Graded Rough Sets based on Neighborhood Operator Over two Different Universes and its Application

The major concern of this paper is to present the notion of rough set based on neighborhood operator on universe set, along with its properties, and examples. Then, we generalize several notions of covering rough sets to neighborhood rough sets with respect to the graded n . Further, we present some notions such as probabilistic neighborhood rough approximations of X , (Type-I / Type-II) probabilistic neighborhood rough approximations of X with error α and β, and (Type-I / Type-II) probabilistic neighborhood rough approximations of X with respect to N . The interesting properties of above notions are investigated in detail. On the other hand, we deﬁne the notion of rough set based on neighborhood operator over two different uni-verses. Subsequently, we present some notions (Type-I / Type-II / Type-III) graded n -neighborhood rough sets and give a two approaches to decision-making problems based on the (Type-II / Type-III) grade n -neighborhood rough sets. Then, we construct the decision steps and give two algorithms of the decision meth-ods. Also, we will give two illustrative examples to show the applicability of the rough set based on neighborhood operator over two different universes to solve the rough decision-making problems. Finally, we give a comparison between the Liu et al.’s approach and our approach.


Introduction
In past few years, Pawlak [1] proposed the rough set theory which can be seen as a mathematical tool to handle with ambiguity and incomplete information systems.The basic notion of rough set, i.e. the (lower / upper) approximations are firstly described through the equivalence classes.In mild of this, such as the graded rough sets [2], similarity or tolerance relations [3][4][5], arbitrary binary relation [6,7], and variable precision rough sets [8,9] in a few extensions of the classical rough sets.On the other hand, the notion of covering-based generalizations of the rough set of a universe are introduced in [10][11][12][13][14][15].The expert researchers studied some notions related of covering rough sets in [16][17][18][19].The notion of neighborhood covering of arbitrary set in an universe is proposed and the basic properties are also investigated in [11,[20][21][22].Several types of neighborhood-related covering rough sets are defined in [23][24][25][26][27].A new notion of complementary neighborhood and some basic of their properties is introduced in [28].
Wong et al. [29] proposed the notion of rough set model on two universes and several applications of the rough set model on two universes are discussed in [30][31][32][33][34][35][36][37][38][39][40].Yao and Lin [41,42] introduced the notion of graded rough sets on one universe.Zhang et al. [43] studied a comparative between the variable precision rough set model and the graded rough set model.Zhang and Miao [44] introduced two basic double-quantitative rough set models of precision and grade and their investigation using granular computing.Based on the work of Yao and Lin [41,42], the notion of graded rough sets on two universes are presented by Liu et al. [45] and some interesting properties are also explored.Yu et al. [46] proposed a variable precision graded rough set model based on two universes.Yu and Wang [47] proposed a kind of graded rough set with variable precision which is over two inconsistent and related universes.
There are some shortcomings in the methods are used to solve the decision-making problem by using the grade rough sets on two universes in [45], as the motivation and objectives of this article, we will introduce a new notion of rough set based on neighborhood operator on universe set U .We will discuss the interesting properties of neighborhood rough sets and also generalized several notions of covering rough sets (i.e., Definitions 2.5 and 2.8-2.11) to neighborhood rough sets with respect to the graded n (see, Theorems 3.10-3.14),respectively.We also prove that graded rough sets based on two universes are neighborhood rough sets (see, Theorem 3.17).Furthermore, we will extend the classical probabilistic rough sets to probabilistic neighborhood rough sets and investigate several of their properties of probabilistic neighborhood rough sets.Meanwhile, the basic properties of the notions (Type-I / Type-II) probabilistic neighborhood rough approximations of X with error α and β are also proposed (see, Definition 3.32).At the same time, the notions (Type-I / Type-II) probabilistic neighborhood rough approximations of X with respect to N are defined (see, Definition 3.34).In other hand, we will present the notion of neighborhood rough sets on two different universes (see, Definition 4.1) and study the basic some of their properties of neighborhood rough set.In addition to, we will propose the notions of (Type-I / Type-II / Type-III) graded n-neighborhood rough set on two different universes (see, Definitions 4.4,4.8,and 4.16), respectively.We then show the upper approximation can't contain the lower approximation (see, Definition 4.4) as illustrated by (Example 4.7).Consequently, according to (Definition 4.8 and Lemma 4.10), the upper approximation contain the lower approximation (see, Theorem 4.11 (1)).Finally, based on Definitions 4.8 and 4.16, we will construct the algorithm 1 (which how it is realize a computation to method of the grade and calculation all the approximations of the sets) and the algorithm 2 (which how it is realize a computation to method of the grade based on relation matrix and calculation all the approximations of the sets).Also, we will present two applications (see subsections 4.2 and 4.3) on (Type-II / Type-III) graded n-neighborhood rough set over two different universes by algorithms 1 and 2, respectively.Finally, we will give a comparison between the Liu et al.'s approach [45] and our approach.
The rest of this paper is organized as follows.In Section 2, we review some basic notions related to Pawlak's rough set, probabilistic rough set, and covering rough sets.In Section 3, we propose the notions of neighborhood operator, neighborhood rough set, and probabilistic neighborhood rough set.Some of their interesting properties of above notions are studied.In Section 4, we present the notions of neighborhood rough set and three types of graded n-neighborhood rough set over two different universes.Based on (Type-II / Type-III) graded n-neighborho od rough set over two different universes, the algorithms 1 and 2, respectively, are constructed and two applications on these notions are given.A comparison between the Liu et al.'s approach [45] and our approach is discussed.Finally, conclusions are given in Section 5.

Pawlak's rough set
Let U be a set and R an equivalence relation on U (in this case (U, R) is called an approximation space). 1 A rough approximation (briefly, a Pawlak rough set) of a subset X ∈ 2 U (the set of all subsets of U ) in (U, R) is a pair (R(X), R(X)), defined by where [u] R is the equivalence class of R containing u ∈ U .The subsets pos(X) = R(X), neg(X) = U − R(X), and bn(X) = R(X)−R(X) are called the positive region, the negative region, and the boundary region of X with respect to (U, R), respectively (cf.[1]).

Probabilistic rough set
1 The idea to define a Pawlak rough set of a subset X is to describe X by U/R (the family of all R-equivalence classes, so-called knowledge).
Let (U, R) be an approximation space and P a probability measure defined on a σ-algebra of subsets of U (in this case (U, R, P ) is called a probabilistic approximation space). 2 For 0 ≤ β < α ≤ 1.A rough approximation with error α and β (briefly, a probabilistic rough set) of a subset are called the positive region, the negative region, and the boundary region of X with respect to (U, R, P ), respectively (cf.[48]).

Covering rough sets
Definition 2.1 (cf.[10,21]).Let U be a universe, and C = {θ i |i ∈ I} be a family of nonempty subsets of U. If ∪ i∈I θ i = U, then C is called a covering of U, and the ordered pair (U, C) is called a covering approximation space.Definition 2.2 (cf.[11,[20][21][22]25]). Let (U, C) be a covering approximation space.For any u ∈ U, we can define the neighborhood of u as Based on Definition 2.2 we present the concept of strong neighborhood covering on a set U as follows: Lemma 2.4.Let (U, C) be a covering approximation space.For all u 1 , u 2 ∈ U and u 1 ∈ N C (u 2 ).Then N C (u 1 ) ⊆ N C (u 2 ) hold.

Proof. clear.
Next, we offer five different types of covering rough sets based on the neighborhood of u (see Definition 2.5) and the complementary neighborhood of u (see Definitions 2.8, 2.10 and 2.11) and adhesion of u (see Definition 2.9) as follows: Definition 2.5 (cf.[25][26][27]).Let U be an universe set, (U, C) be a covering approximation space and X ∈ 2 U .Then are called the lower approximation and the upper approximation of X with respect to C respectively, ) is called the type-1 covering rough approximation of X (or a type-1 covering rough set).
From Definition 2.3, Lemma 2.4, and Definition 2.5 we introduce the following theorem.
Theorem 2.6.Let (U, C) be a covering approximation space, C is a strong neighborhood covering, and X ∈ 2 U .Then the followings hold: The equality (4) of Theorem 2.6 does not hold as the following example.
Example 2.7.Let U = {x 1 , x 2 , x 3 , x 4 , x 5 } and C = {{x 1 , x 2 , x 5 }, {x 3 , x 4 }}.From Definition 2.2, we have Definition 2.8 (cf.[28]).Let (U, C) be a covering approximation space, are called the lower approximation and the upper approximation of X with respect to C respectively, ) is called the type-2 covering rough approximation of X (or a type-2 covering rough set).Definition 2.9 (cf.[11,[20][21][22]25]). Let (U, C) be a covering approximation space, are called the lower approximation and the upper approximation of X with respect to C respectively, ) is called the type-3 covering rough approximation of X (or a type-3 covering rough set).Definition 2.10 (cf.[28]).Let (U, C) be a covering approximation space and X ∈ 2 U .Then are called the lower approximation and the upper approximation of X with respect to C respectively, ) is called the type-4 covering rough approximation of X (or a type-4 covering rough set).Definition 2.11 (cf.[28]).Let (U, C) be a covering approximation space and X ∈ 2 U .Then are called the lower approximation and the upper approximation of X with respect to C respectively, ) is called the type-5 covering rough approximation of X (or a type-5 covering rough set).

A neighborhood operator
In this section we will give a new notion it is called "a neighborhood operator ".Based on a neighborhood operator on a set U, we present some concepts of rough sets and then the basic properties.
We will begin defining the neighborhood operator on a set U , and will subsequently define neighborhood rough set.

Rough sets based on a neighborhood operator
Definition 3.1.A neighborhood operator on a set U is a mapping N from a subset U 0 ⊆ U to 2 U (the set of all subsets of U ) satisfying u ∈ N (u) (∀u ∈ U 0 ).A strong neighborhood operator on a set U is a neighborhood operator N satisfying We note that there is not relation between a neighborhood operator (i.e., N 1 : U 0 −→ 2 U under condition u ∈ N (u) (∀u ∈ U 0 )) as in Definition 3.1 and the neighborhood operator (i.e., N 2 : U −→ 2 U without any condition) as given in [49], for example, let Definition 3.2.Let N be a neighborhood operator on a set U and X ∈ 2 U .Then the triplet (U, U 0 , N ) is called a neighborhood approximation space, are called the lower N -approximation and the upper N -approxima tion of X with respect to N respectively, ( N (X), N (X) ) is called the neighborhood rough approximation of X (or the Nrough approximation of X, briefly, a neighborhood rough set).
To illustrate this idea, let us consider the following example.
The main results are as follows: Theorem 3.6.Let (U, U 0 , N ) be a neighborhood approximation space.Then the followings hold: Proof.Follows from Definition 3.2.
The equality (5) of Theorem 3.6 does not hold as the following example.
(1) N is strong neighborhood operator on U , and (2)

and both
N α and N α preserves orders, i.e.N α (X 1 ) ⊆ N α (X 2 ) and (5) For any countable family {X k } k∈K of disjoin subsets of U , there exists a countable family and N is strong neighborhood operator on U .
Proof.(1) Obviously, N : (2) Follows from Definition 3.5. ( Notice also u ∈∼ The following Definition 3.9, as expected, will generalize 4 almost definitions of covering rough sets.Definition 3.9.Let (U, U 0 , N ) be a neighborhood approximation space and X ∈ 2 U .Then are called the lower N -approximation and the upper N -approxi mation of X with respect to the graded n respectively, and ) is called the neighborhood rough approximation of X (or the N -rough approximation of X, briefly, a neighborhood rough set) with respect to the graded n 5 .
Next, we discuss the connections between neighborhood rough sets with respect to the graded n in Definition 3.9 and covering rough sets presented in Definitions 2.5 and 2.8-2.11.Theorem 3.10.Type-1 covering rough sets (particularly, Pawlak rough sets) are neighborhood rough sets.More precisely, for each cover C of U and each X ∈ 2 U , the pair defined as in Definition 2.5 is the same as defined in the neighborhood approximation space (U, N 1 ) (see Definition 3.9, thus ) is a neighborhood rough set with respect to the graded 0), where Theorem 3.11.Type-2 covering rough sets are neighborhood rough sets.More precisely, for each cover C of U and each X ∈ 2 U , the pair ) is a neighborhood rough set with respect to the graded 0), where Notice that U and V are arbitrary sets. 5We will write Type-3 covering rough sets are also neighborhood rough sets, and have properties that Pawlak rough sets have.
Corollary 3.15.Each kind of rough sets (apart from Type-4 and Type-5 covering rough sets) defined as in Definition 2.10 and Definition 2.11, respectively, are neighborhood rough sets.Definition 3.16.A graded rough set with respect to the grade n (where n is a nonnegative integer) based on two universes V and W 6 (see [45] for nonempty finite set case) is a pair Theorem 3.17.Graded rough sets based on two universes are neighborhood rough sets.
By the way, we show that the rough sets defined in Definitions 2.5 and 2.8-2.11can be constructed in topology-like spaces (called co-knowledge spaces).Definition 3.18.Let (U, K) be a co-knowledge space (i.e.U is a set and K ⊆ 2 U is closed under the operation intersection), and The set of all interior points (resp., all adherent points) of X is denoted by X o K (resp., X − K ).Theorem 3.19.Let (U, C) be a covering rough space and X ∈ 2 U .Then (1) (U, K(C)) is a co-knowledge space, where ) . (4) ) . ( ) . ( ) . (7) ) . 6 We do not discuss graded rough sets based on a family {V } ∪ {W k | k ∈ K} (|K| > 0) of universes (see [45] for a special case) because each of them is actually a graded rough set based on two universes.

Probabilistic rough sets based on neighborhood operator
In this subsection, we will extend the classical probabilistic rough sets to probabilistic neighborhood rough sets.We present two types of probabilistic neighborhood rough sets and then the basic properties.
We begin by proposing the definition of a probabilistic neighborhood rough sets followed by remark.Definition 3.20.Let (U, U 0 , N ) be a neighborhood approximation space, 0 ≤ β < α ≤ 1, and X ∈ 2 U .
are called the lower N -approximation and the upper N -approximation of X with error α and β respectively, and is called the probabilistic neighborhood rough approximation of X (or the (αβ) p -rough approximation of X, briefly, a probabilistic neighborhood rough set).The subset bn αβ p (X) = {u ∈ U 0 | β < P (X|N (u)) < α} is called the boundary region of X with error α and β.Remark 3.21.If α = 1 and β = 0, then (2) The main results are as follows: Theorem 3.22.Let (U, U 0 , N ) be a neighborhood approximation space, 0 ≤ β < α ≤ 1, and X, X 1 ∈ 2 U .Then the followings hold: (2) and (3) follows from Definition 3.32.
The equality ( 5) and ( 6) of Theorem 3.22 does not hold as the following example.
Proof.We only prove (1) and then the proof of (2) can be obtained using similar techniques.
Theorem 3.31.Let (U, U 0 , N ) be a neighborhood approximation space, 0 ≤ β < α ≤ 1, and X, X 1 ∈ 2 U .Then the followings hold: (1) and are called the Type-I lower Ni -approximation and the Type-I upper Ni -approximation of X with error α and β respectively, ) is called the Type-I probabilistic neighborhood rough approximation of X (or the Type-I probabilistic Ni -rough approximation of X) with respect to N . (2) are called the Type-II lower Ni -approximation and the Type-II upper Ni -approximation of X with error α and β respectively, ) is called the Type-II probabilistic neighborhood rough approximation of X (or the Type-II probabilistic Ni -rough approximation of X) with respect to N .
To illustrate Definition 3.32, let us consider the following example. Similarity, If β = 0.3 and α = 0.7.Thus, from Definition 3.24 we obtain Definition 3.34.Let U 0 is subset on a set U and N = { Ni , i ∈ N } is a family of neighborhood operators on U, where Ni : are called the Type-I lower Ni -approximation and the Type-I upper Ni -approximation of X with respect to N respectively, ) is called the Type-I probabilistic neighborhood rough approximation of X (or the Type-I probabilistic Ni -rough approximation of X) with respect to N . (2) are called the Type-II lower Ni -approximation and the Type-II upper Ni -approximation of X with respect to N respectively, ) is called the Type-II probabilistic neighborhood rough approximation of X (or the Type-II probabilistic Ni -rough approximation of X) with respect to N .
The equality ( 4) and ( 5) of Theorem 3.41 does not hold as the following example.
From Examples 3.39 and 3.44, we conclude the following corollary: Corollary 3.46.Let (U, U 0 , N ) be a neighborhood approximation space, N = { Ni , i ∈ N } is a family of neighborhood operators on U, and X, X 1 ∈ 2 U .Then the followings hold: ( Proof.Follows from Definition 3.34. Proof.We only prove (3).
(X) and by Definition 3.32 we have either Corollary 3.49.Let (U, U 0 , N ) be a neighborhood approximation space, N = { Ni , i ∈ N } is a family of neighborhood operators on U, and X ∈ 2 U .Then the followings hold: 4 Rough set based on a neighborhood operator over two different universes In this section we will propose a new notion of rough set based on a neighborhood operator over two different universes U and V. Furthermore, we introduce three types of graded n-neighborhood rough set over two different universes and then study two algorithms and its applications.
Definition 4.4.Let (U V , U 0 , N ) is called a neighborhood approximation space over two different universes U and V, n ∈ N, where N is the set of natural numbers and Y ∈ 2 V .Then are called the lower N -approximation and the upper N -approxim ation of Y with respect to the graded n respectively, and ) is called the Type-I graded n-neighborhood rough approximation of Y on U and V (or the N -Type-I graded n-neighbor hood rough approximation of Y , briefly, Type-I graded n-neighbor hood rough set on U and V ).
Theorem 4.5.Let (U V , U 0 , N ) be a neighborhood approximation space on two different universes U and V and Y ∈ 2 V .Then the followings hold: (1) ( Combining the two formulas, we obtain that

This show N
The above example show that N I n (Y ) ⊆ N I n (Y ) implies that it is a contradiction with Pawlaks rough set theory.So, we present the definition of Type-II graded n-neighborhood rough set on U and V as follows: Algorithm 2. An algorithm for computing approximations.
is the matrix of Y, and n = 0; Output: Approximations of Y with respect to N ; Example 4.17.Let U = {x 1 , x 2 , x 3 , x 4 , x 5 , x 6 , x 7 }, U 0 = {x 1 , x 2 , x 3 , x 4 , x 5 , x 6 } ⊆ U, and V = {y 1 , y 2 , y 3 , y 4 , y 5 , y 6 , y 7 , y 8 }.A neighborhood operator N : U 0 −→ 2 V is defined as follows: N (x 1 ) = {y 2 , y 4 , y 5 }, N (x 2 ) = {y 2 , y 3 , y 6 , y 7 }, N (x 3 ) = {y 1 , y 4 , y 6 , y 8 }, N (x 4 ) = {y 3 , y 8 }, N (x 5 ) = {y 1 , y 4 , y 5 , y 7 }, and N (x 6 ) = {y 2 , y 5 , y 8 }.Consequently, M N = (a ij ) p×m be the neighborhood matrix of N is written as follows: Assume that Y = {y 1 , y 3 , y 4 , y 5 , y 8 }.Then, Assume that we are considering a certain group of sufferers in a medical analysis system.A suffer may show numerous symptoms on the same time and a sickness may be observed by means of numerous symptoms, so as to set up a relation among a sufferer and a sickness.In the current situation, how a specialized doctor could determine what kind of remedies that a patient must be taken based on the prevailing symptoms?This question can be responded through applying the neighborhood matrix model primarily founded on two universes to the medical analysis system.Suppose that U and V are representing the set of patients and the set of symptoms, respectively, and U 0 is a subset of a set U. Then for any patient u ∈ U 0 , there exist some symptoms in V correspond to u.For any Y ∈ 2 V , Y is a positive disorder which contains a few symptoms in V. Afterwards, given a patient, if she or he belongs to set N -POS(Y ), then he or she is really suffered from the sickness denoted by using Y.Hence, all the sufferers belonging to N -POS(Y ), are on appropriate for the remedy to Y right after diagnosis.In other words, if he or she belongs to set N -BND(Y ), then his or her sickness is linked with Y ; however the connection isn't always sure.As a result, the doctor ought to further examine the pathogeny for the current sufferer and undertake the suitable treatment.Moreover, if he or she belongs to set N -NEG (Y ), then his or her sickness has no connection with Y, and the specialized doctor should adopt other techniques.
In the following example, we will show the above discussion: Example 4.18.Let U = {x 1 , x 2 , x 3 , x 4 , x 5 } be a set of five patients, U 0 = {x 1 , x 2 , x 4 } ⊆ U, and V = {y 1 , y 2 , y 3 , y 4 } be a set of four symptoms.Suppose that a neighborhood operator N : U 0 −→ 2 V is defined as follows: N (x 1 ) = {y 1 , y 4 }, N (x 2 ) = {y 2 , y 3 , y 4 }, and N (x 4 ) = {y 1 , y 2 , y 3 }.Then, M N = (a ij ) p×m (where if patient i has symptom j then a ij = 1 else a ij = 0) be the neighborhood matrix of N is written as follows: Then, from Definition 4.16 we can obtain that Thus, we can obtain the subsequent conclusions based on the above demonstrations: (1) It is certain that patient x 4 has sickness Y and the doctor ought to take the corresponding treatment immediately.
(2) The doctor cannot decide whether patients x 1 and x 2 have sickness Y or not based to the symptoms at present.The sufferers should be examined further.
(3) None of the three patients are healthy after diagnosis.
4.5.Comparison between our approach (i.e., Algorithms 1 and 2) and Liu et al.'s approach [45] In this subsection, we compare the proposed approach (i.e., Algorithms 1 and 2) based on (Type II \ Type III) graded nneighborhood rough set over two different universes and Liu et al.'s approach [45] in decision-making problem.
(a) Liu et al. [45] presented an approach using the graded rough set model based on two universes.We compare our approach with that of Liu et al.'s approach [45,Example 4].Thus, we can obtain the subsequent conclusions based on the above demonstrations: (1) It is certain that patient x 2 has disease Y and the doctor should take the corresponding treatment immediately.
(2) The doctor cannot decide whether patient x 3 has disease Y or not according to the symptoms at present.The patient should be examined further.
(3) None of the three patients is healthy after diagnosis.
From above discussion, we can find the the conclusions (1)-(3) by our developed approach (i.e., based on Type II graded n-neighborhood rough set) are in difference with the decision results of proposed by Liu et al.'s approach in [45], which the Example 4 in [45] depend on the following property R(Y ) = {x 3 } ⊆ {x 1 , x 2 , x 3 } = R(Y ) (i.e., it's mean the upper approximation can't contain the lower approximation) which implies that it is a contradiction with Pawlak's rough set theory [1] (i.e., R(Y ) ⊆ R(Y )).This mean there is an error in the approach proposed in [45] based on graded rough set model, which led to the difference in the decision-making results.Therefore, our proposed decision-making approach can successfully avoid and solve the above issue.

