# 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

## 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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