Mathematical Problems in Engineering
Volume 2016 (2016), Article ID 5646927, 3 pages
http://dx.doi.org/10.1155/2016/5646927

Corrigendum

Corrigendum to “On Interval-Valued Hesitant Fuzzy Soft Sets”

Ahmed Mostafa Khalil,¹ Haidong Zhang,² Lianglin Xiong,³ and Weiyuan Ma²

¹Department of Mathematics, Faculty of Science, Al-Azhar University, Assiut 71524, Egypt
²School of Mathematics and Computer Science, Northwest University for Nationalities, Lanzhou, Gansu 730030, China
³School of Mathematics and Computer Science, Yunnan Minzu University, Kunming, Yunnan 650500, China

Received 28 September 2015; Accepted 16 December 2015

Copyright © 2016 Ahmed Mostafa Khalil et al. 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.

1. Introduction

We point out that assertions and of Theorem in the paper titled “On Interval-Valued Hesitant Fuzzy Soft Sets” [1] are not true in general. We verify that the corresponding assertions in Zhang et al. [1] are incorrect by a counterexample. Finally, we introduce reasonable definitions to improve the results.

The soft set theory, proposed by Molodtsov [2], can be used as a general mathematical tool for dealing with uncertainty. Maji et al. [3] presented the concept of fuzzy soft set which is based on a combination of the fuzzy set and soft set models. Later on, Maji et al. [4] defined some operations on soft sets and showed that the distributive law of soft sets is varied. Ali et al. [5] pointed out that the distributive law of soft sets is not true in general. This implies that the distributive law of fuzzy soft sets is not true. In this paper, we show that assertions and of Theorem  36 proposed by Zhang et al. [1] are incorrect by a counterexample and we recorrect Theorem  36 and using the generalized distributive law of interval-valued hesitant fuzzy soft sets.

For the general terminologies in this paper, please refer to [1–4, 6, 7].

Definition 1. Let ; then is called an interval number. In particular, is a real number, if .

Definition 2. Let , and let , and and ; then the degree of possibility of is defined as

Similarly, the degree of possibility of is defined as

Definition 3. Suppose that is an initial universe set, is a set of parameters, is the power set of , and . A pair is called soft set over , where is a mapping given by .

Definition 4. A pair is called a fuzzy soft set over , if and , where is the set of all fuzzy subsets of .

Definition 5. Let be a fixed set, and let be the set of all closed subintervals of . An interval-valued hesitant fuzzy set (IVHFS, for short) on is defined as

where denotes all possible interval-valued membership degrees of the element to .
For convenience, we call an interval-valued hesitant fuzzy element (IVHFE, for short). The set of all interval-valued hesitant fuzzy sets on is denoted by . We can note that an IVHFS can be seen as an interval-valued fuzzy set if there is only one element in , which indicates that interval-valued fuzzy sets are a special type of IVHFSs.

Definition 6. Let and be two IVHFSs; then one has the following:

,.

Definition 7. Let be a soft universe and . A pair is called an interval-valued hesitant fuzzy soft set over , where is a mapping given by .
An interval-valued hesitant fuzzy soft set is a parameterized family of interval-valued hesitant fuzzy subsets of . That is to say, is an interval-valued hesitant fuzzy subset in , . Following the standard notations, can be written as

Sometimes we write as . If , we can also have an interval-valued hesitant fuzzy soft set .

Definition 8. Let be an initial universe and let be a set of parameters. Supposing that and and are two interval-valued hesitant fuzzy soft sets, one says that is an interval-valued hesitant fuzzy soft subset of if and only if

(1),(2),

where, for all , , and stand for the th largest interval number in the IVHFEs and , respectively.
In this case, we write . is said to be an interval-valued hesitant fuzzy soft super set of if is an interval-valued hesitant fuzzy soft subset of . We denote it by .

Definition 9. Let and be two interval-valued hesitant fuzzy soft sets. Now and are said to be interval-valued hesitant fuzzy soft equal if and only if

(1),(2),

which can be denoted by .

Definition 10. Let and be two interval-valued hesitant fuzzy soft sets over . The “ AND ,” denoted by , is defined by

where, for all ,

Definition 11. Let and be two interval-valued hesitant fuzzy soft sets over . The “ OR ,” denoted by , is defined by

where, for all ,

2. Counterexample

We begin this section with Theorem 12 below, originally proposed as Theorem  36 in Zhang et al. [1] and provide a counterexample to show that assertions and are not true.

Theorem 12 (see [1]). Let , , and be three interval-valued hesitant fuzzy soft sets over . Then one has the following:

(1),(2),(3),(4).

The following example shows that assertions and of Theorem 12 above are not true in general.

Example 13. Let be a set of two houses and let be a set of parameters, which stand for expensive, beautiful, cheap, size, location, and in the green surroundings, respectively. Consider , , and to be subsets of , where , , and . Suppose that , , and are three interval-valued hesitant fuzzy soft sets defined by

By Definitions 10 and 11, the interval-valued hesitant fuzzy soft set has the parameter set and interval-valued hesitant fuzzy soft set has a set of parameters as . But we can not find any notion which ensures . Hence Theorem 12 above is not true.

3. Main Results

Definition 14. Let be an initial universe and let be a set of parameters. For subsets and of , let and be interval-valued hesitant fuzzy soft sets. One says that is a generalized interval-valued hesitant fuzzy soft subset of , denoted by , if, for every , there exists such that , where, for all , , , and stand for the th largest interval number in the IVHFEs and , respectively.

Example 15. Let be an initial universe and let be a set of parameters. With and , let and be interval-valued hesitant fuzzy soft sets defined by

Then .

Definition 16. Let and be interval-valued hesitant fuzzy soft sets. One says that and are generalized interval-valued hesitant fuzzy soft set equal, denoted by , if and .

Theorem 17 is the corrected version of assertions and of Theorem 12, originally written as Theorem  36 of Zhang et al. [1].

Theorem 17. Let , , and be three interval-valued hesitant fuzzy soft sets over . Then one has the following:

,.

Proof. For any , , and , we have . Hence conclusion is valid. Similarly, we can prove assertion .

4. Conclusions

Zhang et al. [1] introduced interval-valued hesitant fuzzy soft set based on interval-valued hesitant fuzzy set and proposed several theorems and some operations on an interval-valued hesitant fuzzy soft set. However, we pointed out that assertion and of Theorem  36 [1] are not true. Using the notions of a generalized interval-valued hesitant fuzzy soft subset and fuzzy soft equal, Theorem  36 and of [1] is proposed and proven to be true.

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