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Journal of Applied Mathematics
Volume 2012 (2012), Article ID 870504, 18 pages
http://dx.doi.org/10.1155/2012/870504
Research Article

Generalised Interval-Valued Fuzzy Soft Set

School of Mathematical Sciences, Faculty of Science and Technology, Universiti Kebangsaan Malaysia, 43600 UKM Bangi, Selangor, Malaysia

Received 6 August 2011; Revised 21 January 2012; Accepted 22 February 2012

Academic Editor: Ch. Tsitouras

Copyright © 2012 Shawkat Alkhazaleh and Abdul Razak Salleh. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

We introduce the concept of generalised interval-valued fuzzy soft set and its operations and study some of their properties. We give applications of this theory in solving a decision making problem. We also introduce a similarity measure of two generalised interval-valued fuzzy soft sets and discuss its application in a medical diagnosis problem: fuzzy set; soft set; fuzzy soft set; generalised fuzzy soft set; generalised interval-valued fuzzy soft set; interval-valued fuzzy set; interval-valued fuzzy soft set.

1. Introduction

Molodtsov [1] initiated the theory of soft set as a new mathematical tool for dealing with uncertainties which traditional mathematical tools cannot handle. He has shown several applications of this theory in solving many practical problems in economics, engineering, social science, medical science, and so forth. Presently, work on the soft set theory is progressing rapidly. Maji et al. [2, 3], Roy and Maji [4] have further studied the theory of soft set and used this theory to solve some decision making problems. Maji et al. [5] have also introduced the concept of fuzzy soft set, a more general concept, which is a combination of fuzzy set and soft set and studied its properties. Zou and Xiao [6] introduced soft set and fuzzy soft set into the incomplete environment, respectively. Alkhazaleh et al. [7] introduced the concept of soft multiset as a generalisation of soft set. They also defined the concepts of fuzzy parameterized interval-valued fuzzy soft set [8] and possibility fuzzy soft set [9] and gave their applications in decision making and medical diagnosis. Alkhazaleh and Salleh [10] introduced the concept of a soft expert set, where the user can know the opinion of all experts in one model without any operations. Even after any operation, the user can know the opinion of all experts. In 2011, Salleh [11] gave a brief survey from soft set to intuitionistic fuzzy soft set. Majumdar and Samanta [12] introduced and studied generalised fuzzy soft set where the degree is attached with the parameterization of fuzzy sets while defining a fuzzy soft set. Yang et al. [13] presented the concept of interval-valued fuzzy soft set by combining the interval-valued fuzzy set [14, 15] and soft set models. In this paper, we generalise the concept of fuzzy soft set as introduced by Maji et al. [5] to generalised interval-valued fuzzy soft set. In our generalisation of fuzzy soft set, a degree is attached with the parameterization of fuzzy sets while defining an interval-valued fuzzy soft set. Also, we give some applications of generalised interval-valued fuzzy soft set in decision making problem and medical diagnosis.

2. Preliminary

In this section, we recall some definitions and properties regarding fuzzy soft set and generalised fuzzy soft set required in this paper.

Definition 2.1 (see [15]). An interval-valued fuzzy set 𝑋 on a universe 𝑈 is a mapping such that ([]𝑋𝑈Int0,1),(2.1) where Int([0,1]) stands for the set of all closed subintervals of [0,1], the set of all interval-valued fuzzy sets on 𝑈 is denoted by 𝑃(𝑈).

Suppose that 𝑋𝑃(𝑈), for all 𝑥𝑈, 𝜇𝑥(𝑥)=[𝜇𝑥(𝑥),𝜇+𝑥(𝑥)] is called the degree of membership of an element 𝑥 to 𝑋. 𝜇𝑥(𝑥) and 𝜇+𝑥(𝑥) are referred to as the lower and upper degrees of membership of 𝑥 to 𝑋 where 0𝜇𝑥(𝑥)𝜇+𝑥(𝑥)1.

Definition 2.2 (see [14]). The subset, complement, intersection, and union of the interval-valued fuzzy sets are defined as follows. Let 𝑋,𝑌𝑃(𝑈), then(a)the complement of 𝑋 is denoted by 𝑋𝑐 where 𝜇𝑋𝑐(𝑥)=1𝜇𝑋(𝑥)=1𝜇+𝑋(𝑥),1𝜇𝑋(𝑥),(2.2)(b)the intersection of 𝑋 and 𝑌 is denoted by 𝑌𝑋 where 𝜇𝑋𝑌𝜇(𝑥)=inf𝑋(𝑥),𝜇𝑌=𝜇(𝑥)inf𝑋(𝑥),𝜇𝑌𝜇(𝑥),inf+𝑋(𝑥),𝜇+𝑌,(𝑥)(2.3)(c)the union of 𝑋 and 𝑌 is denoted by 𝑌𝑋 where 𝜇𝑋𝑌𝜇(𝑥)=sup𝑋(𝑥),𝜇𝑌=𝜇(𝑥)sup𝑋(𝑥),𝜇𝑌𝜇(𝑥),sup+𝑋(𝑥),𝜇+𝑌;(𝑥)(2.4)(d)𝑋 is a subset of 𝑌denoted by 𝑋𝑌 if 𝜇𝑋(𝑥)𝜇𝑌(𝑥) and 𝜇+𝑋(𝑥)𝜇+𝑌(𝑥).

Definition 2.3 (see [14]). The compatibility measure 𝜑(𝐴,𝐵) of an interval-valued fuzzy set 𝐴 with an interval-valued fuzzy set 𝐵 (𝐴 is a reference set) is given by 𝜑𝜑(𝐴,𝐵)=(𝐴,𝐵),𝜑+(𝐴,𝐵),(2.5) such that 𝜑𝜑(𝐴,𝐵)=min1(𝐴,𝐵),𝜑2,𝜑(𝐴,𝐵)+(𝜑𝐴,𝐵)=max1(𝐴,𝐵),𝜑2(,𝐴,𝐵)(2.6) where 𝜑1(𝐴,𝐵)=max𝑥𝑋𝜇min𝐴(𝑥),𝜇𝐵(𝑥)max𝑥𝑋𝜇𝐴(,𝜑𝑥)2(𝐴,𝐵)=max𝑥𝑋𝜇min+𝐴(𝑥),𝜇+𝐵(𝑥)max𝑥𝑋𝜇+𝐴.(𝑥)(2.7)

Theorem 2.4 (see [14]). Consider arbitrary, nonempty interval-valued fuzzy sets 𝐴, 𝐵, and 𝐶 from the family of 𝑖𝑣𝑓(𝑋) and a compatibility measure in the sense of Definition 2.3. Then,(a)𝐴𝜑(𝐴,𝐴)=[1,1]={1}, (b)𝜑(𝐴,𝐵)=[0,0]={0}𝐴𝐵=, (c)in general 𝜑(𝐴,𝐵)𝜑(𝐵,𝐴).

