the amplitude term and phase term within a complex fuzzy set with example for periodic data modeling.
Complex Fuzzy Set (CFS)
Introduction
A Complex Fuzzy Set (CFS) extends the traditional fuzzy set by mapping the membership degree to a unit disc in the complex plane. While a standard fuzzy set only measures “how much” an element belongs to a set, a CFS adds a second dimension—the Phase Term. This allows the set to represent data that is periodic or cyclical, such as seasonal weather patterns, financial cycles, or signal processing data.
Definition of Complex Fuzzy Set
A Complex Fuzzy Set A on a universe X is defined by a membership function μA(x) that assigns a complex-valued grade to each element:
The membership function is typically expressed in polar form:
Where:
- rA(x) ∈ [0, 1] is the Amplitude (degree of membership).
- wA(x) ∈ [0, 2π] is the Phase Term (representing periodicity).
- i is the imaginary unit (√-1).
Complex Fuzzy Number (CFN)
A single membership value α = r ⋅ e i w is called a Complex Fuzzy Number (CFN). It is often written as the pair α = (r, w) for simpler calculations.
Detailed Mathematical Operations
Let α₁ = (r₁, w₁) and α₂ = (r₂, w₂) be two CFNs and λ > 0. The operations are defined as follows:
Reverses both the membership degree and the phase position.
Note: Phase sums are typically calculated modulo 2π.
Let α₁ = (0.6, 0.5π) and α₂ = (0.4, 0.3π).
Addition (α₁ ⊕ α₂):
• Amplitude: 0.6 + 0.4 – (0.6 × 0.4) = 1.0 – 0.24 = 0.76
• Phase: 0.5π + 0.3π = 0.8π
Result: (0.76, 0.8π)
Multiplication (α₁ ⊗ α₂):
• Amplitude: 0.6 × 0.4 = 0.24
• Phase: (0.5π × 0.3π) / 2π = 0.15π2 / 2π = 0.075π
Result: (0.24, 0.075π)
Dr. M.U. Mirza
Mathematical Researcher & Educator