The aim of this paper is to correct the assertions (3) and (4) of Theorem 3.23 proposed by Zhang and Shu [Journal of Intelligent and Fuzzy Systems 27 (2014) 2115-2125]. A counterexample illustrates the flaw of the assertions. We introduce the notion of generalized possibility multi-fuzzy soft sets and use it to correct the flaw in those assertions. Finally we introduce two new definitions and five theorems to improve the work of Zhang and Shu, along with two examples.
The soft set theory, proposed by Molodtsov [1], can be used as a general mathematical tool for dealing with uncertainty. Maji et al. [2] 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. [3] defined some operations on soft sets and showed that the distributive law of soft sets is varied. Ali et al. [4] 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 (3) and (4) of Theorem 3.23 proposed by Zhang and Shu [5] are incorrect by a counterexample and correct the Theorem 3.23 (3) and (4) using the generalized distributive law of generalized possibility multi-fuzzy soft sets.
Definition 1.1. (see [6]) Let U be an initial universe set, k be a positive integer and F (U) be the set of all fuzzy subsets of U . A multi-fuzzy set in U is a set of ordered sequences where μi ∈ F (U), i = 1, 2, . . . , k .
The function is called the multi-membership function of multi-fuzzy set k is called the dimension of . The set of all multi-fuzzy sets of dimension k in U is denoted by MkFS (U) .
Definition 1.2. (see [6]) Let and be two multi-fuzzy sets of dimension k in U. The following relations and operations
Definition 1.3. (see [1]) Suppose that U is an initial universe set, E be a set of parameters, P (U) be the power set of U and A ⊂ E . A pair (F, A) is called soft set over U, where F is a mapping given by F : A → P (U) .
Definition 1.4. (see [2]) A pair is called a fuzzy soft set over U, if A ⊆ E and
Definition 1.5. (see [7]) A pair is called a multi-fuzzy soft set of dimension k over U, where is a mapping given by
Definition 1.6. (see [5]) Let U be an initial universe and let E be a set of parameters. The pair (U, E) is called a soft universe. Suppose that and is a multi-fuzzy subset of E ; that is, We say that is is a possibility multi-fuzzy soft set of dimension k (PMkFSS, in short) over the soft universe (U, E) if and only if is a mapping given by where ∀ u ∈ U .
Definition 1.7. (see [5]) Let and be two PMkFSSs over (U, E). Now is said to be a possibility multi-fuzzy soft subset of if and only if
is a multi-fuzzy subset of
is also a multi-fuzzy subset of e ∈ E .
In this case, we write
Definition 1.8. (see [5]) Let and be two PMkFSSs over (U, E). Now is said to be a possibility multi-fuzzy soft equal of if and only if
is a possibility multi-fuzzy soft subset of
is a possibility multi-fuzzy soft subset of
In this case, we write
Definition 1.9. (see [5]) The union operation on the two PMkFSSs and denoted by is defined by a mapping given by such that for all where and
Definition 1.10. (see [5]) The intersection operation on the two PMkFSSs and denoted by is defined by a mapping given by such that for all where and
Definition 1.11. (see [5]) Let and be two PMkFSSs over (U, E). The “ AND ” denoted by is defined by where for all (α, β) ∈ A × B, such that and
Definition 1.12. (see [5]) Let and be two PMkFSSs over (U, E). The “ OR ” denoted by is defined by where for all (α, β) ∈ A × B, such that and
Counterexample
We begin this section with Theorem 2.1 below, originally proposed as Theorem 3.23 in Zhang and Shu [5] and provide a counterexample to show that assertions (3) and (4) are not true.
Theorem 2.1.(see [5]) Let and be any three PMkFSSs over (U, E) . Then we have
The following example shows that assertions (3) and (4) of Theorem 2.1 above are not true in general.
Example 2.2. Let U = {u1, u2} be a set of two houses and E = {e1, e2, e3, e4, e5, e6} be a set of parameters, which stand for expensive, beautiful, cheap, size, location and in the green surroundings respectively. Consider A, B and C to be subsets of E, where A = {e1, e2} = {expensive, beautiful} , B = {e3, e4} = {cheap, size} and C = {e5, e6} = {location, in the green surroundings} . Suppose that and be three PMkFSSs defined by
By Definitions 1.11 and 1.12, the possibility multi-fuzzy soft set has the parameter set A × (B × C) and possibility multi-fuzzy soft set has a set of parameters as (A × B) × (A × C) . But we can not find any notion which ensure A × (B × C) = (A × B) × (A × C) . Hence Theorem 2.1 above is not true.
Main results
In this section we introduce the concept of a generalized possibility multi-fuzzy soft subset and fuzzy soft equal and give some theorems and examples.
We begin by proposing the definition of a generalized possibility multi-fuzzy soft subset followed by an example.
Definition 3.1. Let U be an initial universe and E be a set of parameters. For subsets A and B of E, let and be two PMkFSSs over (U, E) . is generalized possibility multi-fuzzy soft subset of , denoted by if, for every α ∈ A there exists β ∈ B such that where for all in PMkFSSs and respectively.
Example 3.2. Let U = {u1, u2, u3} be an initial universe and E = {e1, e2, e3, e4} be a set of parameters. With A = {e1, e3} and B = {e2, e4} , let and be two PMkFSSs over (U, E) defined by
Then
We establish the definition of generalized possibility multi-fuzzy soft set equal below followed by a theorem on its equality.
Definition 3.3. Let and be two PMkFSSs over (U, E). and are generalized possibility multi-fuzzy soft set equal, denoted by if and
Theorem 3.4.Let and be generalized possibility multi-fuzzy soft sets. Then if and only if
Proof. If then and For any there exists βα ∈ B such that since Hence we obtain In a similar fashion, we can show that since Conversely, assume that Then for every Hence there exists βα ∈ B such that This implies that Similarly, we can show that and so we deduce that as required.
In the following Theorem 3.5 is the corrected version of assertions (3) and (4) of Theorem 2.1, originally written as Theorem 3.23 of Zhang and Shu [5].
Theorem 3.5.Let and be any three PMkFSSs over (U, E) . Then we have
Proof. We only show the validity of (1); then the proof of (2) can be obtained using similar techniques. Now, let us write for where for all (β, γ) ∈ B × C . Then let where for all (α, (β, γ)) ∈ A × (B × C) . In a similar fashion, we can show that for all ((α, β), (α, γ)) ∈ (A × B) × (A × C) . Hence for every (α, (β, γ)) ∈ A × (B × C), there exists ((α, β), (α, γ)) ∈ (A × B) × (A × C) such that By Definition 3.1, we obtain
The equalities in Theorem 3.5 are not true in general as illustrated by the following example.
Example 3.6. Let U = {u1, u2} be a set of two houses and E = {e1, e2, e3, e4} be a set of parameters, which stand for expensive, beautiful, cheap and size, respectively. Consider A, B and C to be subsets of E, where A = {e1, e2} = {expensive, beautiful} , B = {e3} = {cheap} and C = {e4} = { size} . Suppose that and be three PMkFSSs defined by
Let us write for where for all (β, γ) ∈ B × C . Then where for all (α, (β, γ)) ∈ A × (B × C) . It is easily seen that A × (B × C) = {(e1, (e3, e4)), (e2, (e3, e4))} .
By calculation we obtain
and
Next, we write for all ((α, β), (α, γ)) ∈ (A × B) × (A × C) . Hence A × B = {(e1, e3), (e2, e3)} and A × C = {(e1, e4), (e2, e4)} . Thus we have
By calculation we obtain
and
Clearly, we have
and
This shows that does not hold. Using similar techniques, we can also show that the second statement of Theorem 3.5 does not hold.
In order to give a deeper insight on associativity, intersection and union of generalized possibility multi-fuzzy soft sets equal, we propose the following Theorems 3.7, 3.8 and 3.9 along with their corresponding proofs.
Theorem 3.7.Let and be any three PMkFSSs over (U, E) . Then we have
Proof. (1) and (2) are similar to the proof of Theorem 3.5 (1) and (2) respectively.
Theorem 3.8.Let and be two PMkFSSs over (U, E) . Then
Proof. To prove the first assertion, Let us write for where for all (α, β) ∈ A × B . We denote by where for all (β, α) ∈ B × A . For any (α, β) ∈ A × B, there exists (β, α) ∈ B × A such that Hence, by Definition 3.1, we have Similarly, we can show that Therefore, we have The second assertion can be proven in a similar fashion and thus omitted.
Theorem 3.9.Let and be any three PMkFSSs over (U, E) . Then we have
Proof. We only show the validity of (1); then the proof of (2) can be obtained using similar techniques. Now, let us write for where for all (β, γ) ∈ B × C . Then let where for all (α, (β, γ)) ∈ A × (B × C) . On the other hand, let us write for where for all (α, β) ∈ A × B . Then let where for all ((α, β), γ) ∈ (A × B) × C . Since we deduce that and are indeed the same set. Hence by Theorem 3.4, we can conclude that as required.
Conclusion
Zhang and Shu [5] introduced the concept of possibility multi-fuzzy soft sets based on a generalization parameter which itself is multi-fuzzy and proposed several theorems and some operations on a possibility multi-fuzzy soft set. However, we pointed out that assertion in Theorem 3.23 (3) and (4) of [5] is flawed. Using the notions of a generalized possibility multi-fuzzy soft subset and fuzzy soft equal, the Theorem 3.23 (3) and (4) of [5] is corrected, along with the introduction of a few new theorems and definitions.
Footnotes
Acknowledgments
We are indebted to Universiti Kebangsaan Malaysia for providing financial support and facilities for this research under the grant DPP-2015-FST.
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