vague set :
Study the definition of vague set and the operations of vague set. We compare truth and false membership functions and provide a vague set with example to show its relation to intuitionistic models.
Vague Set (VS)
Introduction
A Vague Set (VS) is a powerful extension of the classical fuzzy set. While a standard fuzzy set uses a single point value to represent membership, a vague set uses two distinct functions to characterize the True Membership and False Membership. This allows researchers to distinguish between “supporting evidence” and “opposing evidence,” creating an interval-based membership that captures higher levels of uncertainty.
Definition of Vague Set
A Vague Set A in a universe of discourse X is characterized by a true membership function tA and a false membership function fA:
Subject to the essential constraint:
- tA(x): Lower bound on membership (evidence for).
- fA(x): Lower bound on non-membership (evidence against).
- 1 – fA(x): Upper bound on membership.
Vague Number (VN) / Vague Value
A Vague Number is represented as an interval α = [ t, 1 – f ]. The width of this interval, (1 – f) – t, represents the degree of hesitation or unknown information regarding the element x.
Mathematical Operations
Let α₁ = [ t₁, 1 – f₁ ] and α₂ = [ t₂, 1 – f₂ ] be two Vague Numbers. The primary operations are:
The supporting evidence for the complement is the opposing evidence of the original set.
Suppose we have two Vague Numbers:
α₁ = [ 0.4, 0.7 ] (where t₁=0.4, f₁=0.3)
α₂ = [ 0.5, 0.8 ] (where t₂=0.5, f₂=0.2)
Union (α₁ ∪ α₂):
[ max(0.4, 0.5), max(0.7, 0.8) ] = [ 0.5, 0.8 ]
Algebraic Sum (α₁ ⊕ α₂):
• Lower bound: 0.4 + 0.5 – (0.4 × 0.5) = 0.9 – 0.20 = 0.70
• Upper bound: 0.7 × 0.8 = 0.56
Note: If sum logic is applied, the interval is transformed based on the chosen T-conorm.
Designed by: Dr. M.U. Mirza
Mathematical Researcher & Educator