Interval Valued Intuitionistic Fuzzy Sets: A Comprehensive Guide

Interval valued intuitionistic fuzzy sets (IVIFSs) are a type of fuzzy set that can be used to model uncertainty and vagueness in various fields. They are an extension of intuitionistic fuzzy sets (IFSs),which were introduced by Atanassov in 1986. IVIFSs were first introduced by Atanassov and Gargov in 1989.

Interval-Valued Intuitionistic Fuzzy Sets (Studies in Fuzziness and Soft Computing Book 388)

by Krassimir T. Atanassov

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An IVIFS is characterized by three functions: a membership function, a non-membership function, and a hesitation function. The membership function maps elements of the universe of discourse to an interval of values in the unit interval [0, 1]. The non-membership function maps elements of the universe of discourse to an interval of values in the unit interval [0, 1]. The hesitation function is the difference between the membership function and the non-membership function.

IVIFSs have been used in a variety of applications, including:

• Decision making
• Pattern recognition
• Image processing
• Medical diagnosis
• Financial forecasting

In this article, we will provide a comprehensive to IVIFSs. We will cover their definition, operations, and applications. We will also provide some examples of how IVIFSs can be used to solve real-world problems.

Definition of Interval Valued Intuitionistic Fuzzy Sets

An interval valued intuitionistic fuzzy set (IVIFS) on a universe of discourse X is characterized by three functions:

• A membership function μ_A(x): X → [0, 1]
• A non-membership function ν_A(x): X → [0, 1]
• A hesitation function π_A(x): X → [0, 1]

The membership function μ_A(x) maps each element x of X to an interval of values in the unit interval [0, 1]. The non-membership function ν_A(x) maps each element x of X to an interval of values in the unit interval [0, 1]. The hesitation function π_A(x) is the difference between the membership function and the non-membership function, i.e., π_A(x) = μ_A(x) - ν_A(x).

The following conditions must be satisfied for an IVIFS to be valid:

• 0 ≤ μ_A(x) + ν_A(x) ≤ 1 for all x ∈ X
• 0 ≤ π_A(x) ≤ 1 for all x ∈ X

The first condition ensures that the sum of the membership and non-membership degrees is always less than or equal to 1. The second condition ensures that the hesitation degree is always less than or equal to 1.

Operations on Interval Valued Intuitionistic Fuzzy Sets

There are a number of operations that can be performed on IVIFSs. These operations include:

• Union
• Intersection
• Complement
• Scalar multiplication
• Fuzzy implication

The union of two IVIFSs A and B is an IVIFS C such that:

• μ_C(x) = [max(μ_A(x),μ_B(x)),min(μ_A(x),μ_B(x))]
• ν_C(x) = [min(ν_A(x),ν_B(x)),max(ν_A(x),ν_B(x))]
• π_C(x) = [min(π_A(x),π_B(x)),max(π_A(x),π_B(x))]

The intersection of two IVIFSs A and B is an IVIFS C such that:

• μ_C(x) = [min(μ_A(x),μ_B(x)),max(μ_A(x),μ_B(x))]
• ν_C(x) = [max(ν_A(x),ν_B(x)),min(ν_A(x),ν_B(x))]
• π_C(x) = [max(π_A(x),π_B(x)),min(π_A(x),π_B(x))]

The complement of an IVIFS A is an IVIFS B such that:

• μ_B(x) = [1 - ν_A(x),1 - μ_A(x)]
• ν_B(x) = [1 - μ_A(x),1 - ν_A(x)]
• π_B(x) = [π_A(x),π_A(x)]

The scalar multiplication of an IVIFS A by a scalar α is an IVIFS B such that:

• μ_B(x) = [αμ_A(x),αμ_A(x)]
• ν_B(x) = [αν_A(x),αν_A(x)]
• π_B(x) = [1 - α

Interval-Valued Intuitionistic Fuzzy Sets (Studies in Fuzziness and Soft Computing Book 388)

by Krassimir T. Atanassov

5 out of 5

Language	:	English
File size	:	97769 KB
Text-to-Speech	:	Enabled
Screen Reader	:	Supported
Enhanced typesetting	:	Enabled
Print length	:	349 pages

FREE