What is an Intuitionistic Fuzzy Set (IFS)?

In standard fuzzy sets, we only consider the degree of membership. However, in the real world, there is often a degree of uncertainty or hesitation. Intuitionistic Fuzzy Sets, introduced by Krassimir Atanassov in 1986, extend fuzzy sets by adding a degree of non-membership.

The Formal Definition

An Intuitionistic Fuzzy Set A in a universe U is characterized by two functions:

• μ_A(x): Degree of Membership
• ν_A(x): Degree of Non-membership

The Constraint: For every element, the sum of membership and non-membership must be between 0 and 1:
0 ≤ μ_A(x) + ν_A(x) ≤ 1

Operations of Intuitionistic Fuzzy Sets

Given two Intuitionistic Fuzzy Sets A and B, the following operations apply:

• Union (OR): Takes the max of membership and min of non-membership.
  A ∪ B = { max(μ_A, μ_B), min(ν_A, ν_B) }
• Intersection (AND): Takes the min of membership and max of non-membership.
  A ∩ B = { min(μ_A, μ_B), max(ν_A, ν_B) }
• Complement: Swaps the membership and non-membership values.
  A^c = { ν_A, μ_A }

Intuitionistic Fuzzy Numbers (IFN)

An Intuitionistic Fuzzy Number is usually represented as a Triangular Intuitionistic Fuzzy Number (TIFN). It is defined by its membership and non-membership boundaries.

A TIFN is often written as: A = < (a, b, c); (a', b, c') > where:

• (a, b, c) defines the Membership triangle.
• (a’, b, c’) defines the Non-membership triangle.
• Constraint: a’ ≤ a and c’ ≥ c.

Operations of Intuitionistic Fuzzy Numbers

For two TIFNs, A = (a₁, b₁, c₁) and B = (a₂, b₂, c₂), the arithmetic is performed component-wise: