Neutrosophic set:
Dive into the definition of neutrosophic set and the specialized operations of neutrosophic set. This post details the truth-membership, indeterminacy-membership, and falsity-membership within a neutrosophic set with example.
What is a Neutrosophic Set (NS)?
Introduced by Florentin Smarandache in 1995, a Neutrosophic Set is a branch of philosophy that studies the origin, nature, and scope of neutralities. It is a powerful extension of fuzzy logic that handles Indeterminacy independently.
• T (Truth): The degree of membership.
• I (Indeterminacy): The degree of uncertainty or “neutral” thought.
• F (Falsity): The degree of non-membership.
Note: In Single-Valued Neutrosophic Sets (SVNS), the sum of T, I, and F must be:
The “Soccer Match” Example
Imagine you ask a sports analyst about a team’s chance of winning:
• Falsity (F) = 0.3: 30% chance they lose.
• Indeterminacy (I) = 0.2: 20% chance the game is canceled or ends in a way we cannot predict (like a riot or weather issue).
Operations of Neutrosophic Sets
Let’s take two Neutrosophic values: A = (T₁, I₁, F₁) and B = (T₂, I₂, F₂).
1. Union (OR Operation)
Takes the maximum truth, the minimum indeterminacy, and the minimum falsity.
Calculation:
Max Truth: max(0.7, 0.5) = 0.7
Min Indeterminacy: min(0.4, 0.2) = 0.2
Min Falsity: min(0.2, 0.6) = 0.2
Result: (0.7, 0.2, 0.2)
2. Intersection (AND Operation)
Takes the minimum truth, the maximum indeterminacy, and the maximum falsity.
Calculation:
Min Truth: min(0.7, 0.5) = 0.5
Max Indeterminacy: max(0.4, 0.2) = 0.4
Max Falsity: max(0.2, 0.6) = 0.6
Result: (0.5, 0.4, 0.6)
Single-Valued Neutrosophic Numbers (SVNN)
When Neutrosophic sets are used as individual data points for calculation, they are called Neutrosophic Numbers.
| Operation | Arithmetic Formula |
|---|---|
| Addition (⊕) | ( T₁+T₂ – T₁T₂, I₁I₂, F₁F₂ ) |
| Multiplication (⊗) | ( T₁T₂, I₁+I₂ – I₁I₂, F₁+F₂ – F₁F₂ ) |
Arithmetic Step-by-Step Example
Let A = (0.6, 0.3, 0.4) and B = (0.5, 0.2, 0.1).
I: 0.3 × 0.2 = 0.06
F: 0.4 × 0.1 = 0.04
Sum Result: (0.8, 0.06, 0.04)
I: 0.3 + 0.2 – (0.3 × 0.2) = 0.5 – 0.06 = 0.44
F: 0.4 + 0.1 – (0.4 × 0.1) = 0.5 – 0.04 = 0.46
Product Result: (0.3, 0.44, 0.46)
Designed by: Dr. M.U. Mirza
Mathematical Researcher & Educator