Let 𝑈 be a universal set and 𝐸 a set of parameters. Let 𝑃(𝑈) denote the power set of 𝑈 and 𝐴𝐸. Molodtsov [1] defined soft set as follows.

Definition 2.5. A pair (𝐹,𝐸) is called a soft set over 𝑈, where 𝐹 is a mapping given by 𝐹𝐸𝑃(𝑈). In other words, a soft set over 𝑈 is a parameterized family of subsets of the universe 𝑈.

Definition 2.6 (see [5]). Let 𝑈 be a universal set, and let 𝐸 be a set of parameters. Let 𝐼𝑈 denote the power set of all fuzzy subsets of 𝑈. Let 𝐴𝐸. A pair (𝐹,𝐸) is called a fuzzy soft set over 𝑈 where 𝐹 is a mapping given by 𝐹𝐴𝐼𝑈.(2.8)

Definition 2.7 (see [12]). Let 𝑈={𝑥1,𝑥2,,𝑥𝑛} be the universal set of elements and 𝐸={𝑒1,𝑒2,,𝑒𝑚} be the universal set of parameters. The pair (𝑈,𝐸) will be called a soft universe. Let 𝐹𝐸𝐼𝑈, where 𝐼𝑈 is the collection of all fuzzy subsets of 𝑈, and let 𝜇 be a fuzzy subset of 𝐸. Let 𝐹𝜇𝐸𝐼𝑈×𝐼 be a function defined as follows: 𝐹𝜇(𝑒)=(𝐹(𝑒),𝜇(𝑒)).(2.9) Then, 𝐹𝜇 is called a generalised fuzzy soft set (GFSS in short) over the soft set (𝑈,𝐸). Here, for each parameter 𝑒𝑖, 𝐹𝜇(𝑒𝑖)=(𝐹(𝑒𝑖),𝜇(𝑒𝑖)) indicates not only the degree of belongingness of the elements of 𝑈 in 𝐹(𝑒𝑖) but also the degree of possibility of such belongingness which is represented by 𝜇(𝑒𝑖). So we can write 𝐹𝜇(𝑒𝑖) as follows: 𝐹𝜇𝑒𝑖=𝑥1𝐹𝑒𝑖𝑥1,𝑥2𝐹𝑒𝑖𝑥2𝑥,,𝑛𝐹𝑒𝑖𝑥𝑛𝑒,𝜇𝑖,(2.10) where 𝐹(𝑒𝑖)(𝑥1),𝐹(𝑒𝑖)(𝑥2),,𝐹(𝑒𝑖)(𝑥𝑛) are the degree of belongingness and 𝜇(𝑒𝑖) is the degree of possibility of such belongingness.

Definition 2.8 (see [13]). Let 𝑈 be an initial universe and 𝐸 a set of parameters. 𝑃(𝑈) denotes the set of all interval-valued fuzzy sets of 𝑈. Let 𝐴𝐸. A pair (𝐹,𝐴) is an interval-valued fuzzy soft set over 𝑈, where 𝐹 is a mapping given by 𝐹𝐴𝑃(𝑈).

3. Generalised Interval-Valued Fuzzy Soft Set

In this section, we generalise the concept of interval-valued fuzzy soft sets as introduced in [13]. In our generalisation of interval-valued fuzzy soft set, a degree is attached with the parameterization of fuzzy sets while defining an interval-valued fuzzy soft set.

Definition 3.1. Let 𝑈={𝑥1,𝑥2,,𝑥𝑛} be the universal set of elements and 𝐸={𝑒1,𝑒2,,𝑒𝑚} the universal set of parameters. The pair (𝑈,𝐸) will be called a soft universe. Let 𝐹𝐸𝑃(𝑈) and 𝜇 be a fuzzy set of 𝐸, that is,𝜇𝐸𝐼=[0,1], where 𝑃(𝑈) is the set of all interval-valued fuzzy subsets on 𝑈. Let 𝐹𝜇𝐸𝑃(𝑈)×𝐼 be a function defined as follows: 𝐹𝜇(𝑒)=𝐹(𝑒),𝜇(𝑒).(3.1) Then, 𝐹𝜇 is called a generalised interval-valued fuzzy soft set (GIVFSS in short) over the soft universe (𝑈,𝐸). For each parameter 𝑒𝑖, 𝐹𝜇(𝑒𝑖)=(𝐹(𝑒𝑖)(𝑥),𝜇(𝑒𝑖)) indicates not only the degree of belongingness of the elements of 𝑈 in 𝐹(𝑒𝑖) but also the degree of possibility of such belongingness which is represented by 𝜇(𝑒𝑖). So we can write 𝐹𝜇(𝑒𝑖) as follows: 𝐹𝜇𝑒𝑖=𝑥1𝐹𝑒𝑖𝑥1,𝑥2𝐹𝑒𝑖𝑥2𝑥,,𝑛𝐹𝑒𝑖𝑥𝑛𝑒,𝜇𝑖.(3.2)

Example 3.2. Let 𝑈={𝑥1,𝑥2,𝑥3} be a set of universe, 𝐸={𝑒1,𝑒2,𝑒3} a set of parameters, and let 𝜇𝐸𝐼. Define a function 𝐹𝜇𝐸𝑃(𝑈)×𝐼 as follows: 𝐹𝜇𝑒1=𝑥1[],𝑥0.3,0.62[],𝑥0.7,0.83[],𝐹0.5,0.8,0.6𝜇𝑒2=𝑥1[],𝑥0.1,0.42[],𝑥0,0.33[],𝐹0.1,0.5,0.5𝜇𝑒3=𝑥1[],𝑥0.7,0.82[],𝑥0.1,0.23[].0,0.4,0.3(3.3) Then, 𝐹𝜇 is a GIVFSS over (𝑈,𝐸).
In matrix notation, we write𝐹𝜇=[][][]0.3,0.6][0.7,0.8][0.5,0.80.60.1,0.4][0,0.3][0.1,0.50.50.7,0.8][0.1,0.2][0,0.40.3.(3.4)

Definition 3.3. Let 𝐹𝜇 and 𝐺𝛿 be two GIVFSSs over (𝑈,𝐸). 𝐹𝜇 is called a generalised interval-valued fuzzy soft subset of 𝐺𝛿, and we write 𝐹𝜇𝐺𝛿 if(a)𝜇(𝑒) is a fuzzy subset of 𝛿(𝑒) for all 𝑒𝐸,(b)𝐹(𝑒) is an interval-valued fuzzy subset of 𝐺(𝑒) for all 𝑒𝐸.

