Decomposition Theorem of Generalized Interval-Valued Intuitionistic Fuzzy Sets

Decomposition Theorem of Generalized Interval-Valued Intuitionistic Fuzzy Sets

Amal Kumar Adak (Vidyasagar University, India), Monoranjan Bhowmik (VTT College, India) and Madhumangal Pal (Vidyasagar University, India)
DOI: 10.4018/978-1-4666-6252-0.ch009
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Abstract

In this chapter, the authors establish decomposition theorems of Generalized Interval-Valued Intuitionistic Fuzzy Sets (GIVIFS) by use of cut sets of generalized interval-valued intuitionistic fuzzy sets. First, new definitions of eight kinds of cut sets generalized interval-valued intuitionistic fuzzy sets are introduced. Second, based on these new cut sets, the decomposition generalized interval-valued intuitionistic fuzzy sets are established. The authors show that each kind of cut sets corresponds to two kinds of decomposition theorems. These results provide a fundamental theory for the research of generalized interval-valued intuitionistic fuzzy sets.
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2. Preliminaries

In this section, we recalled some preliminaries and the definition of IVIFS and GIVIFS.

  • Definition 2.1: An IVIFS A over X (universe of discourse) is an object having the form A = {〈x,MA(x),NA(x)〉 | x ∈ X}, where MA(x): X → [I] and NA(x): X → [I]. The intervals MA(x) and NA(x) denote the intervals of the degree of membership and degree of non-membership of the element x to the set A, where MA(x) = [MAL(x),MAU(x)] and NA(x) = [NAL(x),NAU(x)], for all x ∈ X, with the condition 0 ≤ MAU(x)+NAU(x) ≤ 1. For simplicity, we denote A = {〈x, [A-(x),A+(x)], [B-(x),B+(x)]〉 | x ∈ X}.

  • Definition 2.2: If the IVIFS A = {〈x,MA(x),NA(x)〉 | x ∈ X}, satisfying the condition MAU(x) ∧ NAU(x) ≤ 0.5 for all x ∈ X then A is called generalized interval-valued intuitionistic fuzzy set (GIVIFS). The condition MAU(x) ∧NAU(x) ≤ 0.5 is called generalized interval-valued intuitionistic fuzzy condition (GIVIFC). The maximum value of MAU(x) and NAU(x) is 1.0, therefore GIVIFC imply that

0 ≤ MAU(x) + NAU(x) ≤ 1.5.

It may be noted that all IVIFS are GIVIFS but the converse is not true.

Let F(X) be the set of all GIVIFSs defined on X.

2.1. Some Operations on GIVIFSs

In [2], Bhowmik and Pal defined some relational operations on GIVIFSs. Let A and B be two

GIVIFSs on X, where

A = {〈[MAL(x),MAU], [NAL(x),NAU(x)]: x ∈ X〉} and B = {〈[MBL(x),MBU], [NBL(x),NBU(x)]: x ∈ X〉}.

Then,

  • 1.

    A ⊆ B iff {(MAU(x) ≤ MBU(x) and MAL(x) ≤ MBL(x))} and{(NAU(x) ≥ NBU(x) and NAL(x) ≥ NBL(x))}, for all x ∈ X.

  • 2.

    A ∩ B = {〈[min{MAL(x),MBL(x)}, min{MAU(x),MBU(x)}],[max{NAL(x),NBL(x)}, max{NAU(x),NBU(x)}]〉: x ∈ X}.

  • 3.

    A ∪ B = {〈[max{MAL(x),MBL(x)}, max{MAU(x),MBU(x)}],[min{NAL(x),NBL(x)}, min{NAU(x),NBU(x)}]〉: x ∈ X}.

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