Bipolar fuzzy set :
Discover the definition of bipolar fuzzy set and the operations of bipolar fuzzy set. We examine positive and negative membership grades and provide a bipolar fuzzy set with example to model bipolarity in human reasoning.
Bipolar Fuzzy Set (BFS)
Introduction
A Bipolar Fuzzy Set (BFS) is an extension of Zadeh’s traditional fuzzy set. While a standard fuzzy set handles membership in the range [0, 1], BFS acknowledges that many real-world properties have a “counter-property.” It uses a dual-scale: Positive membership for the property and Negative membership for its opposite, mapping values to the interval [-1, 1].
Definition of Bipolar Fuzzy Set
A Bipolar Fuzzy Set B in a universe of discourse X is defined as:
Where:
- μ+B(x) : X → [0, 1] represents the positive satisfaction degree.
- μ–B(x) : X → [-1, 0] represents the negative satisfaction degree (dissatisfaction).
Bipolar Fuzzy Number (BFN)
A Bipolar Fuzzy Number is represented as a pair α = (μ+, μ–) where μ+ ∈ [0, 1] and μ– ∈ [-1, 0].
Mathematical Operations
Let α₁ = (μ₁+, μ₁–) and α₂ = (μ₂+, μ₂–) be two Bipolar Fuzzy Numbers and λ > 0. The operations are defined as follows:
Let α₁ = (0.4, -0.3) and α₂ = (0.5, -0.2).
Addition (α₁ ⊕ α₂):
• Positive part: 0.4 + 0.5 – (0.4 × 0.5) = 0.9 – 0.20 = 0.7
• Negative part: – |(-0.3) × (-0.2)| = – (0.06) = -0.06
Result: (0.7, -0.06)
Multiplication (α₁ ⊗ α₂):
• Positive part: 0.4 × 0.5 = 0.2
• Negative part: – (0.3 + 0.2 – (0.3 × 0.2)) = – (0.5 – 0.06) = -0.44
Result: (0.2, -0.44)
Dr. M.U. Mirza
Mathematical Researcher & Educator