Example 3.4. Let 𝑈={𝑥1,𝑥2,𝑥3} be a set of three cars, and let 𝐸={𝑒1,𝑒2,𝑒3} be a set of parameters where 𝑒1 = cheap, 𝑒2 = expensive, 𝑒3 = red. Let 𝐹𝜇 be a GIVFSS over (𝑈,𝐸) defined as follows: 𝐹𝜇𝑒1=𝑥1[],𝑥0.1,0.32[],𝑥0.5,0.73[],𝐹0.3,0.5,0.4𝜇𝑒2=𝑥1[],𝑥0,0.32[],𝑥0,0.23[],𝐹0.1,0.3,0.4𝜇𝑒3=𝑥1[],𝑥0.5,0.62[],𝑥0.1,0.13[].0.1,0.3,0.1(3.5) Let 𝐺𝛿 be another GIVFSS over (𝑈,𝐸) defined as follows: 𝐺𝛿𝑒1=𝑥1[],𝑥0.3,0.62[],𝑥0.7,0.83[],𝐺0.5,0.8,0.6𝛿𝑒2=𝑥1[],𝑥0.2,0.42[],𝑥0.2,0.33[],𝐺0.3,0.5,0.5𝛿𝑒3=𝑥1[],𝑥0.7,0.82[],𝑥0.2,0.43[].0.2,0.5,0.3(3.6) It is clear that 𝐹𝜇 is a GIVFS subset of 𝐺𝛿.

Definition 3.5. Two GIVFSSs 𝐹𝜇 and 𝐺𝛿 over (𝑈,𝐸) are said to be equal, and we write 𝐹𝜇=𝐺𝛿 if 𝐹𝜇 is a GIVFS subset of 𝐺𝛿 and 𝐺𝛿 is a GIVFS subset of 𝐹𝜇. In other words, 𝐹𝜇=𝐺𝛿 if the following conditions are satisfied: (a)𝜇(𝑒) is equal to 𝛿(𝑒) for all 𝑒𝐸,(b)𝐹(𝑒) is equal to 𝐺(𝑒) for all 𝑒𝐸.

Definition 3.6. A GIVFSS is called a generalised null interval-valued fuzzy soft set, denoted by 𝜇 if 𝜇𝐸𝑃(𝑈)×𝐼 such that 𝜇(𝑒)=𝐹(𝑒)(𝑥),𝜇(𝑒),(3.7) where 𝐹(𝑒)=[0,0]=[0] and 𝜇(𝑒)=0 for all 𝑒𝐸.

Definition 3.7. A GIVFSS is called a generalised absolute interval-valued fuzzy soft set, denoted by 𝐴𝜇 if 𝐴𝜇𝐸𝑃(𝑈)×𝐼 such that 𝐴𝜇(𝑒)=𝐹(𝑒)(𝑥),𝜇(𝑒),(3.8) where 𝐹(𝑒)=[1,1]=[1], and 𝜇(𝑒)=1 for all 𝑒𝐸.

4. Basic Operations on GIVFSS

In this section, we introduce some basic operations on GIVFSS, namely, complement, union and intersection and we give some properties related to these operations.

Definition 4.1. Let 𝐹𝜇 be a GIVFSS over (𝑈,𝐸). Then, the complement of 𝐹𝜇, denoted by 𝐹𝑐𝜇 and is defined by 𝐹𝑐𝜇=𝐺𝛿, such that 𝛿(𝑒)=𝑐(𝜇(𝑒)) and 𝐺(𝑒)=̃𝑐(𝐹(𝑒)) for all 𝑒𝐸, where 𝑐 is a fuzzy complement and ̃𝑐 is an interval-valued fuzzy complement.

Example 4.2. Consider a GIVFSS 𝐹𝜇 over (𝑈,𝐸) as in Example 3.2: 𝐹𝜇=[][][]0.3,0.6][0.7,0.8][0.5,0.80.60.1,0.4][0,0.3][0.1,0.50.50.7,0.8][0.1,0.2][0,0.40.3.(4.1) By using the basic fuzzy complement for 𝜇(𝑒) and interval-valued fuzzy complement for 𝐹(𝑒), we have 𝐹𝑐𝜇=𝐺𝛿 where 𝐺𝛿=[][][]0.4,0.7][0.2,0.3][0.2,0.50.40.6,0.9][0.7,1][0.5,0.90.50.2,0.3][0.8,0.9][0.6,10.7.(4.2)

Proposition 4.3. Let 𝐹𝜇 be a GIVFSS over (𝑈,𝐸). Then, the following holds: 𝐹𝑐𝜇𝑐=𝐹𝜇.(4.3)

Proof. Since 𝐹𝑐𝜇=𝐺𝛿, then 𝐹𝑐𝜇𝑐=𝐺𝑐𝛿(4.4) but, from Definition 4.1,  𝐺𝛿=(̃𝑐(𝐹(𝑒)),𝑐(𝜇(𝑒))), then 𝐺𝑐𝛿===𝐹̃𝑐̃𝑐𝐹(𝑒),𝑐(𝑐(𝜇(𝑒)))𝐹(𝑒),𝜇(𝑒)𝜇.(4.5)

Definition 4.4. Union of two GIVFSSs (𝐹𝜇,𝐴) and (𝐺𝛿,𝐵), denoted by 𝐹𝜇𝐺𝛿, is a GIVFSS (𝐻𝜈,𝐶) where 𝐶=𝐴𝐵 and 𝐻𝜈𝐸𝑃(𝑈)×𝐼 is defined by 𝐻𝜈(𝑒)=𝐻(𝑒),𝜈(𝑒)(4.6) such that 𝐻(𝑒)=𝐹(𝑒)𝐺(𝑒) and 𝜈(𝑒)=𝑠(𝜇(𝑒),𝛿(𝑒)), where 𝑠 is an 𝑠-norm and 𝐻(𝑒)=[sup(𝜇𝐹(𝑒),𝜇𝐺(𝑒)),sup(𝜇+𝐹(𝑒),𝜇+𝐺(𝑒))].

Example 4.5. Consider GIVFSS 𝐹𝜇 and 𝐺𝛿 as in Example 3.4. By using the interval-valued fuzzy union and basic fuzzy union, we have 𝐹𝜇𝐺𝛿=𝐻𝑣, where 𝐻𝑣𝑒1=𝑥1,𝑥sup(0.1,0.3),sup(0.3,0.6)2,𝑥sup(0.5,0.7),sup(0.7,0.8)3=𝑥sup(0.3,0.5),sup(0.5,0.8),max(0.4,0.6)1[],𝑥0.3,0.62[],𝑥0.7,0.83[].0.5,0.8,0.6(4.7) Similarly, we get 𝐻𝑣𝑒2=𝑥1[],𝑥0.2,0.42[],𝑥0.2,0.33[],𝐻0.3,0.5,0.5𝑣𝑒3=𝑥1[],𝑥0.7,0.82[],𝑥0.2,0.43[].0.2,0.5,0.3(4.8) In matrix notation, we write 𝐻𝑣[][][](𝑒)=0.3,0.6][0.7,0.8][0.5,0.80.60.2,0.4][0.2,0.3][0.3,0.50.50.7,0.8][0.2,0.4][0.2,0.50.3.(4.9)

Proposition 4.6. Let 𝐹𝜇, 𝐺𝛿, and 𝐻𝑣 be any three GIVFSSs. Then, the following results hold.(a)𝐹𝜇𝐺𝛿=𝐺𝛿𝐹𝜇.(b)𝐹𝜇𝐺(𝛿𝐻𝑣𝐹)=(𝜇𝐺𝛿𝐻)𝑣. (c)𝐹𝜇𝐹𝜇𝐹𝜇. (d)𝐹𝜇𝐴𝜇=𝐴𝜇. (e)𝐹𝜇𝜇=𝐹𝜇.

Proof. (a) 𝐹𝜇𝐺𝛿=𝐻𝜈.
From Definition 4.4, we have 𝐻𝜈(𝑒)=(𝐻(𝑒)(𝑥),𝜈(𝑒)) such that 𝐻(𝑒)=𝐹(𝑒)𝐺(𝑒) and 𝜈(𝑒)=𝑠(𝜇(𝑒),𝛿(𝑒)).
But 𝐻(𝑒)=𝐹(𝑒)𝐺(𝑒)=𝐺(𝑒)𝐹(𝑒) (since union of interval-valued fuzzy sets is commutative) and 𝜈(𝑒)=𝑠(𝜇(𝑒),𝛿(𝑒))=𝑠(𝛿(𝑒),𝜇(𝑒)) (since 𝑠-norm is commutative), then 𝐺𝛿𝐹𝜇=𝐻𝜈.
(b) The proof is straightforward from Definition 4.4.
(c) The proof is straightforward from Definition 4.4.
(d) The proof is straightforward from Definition 4.4.
(e) The proof is straightforward from Definition 4.4.

Definition 4.7. Intersection of two GIVFSSs (𝐹𝜇,𝐴) and (𝐺𝛿,𝐵), denoted by 𝐹𝜇𝐺𝛿, is a GIVFSS (𝐻𝜈,𝐶) where 𝐶=𝐴𝐵 and 𝐻𝜈𝐸𝑃(𝑈)×𝐼 is defined by 𝐻𝜈(𝑒)=𝐻(𝑒),𝜈(𝑒)(4.10) such that 𝐻(𝑒)=𝐹(𝑒)𝐺(𝑒) and 𝜈(𝑒)=𝑡(𝜇(𝑒),𝛿(𝑒)), where 𝑡 is a 𝑡-norm and 𝐻(𝑒)=[inf(𝜇𝐹(𝑒),𝜇𝐺(𝑒)),inf(𝜇+𝐹(𝑒),𝜇+𝐺(𝑒))].

Example 4.8. Consider GIVFSS 𝐹𝜇 and 𝐺𝛿 as in Example 4.5. By using the interval-valued fuzzy intersection and basic fuzzy intersection, we have 𝐹𝜇𝐺𝛿=𝐻𝑣, where 𝐻𝑣𝑒1=𝑥1[],𝑥inf(0.1,0.3),inf(0.3,0.6)2[],𝑥inf(0.5,0.7),inf(0.7,0.8)3[]=𝑥inf(0.3,0.5),inf(0.5,0.8)min(0.4,0.6)1[],𝑥0.1,0.32[],𝑥0.5,0.73[].0.3,0.5,0.4(4.11) Similarly, we get 𝐻𝑣𝑒2=𝑥1[],𝑥0,0.32[],𝑥0,0.23[],𝐻0.1,0.3,0.4𝑣𝑒3=𝑥1[],𝑥0.5,0.62[],𝑥0.1,0.13[].0.1,0.3,0.1(4.12) In matrix notation, we write 𝐻𝑣[][][](𝑒)=0.1,0.3][0.5,0.7][0.3,0.50.40,0.3][0,0.2][0.1,0.30.40.5,0.6][0.1,0.1][0.1,0.30.1.(4.13)

Proposition 4.9. Let 𝐹𝜇, 𝐺𝛿, and 𝐻𝑣 be any three GIVFSSs. Then, the following results hold.(a)𝐹𝜇𝐺𝛿=𝐺𝛿𝐹𝜇.(b)𝐹𝜇𝐺(𝛿𝐻𝑣𝐹)=(𝜇𝐺𝛿𝐻)𝑣. (c)𝐹𝜇𝐹𝜇𝐹𝜇. (d)𝐹𝜇𝐴𝜇=𝐹𝜇. (e)𝐹𝜇𝜇=𝜇.

Proof. (a) 𝐹𝜇𝐺𝛿=𝐻𝜈.
From Definition 4.7, we have 𝐻𝜈(𝑒)=(𝐻(𝑒)(𝑥),𝜈(𝑒)) such that 𝐻(𝑒)=𝐹(𝑒)𝐺(𝑒) and 𝜈(𝑒)=𝑡(𝜇(𝑒),𝛿(𝑒)).
But 𝐻(𝑒)=𝐹(𝑒)𝐺(𝑒)=𝐺(𝑒)𝐹(𝑒) (since intersection of interval-valued fuzzy sets is commutative) and 𝜈(𝑒)=𝑡(𝜇(𝑒),𝛿(𝑒))=𝑡(𝛿(𝑒),𝜇(𝑒)) (since 𝑡-norm is commutative), then 𝐺𝛿𝐹𝜇=𝐻𝜈.
(b) The proof is straightforward from Definition 4.7.
(c) The proof is straightforward from Definition 4.7.
(d) The proof is straightforward from Definition 4.7.
(e) The proof is straightforward from Definition 4.7.

Proposition 4.10. Let 𝐹𝜇 and 𝐺𝛿 be any two GIVFSSs. Then the DeMorgan’s Laws hold:(a)(𝐹𝜇𝐺𝛿)𝑐=𝐺𝑐𝛿𝐹𝑐𝜇.(b)(𝐹𝜇𝐺𝛿)𝑐=𝐺𝑐𝛿𝐹𝑐𝜇.

Proof. (a) Consider 𝐹𝑐𝜇𝐺𝑐𝛿==((̃𝑐(𝐹(𝑒)),𝑐(𝜇(𝑒)))(̃𝑐(𝐺(𝑒)),𝑐(𝛿(𝑒))))=((̃𝑐(𝐹(𝑒))̃𝑐(𝐺(𝑒))),(𝑐(𝜇(𝑒))𝑐(𝛿(𝑒))))((𝐹(𝑒))(𝐺(𝑒)))̃𝑐,(𝜇(𝑒)𝛿(𝑒))𝑐=𝐹𝜇𝐺𝛿𝑐.(4.14)
(b) The proof is similar to the above.

Proposition 4.11. Let 𝐹𝜇, 𝐺𝛿, and 𝐻𝑣 be any three GIVFSSs. Then, the following results hold. (a)𝐹𝜇𝐺(𝛿𝐻𝑣𝐹)=(𝜇𝐺𝛿𝐹)(𝜇𝐻𝑣). (b)𝐹𝜇𝐺(𝛿𝐻𝑣𝐹)=(𝜇𝐺𝛿𝐹)(𝜇𝐻𝑣).

Proof. (a) For all 𝑥𝐸, 𝜆𝐹(𝑥)(𝐺(𝑥)𝐻(𝑥))𝜆(𝑥)=sup𝐹(𝑥)(𝑥),𝜆𝐺(𝑥)𝐻(𝑥)𝜆(𝑥),sup+𝐹(𝑥)(𝑥),𝜆+𝐺(𝑥)𝐻(𝑥)=𝜆(𝑥)sup𝐹(𝑥)𝜆(𝑥),inf𝐺(𝑥)(𝑥),𝜆𝐻(𝑥),𝜆(𝑥)sup+𝐹(𝑥)𝜆(𝑥),inf+𝐺(𝑥)(𝑥),𝜆+𝐻(𝑥)=𝜆(𝑥)infsup𝐹(𝑥)(𝑥),𝜆𝐺(𝑥)(𝜆𝑥),sup𝐹(𝑥)(𝑥),𝜆𝐻(𝑥)(,𝜆𝑥)infsup+𝐹(𝑥)(𝑥),𝜆+𝐺(𝑥)𝜆(𝑥),sup+𝐹(𝑥)(𝑥),𝜆+𝐻(𝑥)(𝑥)=𝜆(𝐹(𝑥)𝐺(𝑥))(𝐹(𝑥)𝐻(𝑥))(𝛾𝑥),𝜇(𝑥)(𝛿(𝑥)𝜈(𝑥))𝛾(𝑥)=max𝜇(𝑥)(𝑥),𝛾𝛿(𝑥)𝜈(𝑥)𝛾(𝑥)=max𝜇(𝑥)𝛾(𝑥),min𝛿(𝑥)(𝑥),𝛾𝜈(𝑥)𝛾(𝑥)=minmax𝜇(𝑥)(𝑥),𝛾𝛿(𝑥)𝛾(𝑥),max𝜇(𝑥)(𝑥),𝛾𝜈(𝑥)𝛾(𝑥)=min𝜇(𝑥)𝛿(𝑥)(𝑥),𝛾𝜇(𝑥)𝜈(𝑥)(𝑥)=𝛾(𝜇(𝑥)𝛿(𝑥))(𝜇(𝑥)𝜈(𝑥))(𝑥).(4.15)
(b) Similar to the proof of (a).

5. AND and OR Operations on GIVFSS with Application

In this section, we give the definitions of AND and OR operations on GIVFSS. An application of this operations in decision making problem has been shown.

Definition 5.1. If (𝐹𝜇,𝐴) and (𝐺𝛿,𝐵) are two GIVFSSs, then “(𝐹𝜇,𝐴) AND (𝐺𝛿,𝐵)” denoted by (𝐹𝜇𝐺,𝐴)(𝛿,𝐵) is defined by 𝐹𝜇𝐺,𝐴𝛿=𝐻,𝐵𝜆,𝐴×𝐵,(5.1) where 𝐻𝜆(𝛼,𝛽)=(𝐻(𝛼,𝛽),𝜆(𝛼,𝛽)) for all (𝛼,𝛽)𝐴×𝐵, such that 𝐻(𝛼,𝛽)=𝐹(𝛼)𝐺(𝛽) and 𝜆(𝛼,𝛽)=𝑡(𝜇(𝛼),𝛿(𝛽)), for all (𝛼,𝛽)𝐴×𝐵, where 𝑡 is a 𝑡-norm.

Example 5.2. Suppose the universe consists of three machines 𝑥1,𝑥2,𝑥3, that is,𝑈={𝑥1,𝑥2,𝑥3}, and consider the set of parameters 𝐸={𝑒1,𝑒2,𝑒3} which describe their performances according to certain specific task. Suppose a firm wants to buy one such machine depending on any two of the parameters only. Let there be two observations 𝐹𝜇 and 𝐺𝛿 by two experts 𝐴 and 𝐵, respectively, defined as follows: 𝐹𝜇𝑒1=𝑥1[],𝑥0.1,0.32[],𝑥0.5,0.73[],𝐹0.3,0.5,0.4𝜇𝑒2=𝑥1[],𝑥0,0.32[],𝑥0,0.23[],𝐹0.1,0.3,0.4𝜇𝑒3=𝑥1[],𝑥0.5,0.62[],𝑥0.1,0.13[],𝐺0.1,0.3,0.1𝜇𝑒1=𝑥1[],𝑥0.3,0.52[],𝑥0.2,0.63[],𝐺0.4,0.5,0.3𝜇𝑒2=𝑥1[],𝑥0.3,0.52[],𝑥0.4,0.63[],𝐺0,0.3,0.1𝜇𝑒3=𝑥1[],𝑥0.1,0.62[],𝑥0.4,0.73[].0.2,0.3,0.2(5.2) To find the AND between the two GIVFSSs, we have (𝐹𝜇,𝐴) AND (𝐺𝛿𝐻,𝐵)=(𝜆,𝐴×𝐵), where 𝐻𝜆𝑒1,𝑒1=𝑥1[],𝑥0.1,0.32[],𝑥0.2,0.63[],𝐻0.3,0.5,0.3𝜆𝑒1,𝑒2=𝑥1[],𝑥0.1,0.32[],𝑥0.4,0.63[],𝐻0,0.3,0.1𝜆𝑒1,𝑒3=𝑥1[],𝑥0.1,0.32[],𝑥0.4,0.73[],𝐻0.2,0.3,0.2𝜆𝑒2,𝑒1=𝑥1[],𝑥0,0.32[],𝑥0,0.23[],𝐻0.1,0.3,0.3𝜆𝑒2,𝑒2=𝑥1[],𝑥0,0.32[],𝑥0,0.23[],𝐻0,0.3,0.1𝜆𝑒2,𝑒3=𝑥1[],𝑥0,0.32[],𝑥0,0.23[],𝐻0.1,0.3,0.2𝜆𝑒3,𝑒1=𝑥1[],𝑥0.3,0.52[],𝑥0.1,0.13[],𝐻0.1,0.3,0.1𝜆𝑒3,𝑒2=𝑥1[],𝑥0.3,0.52[],𝑥0.1,0.13[],𝐻0,0.3,0.1𝜆𝑒3,𝑒3=𝑥1[],𝑥0.1,0.62[],𝑥0.1,0.13[].0.1,0.3,0.1(5.3) Now, to determine the best machine, we first compute the numerical grade 𝑟𝑝𝑃(𝑥𝑖) for each 𝑝𝑃 such that 𝑟𝑝𝑃𝑥𝑖=𝑥𝑈𝑐𝑖𝜇𝐻𝑝𝑖+𝑐(𝑥)+𝑖𝜇+𝐻𝑝𝑖(𝑥).(5.4) The result is shown in Tables 1 and 2. Let 𝑃={𝑝1=(𝑒1,𝑒1),𝑝2=(𝑒1,𝑒2),,𝑝9=(𝑒3,𝑒3)}. Now, we mark the highest numerical grade (indicated in parenthesis) in each row excluding the last row which is the grade of such belongingness of a machine against each pair of parameters (see Table 3). Now, the score of each such machine is calculated by taking the sum of the products of these numerical grades with the corresponding value of 𝜇. The machine with the highest score is the desired machine. We do not consider the numerical grades of the machine against the pairs (𝑒𝑖,𝑒𝑖), 𝑖=1,2,3, as both the parameters are the same:Score (𝑥1)=(10.1)+(1.10.1)=0.21, Score (𝑥2)=(1.30.1)+(1.30.2)=0.39,Score (𝑥3)=(0.30.3)+(0.30.2)=0.15.The firm will select the machine with the highest score. Hence, they will buy machine 𝑥2.

tab1
Table 1: (𝐻𝜆,𝐴×𝐵).
tab2
Table 2: Numerical grade 𝑟𝑝𝑃(𝑥𝑖).
tab3
Table 3: Grade table.

Definition 5.3. If (𝐹𝜇,𝐴) and (𝐺𝛿,𝐵) are two GIVFSSs, then “(𝐹𝜇,𝐴) OR (𝐺𝛿,𝐵)” denoted by (𝐹𝜇𝐺,𝐴)(𝛿,𝐵) is defined by 𝐹𝜇𝐺,𝐴𝛿=𝐻,𝐵𝜆,𝐴×𝐵,(5.5) where 𝐻𝜆(𝛼,𝛽)=(𝐻(𝛼,𝛽),𝜆(𝛼,𝛽)) for all (𝛼,𝛽)𝐴×𝐵, such that 𝐻(𝛼,𝛽)=𝐹(𝛼)𝐺(𝛽) and 𝜆(𝛼,𝛽)=𝑠(𝜇(𝛼),𝛿(𝛽)), for all (𝛼,𝛽)𝐴×𝐵, where 𝑠 is an 𝑠-norm.

Example 5.4. Consider Example 5.2. To find the OR between the two GIVFSSs, we have (𝐹𝜇,𝐴) OR (𝐺𝛿𝐻,𝐵)=(𝜆,𝐴×𝐵), where 𝐻𝜆𝑒1,𝑒1=𝑥1[],𝑥0.3,0.52[],𝑥0.5,0.73[],𝐻0.4,0.5,0.4𝜆𝑒1,𝑒2=𝑥1[],𝑥0.3,0.52[],𝑥0.5,0.73[],𝐻0.4,0.5,0.4𝜆𝑒1,𝑒3=𝑥1[],𝑥0.1,0.62[],𝑥0.5,0.73[],𝐻0.3,0.5,0.4𝜆𝑒2,𝑒1=𝑥1[],𝑥0.3,0.52[],𝑥0.,0.63[],𝐻0.4,0.5,0.4𝜆𝑒2,𝑒2=𝑥1[],𝑥0.3,0.52[],𝑥0.4,0.63[],𝐻0.1,0.3,0.4𝜆𝑒2,𝑒3=𝑥1[],𝑥0.1,0.62[],𝑥0.4,0.73[],𝐻0.2,0.3,0.4𝜆𝑒3,𝑒1=𝑥1[],𝑥0.5,0.62[],𝑥0.2,0.63[],𝐻0.4,0.5,0.3𝜆𝑒3,𝑒2=𝑥1[],𝑥0.5,0.62[],𝑥0.4,0.63[],𝐻0.1,0.3,0.1𝜆𝑒3,𝑒3=𝑥1[],𝑥0.5,0.62[],𝑥0.4,0.73[].0.2,0.3,0.2(5.6)

Remark 5.5. We use the same method in Example 5.2 for the OR operation if the firm wants to buy one such machine depending on any one of the parameters only.

Proposition 5.6. Let (𝐹𝜇,𝐴) and (𝐺𝛿,𝐵) be any two GIVFSSs. Then, the following results hold:(a)𝐹((𝜇𝐺,𝐴)(𝛿,𝐵))𝑐𝐹=(𝜇,𝐴)𝑐𝐺(𝛿,𝐵)𝑐, (b)𝐹((𝜇𝐺,𝐴)(𝛿,𝐵))𝑐𝐹=(𝜇,𝐴)𝑐𝐺(𝛿,𝐵)𝑐.

Proof. Straightforward from Definitions 4.1, 5.1, and 5.3.

Proposition 5.7. Let (𝐹𝜇,𝐴), (𝐺𝛿,𝐵), and (𝐻𝜆,𝐶) be any three GIVFSSs. Then, the following results hold:(a)(𝐹𝜇𝐺,𝐴)((𝛿𝐻,𝐵)(𝜆𝐹,𝐶))=((𝜇𝐺,𝐴)(𝛿𝐻,𝐵))(𝜆,𝐶), (b)(𝐹𝜇𝐺,𝐴)((𝛿𝐻,𝐵)(𝜆𝐹,𝐶))=((𝜇𝐺,𝐴)(𝛿𝐻,𝐵))(𝜆,𝐶), (c)(𝐹𝜇𝐺,𝐴)((𝛿𝐻,𝐵)(𝜆𝐹,𝐶))=((𝜇𝐺,𝐴)(𝛿𝐹,𝐵))((𝜇𝐻,𝐴)(𝜆,𝐶)), (d)(𝐹𝜇𝐺,𝐴)((𝛿𝐻,𝐵)(𝜆𝐹,𝐶))=((𝜇𝐺,𝐴)(𝛿𝐹,𝐵))((𝜇𝐻,𝐴)(𝜆,𝐶)).

Proof. Straightforward from Definitions 5.1 and 5.3.

Remark 5.8. The commutativity property does not hold for AND and OR operations since 𝐴×𝐵𝐵×𝐴.

6. Similarity between Two GIVFSS

In this section, we give a measure of similarity between two GIVFSSs. We are taking the set theoretic approach because it is easier to calculate on and is a very popular method too.

Definition 6.1. Similarity between two GIVFSSs 𝐹𝜇 and 𝐺𝛿, denoted by 𝐹𝑆(𝜇,𝐺𝛿), is defined by 𝑆𝐹𝜇,𝐺𝛿=𝜑𝐺𝐹,𝑚(𝜇,𝛿),𝜑+𝐺𝜑𝐹,𝑚(𝜇,𝛿),suchthat𝐺𝜑𝐹,=min1𝐺𝐹,,𝜑2𝐺,𝜑𝐹,+𝐺𝜑𝐹,=max1𝐺𝐹,,𝜑2𝐺,𝐹,(6.1) where 𝜑1𝐺=𝐹,0,if𝜇𝐹𝑖(𝑥)=0,𝑖,𝑛𝑖max𝑥𝑋𝜇min𝐹𝑖(𝑥),𝜇𝐺𝑖(𝑥)𝑛𝑖max𝑥𝑋𝜇𝐹𝑖𝜑(𝑥),otherwise,2𝐺=𝐹,𝑛𝑖max𝑥𝑋𝜇min+𝐹𝑖(𝑥),𝜇+𝐺𝑖(𝑥)𝑛𝑖max𝑥𝑋𝜇+𝐹𝑖,𝑚||||(𝑥)(𝜇(𝑒),𝛿(𝑒))=1𝜇(𝑒)𝛿(𝑒)||||.𝜇(𝑒)+𝛿(𝑒)(6.2)

Definition 6.2. Let 𝐹𝜇 and 𝐺𝛿 be two GIVFSSs over the same universe (𝑈,𝐸). We say that two GIVFSS are significantly similar if 𝜑(𝐹,𝐺)𝑚(𝜇,𝛿)1/2.

Theorem 6.3. Let 𝐹𝜇, 𝐺𝛿, and 𝐻𝜆 be any three GIVFSSs over (𝑈,𝐸). Then, the following hold: (a)in general 𝐹𝑆(𝜇,𝐺𝛿𝐺)𝑆(𝛿,𝐹𝜇), (b)𝜑(𝐹,𝐺)0 and 𝜑+(𝐹,𝐺)1, (c)𝐹𝜇=𝐺𝛿𝐹𝑆(𝜇,𝐺𝛿)=1, (d)𝐹𝜇𝐺𝛿𝐻𝜆𝐹𝑆(𝜇,𝐻𝜆𝐺)𝑆(𝛿,𝐻𝜆), (e)𝐹𝜇𝐺𝛿𝐹=𝑆(𝜇,𝐺𝛿)=0.

Proof. (a) The proof is straightforward and follows from Definition 6.1.
(b) From Definition 6.1, we have𝜑1𝐺=𝐹,0,if𝜇𝐹𝑖(𝑥)=0,𝑖,𝑛𝑖max𝑥𝑋𝜇min𝐹𝑖(𝑥),𝜇𝐺𝑖(𝑥)𝑛𝑖max𝑥𝑋𝜇𝐹𝑖(𝑥),otherwise.(6.3)
If 𝜇𝐹𝑖(𝑥)=0, for all 𝑖, then 𝜑(𝐹,𝐺)=0, and, if 𝜇𝐹𝑖(𝑥)0, for some 𝑖, then it is clear that 𝜑(𝐹,𝐺)0.
Also since 𝜑+(𝐹,𝐺)=max(𝜑1(𝐹,𝐺),𝜑2(𝐹,𝐺)), suppose that 𝜑1(𝐹,𝐺)=1 and 𝜑2(𝐹,𝐺)=1, then 𝜑+(𝐹,𝐺)=1, that means, if 𝜑1(𝐹,𝐺)<1 and 𝜑2(𝐹,𝐺)<1, then 𝜑+(𝐹,𝐺)1.
(c) The proof is straightforward and follows from Definition 6.1.
(d) The proof is straightforward and follows from Definition 6.1.
(e) The proof is straightforward and follows from Definition 6.1.