Conclusions
This study aims to propose a new notion of neighborhood rough set on universe set and the interesting properties of neighborhood rough sets are discussed.Several notions of covering rough sets to neighborhood rough sets with respect to the graded n are generalized.We also proved a graded rough sets based on two universes are neighborhood rough sets.Furthermore, we extend the classical probabilistic rough sets to probabilistic rough sets based on neighborhood operator and investigate some basic of their properties of the probabilistic neighborhood rough sets.
Additionally, we propose some notions such as (Type-I / Type-II) probabilistic neighborhood rough approximations of X with error α and β and (Type-I / Type-II) probabilistic neighborhood rough approximations of X with respect to N .Besides defining the notion of rough set based on neighborhood operator over two different universes U and V, three notions (Type-I / Type-II / Type-III) graded n-neighborhood rough sets so far are proposed.The basic properties of above notions are then explored.Finally, we give two applications based on (Type-II / Type-III) graded nneighborhood rough sets over two different universes for a rough decision-making problem.
In the future, we can extend the results of this study as new notions to define the properties of the previous studies [13-15, 21-25, 32, 34, 38-40, 50-56].In addition to, one may investigate further based on neighborhood operator with some links to topology.

Theorem 3 . 12 .
Type-3 covering rough sets are Pawlak rough sets determined by the partition {P C (u)} u∈U of U .Therefore, 4

4. 1 .
Rough set based on a neighborhood operator over two different universes Definition 4.1.Let U 0 is subset on a set U, N : U 0 −→ 2 V

}
are called the lower N -approximation and the upper N -approxima tion of Y with respect to N respectively, ( N (Y ), N (Y ) ) is called the neighborhood rough approximation of Y on U and V (or the N -rough approximation of Y , briefly, a neighborhood rough set on U and V ).
Y ), N (Y ) ) , and N I n (Y ) ⊆ N I n (Y ), and directly from Definition 4.4, we have N I n (∅) = ∅ and N I n (V ) = U 0 .Now, we only prove that N

Remark 4 . 6 .
Let Y ∈ 2 V .Then N I n (Y ) ⊆ N I n (Y )does not hold by the following example.