Example 6.4. Let 𝐹𝜇 be GIVFSS over (𝑈,𝐸) defined as follows: 𝐹𝜇𝑒1=𝑥1[],𝑥0.3,0.72[],𝑥0.4,0.83[],𝐹0.1,0.3,0.4𝜇𝑒2=𝑥1[],𝑥0.5,0.62[],𝑥0.1,0.33[],𝐹0,0.4,0.6𝜇𝑒3=𝑥1[],𝑥0.7,0.92[],𝑥0.1,0.53[].0.8,1,0.8(6.4) Let 𝐺𝛿 be another GIVFSS over (𝑈,𝐸) defined as follows: 𝐺𝛿𝑒1=𝑥1[],𝑥0.1,0.42[],𝑥0.5,0.73[],𝐺0.2,0.3,0.3𝛿𝑒2=𝑥1[],𝑥0.6,0.82[],𝑥0.5,0.63[],𝐺0.4,0.8,0.7𝛿𝑒3=𝑥1[],𝑥0.4,0.72[],𝑥0.3,0.53[].0.5,0.7,0.6(6.5) Here, ||||𝑚(𝜇(𝑒),𝛿(𝑒))=1𝜇(𝑒)𝛿(𝑒)||||||||+||||+||||𝜇(𝑒)+𝛿(𝑒)=1(0.40.3)(0.60.7)(0.80.6)||||+||||+||||𝜑(0.4+0.3)(0.6+0.7)(0.80+0.6)0.8824,1𝐺=(𝐹,max{min(0.3,0.1),min(0.4,0.5),min(0.1,0.2)}+max(0.3,0.4,0.1)+max(0.5,0.1,0)+max(0.7,0.1,0.8)max{min(0.5,0.6),min(0.1,0.5),min(0,0.4)}+max(0.3,0.4,0.1)+max(0.5,0.1,0)+max(0.7,0.1,0.8)max{min(0.7,0.4),min(0.1,0.3),min(0.8,0.5)})=max(0.3,0.4,0.1)+max(0.5,0.1,0)+max(0.7,0.1,0.8)max{0.1,0.4,0.1}+max{0.5,0.1,0}+max{0.4,0.1,0.5}=max(0.3,0.4,0.1)+max(0.5,0.1,0)+max(0.7,0.1,0.8)0.4+0.5+0.5𝜑0.4+0.5+0.8=0.824,2𝐺=𝐹,(max{min(0.7,0.4),min(0.8,0.7),min(0.3,0.3)}+max(0.7,0.8,0.3)+max(0.6,0.3,0.4)+max(0.9,0.5,1)max{min(0.6,0.8),min(0.3,0.6),min(0.4,0.8)}+max(0.7,0.8,0.3)+max(0.6,0.3,0.4)+max(0.9,0.5,1)max{min(0.9,0.7),min(0.5,0.5),min(1,0.7)})=max(0.7,0.8,0.3)+max(0.6,0.3,0.4)+max(0.9,0.5,1)max{0.4,0.7,0.3}+max{0.6,0.3,0.4}+max{0.7,0.5,0.7}=max(0.7,0.8,0.3)+max(0.6,0.3,0.4)+max(0.9,0.5,1)0.7+0.6+0.70.8+0.6+1=0.833.(6.6) Then, 𝜑𝐺𝜑𝐹,=min1𝐺𝐹,,𝜑2𝐺𝜑𝐹,=min(0.824,0.833)=0.824,+𝐺𝜑𝐹,=max1𝐺𝐹,,𝜑2𝐺𝐹,=max(0.824,0.833)=0.833.(6.7) Hence, the similarity between the two GIVFSSs 𝐹𝜇 and 𝐺𝛿 will be 𝑆𝐹𝜇,𝐺𝛿=𝜑𝐺𝐹,𝑚(𝜇,𝛿),𝜑+𝐺=[]=[].𝐹,𝑚(𝜇,𝛿)(0.824)(0.8824),(0.833)(0.8824)0.727,0.735(6.8) Therefore, 𝐹𝜇 and 𝐺𝛿 are significantly similar.

7. Application of Similarity Measure in Medical Diagnosis

In this section, we will try to estimate the possibility that a sick person having certain visible symptoms is suffering from dengue fever. For this, we first construct a GIVFSS model for dengue fever and the GIVFSS of symptoms for the sick person. Next, we find the similarity measure of these two sets. If they are significantly similar, then we conclude that the person is possibly suffering from dengue fever. Let our universal set contain only two elements “yes” and “no,” that is, 𝑈={𝑦,𝑛}. Here, the set of parameters 𝐸 is the set of certain visible symptoms. Let 𝐸={𝑒1,𝑒2,𝑒3,𝑒4,𝑒5,𝑒6,𝑒7}, where 𝑒1 = body temperature, 𝑒2 = cough with chest congestion, 𝑒3 = loose motion, 𝑒4 = chills, 𝑒5 = headache, 𝑒6 = low heart rate (bradycardia), and 𝑒7 = pain upon moving the eyes. Our model GIVFSS for dengue fever 𝑀𝜇 is given in Table 4, and this can be prepared with the help of a physician.

tab4
Table 4: Model GIVFSS for dengue fever.

Now, after talking to the sick person, we can construct his GIVFSS 𝐺𝛿 as in Table 5.

tab5
Table 5: GIVFSS for the sick person.

Now, we find the similarity measure of these two sets by using the same method as in Example 6.4, where, after the calculation, we get 𝜑(𝑀,𝐺)𝑚(𝜇,𝛿)0.22<1/2. Hence the two GIVFSSs are not significantly similar. Therefore, we conclude that the person is not suffering from dengue fever.

8. Conclusion

In this paper, we have introduced the concept of generalised interval-valued fuzzy soft set and studied some of its properties. The complement, union, intersection, “AND,” and “OR” operations have been defined on the interval-valued fuzzy soft sets. An application of this theory is given in solving a decision making problem. Similarity measure of two generalised interval-valued fuzzy soft sets is discussed, and its application to medical diagnosis has been shown.

Acknowledgment

The authors would like to acknowledge the financial support received from Universiti Kebangsaan Malaysia under the Research Grants UKM-ST-06-FRGS0104-2009 and UKM-DLP-2011-038.

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