SH Fuzzy Partition and Fuzzy Equivalence

Int. J. Contemp. Math. Sciences, Vol. 5, 2010, no. 8, 381 - 392

S-H Fuzzy Partition and Fuzzy Equivalence Relation M. A. Shakhatreh and M. A. Hayajneh Yarmouk University, Department of Mathematics, Irbid, Jordan [email protected], [email protected] Abstract In this paper we will introduce a certain collection of fuzzy subsets of a nonempty set X. This certain collection gives raise to a useful new fuzzy partition. This new fuzzy partition gives us a way of generating a fuzzy equivalence relation as in the ordinary case; any partition generates an equivalence relation.

Mathematics Subject Classification: 03E72, 46S40, 26E50 Keywords: Fuzzy sets, Fuzzy Equivalence Class, Fuzzy Equivalence Relation, Fuzzy Partition

1

Introduction

The fuzzy set theory was introduced by Zedeh (1965) in [8] in order to provide a scheme for handling a variety of problems in which a fundamental role is played by an indefiniteneis arising from a sort of intrinsic ambiguity. As fuzzy equivalence classes and fuzzy partitions plays a major role in many topics in fuzzy theory, such as fuzzy measure, fuzzy integration and fuzzy algebra, see [4] and [5]. In [2], the authors proposed a new definition of a fuzzy equivalence class such that it has built a certain partition which has been discussed in reference [4]. In the ordinary case we know that each partition T of X gives rise to an equivalence relation R defined by xRy ⇔ x, y ∈ P for some P ∈ T . So, this leads us to the following question : Can we define a way to obtain the fuzzy eqivalence relation that was introduced in [7] from the fuzzy partition that was introduced in [4]? The answer of this question is that when the definition of the fuzzy partition was designed, it wasn’t taken care of it to be fitting with the definition of the fuzzy equivalence relation. So, in this article, we will introduce a new definition of a fuzzy partition which is the same partition introduced in [4] and [2] containing an additional condition. For the importance of this partition, we will call it the Shakhatreh-Hayajneh (S-H fuzzy partition). This additional condition gives us the ability to generate a fuzzy equivalence relation.

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In [1], the authors introduce some new properties of the set of all fuzzy equivalence  which were introduced in [2]. In this article, the authors were amazed by classes [X]  form the S-H fuzzy partition of noticing that the set of all fuzzy equivalence classes [X] X. This fact gives us a feeling of the suitability of the definition of the S-H fuzzy partition. Throughout the proof of this fact, we will use a useful property of the set of all fuzzy  introduced in [1] which says that [x]  ∩ [y]  ≤ [x](y).  equivalence classes [X]

2

Definition and preliminaries

Let X be a universal set and P a subset of X. Then the characteristic function χP of the set P is defined to be the function χP : X → [0, 1] given by  1 if x ∈ P , χP (x) = 0 if x ∈ /P . Throughout this paper, one can think of the fuzzy set λ in X as a function from X to [0, 1]. Sometimes one can keep the symbol λ for the fuzzy set and associate it with a function μλ from X to [0, 1] called the membership function. We can also write λ = (x, μλ (x)). The fuzzy set λ is said to be contained, ⊆, in the fuzzy set μ if and only if λ(x) ≤ μ(x) for each x ∈ X. Let A := {λγ : γ ∈ Γ} bea collection of fuzzy sets. Then  the union and the intersection are defined as follows: ( γ∈Γ )(x) = supγ∈Γ (x) and ( γ∈Γ )(x) = infγ∈Γ (x). Let T := {Pγ : γ ∈ Γ} be a collection of subsets of X. Then T is called an ordinary partition  of X if and only if Pγ = φ for each γ ∈ Γ, Pγ ∩ Pη = φ whenever γ = η and X = γ∈Γ Pγ . Now, we will state some definitions and theorems which were given in [7], [4], [6], and [2]. Definition 2.1 (Weak-Separated Fuzzy Subsets, see [6]) Let F be a collection of fuzzy B  ∈ Fwith A  = B.  subsets of a nonempty set X and A,  and B  are called Weak-Separated fuzzy subsets. If μA∩B (x) < 0.5 , ∀x ∈ X , then A  be a fuzzy relation on a Definition 2.2 (Fuzzy Equivalence Relation, see [7]) Let R nonempty set X.  is reflexive on X iff μ (x, x) = 1 , ∀(x, x) ∈ X × X. (a) R R  is symmetric on X iff μ (x, y) = μ (y, x) , ∀(x, y), (y, x) ∈ X × X. (b) R R R     is transitive on X iff μ (x, z) ≥ maxy min μ (x, y), μ (y, z) , ∀(x, z), (x, y), (y, z) ∈ (c) R R R R X × X. Definition 2.3 (Fuzzy Partition, see [4]) Let X be a nonempty set. A fuzzy partition T of X is a set of nonempty fuzzy subsets of X such that

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B  ∈ T and A  = B,  then (A  ∩ B)(x)  (a) If A, < 0.5. (b)

 w∈T

w  = X.

 be a fuzzy Definition 2.4 (Elements which have Strong Bond with x, see [2]) Let R relation on a nonempty set X and x ∈ X. Then B(x) = y ∈ X : μR (y, x) ≥ 0.5 is called the set of all elements which has strong bond with x.  be a fuzzy equivalence relation Definition 2.5 (Fuzzy Equivalence Classes, see [1]) Let R on a nonempty set X and x ∈ X. Then

 (a) [x] = (y, μ[x]  (y)) : y ∈ X , where  μ[x]  (y) =

1 if y ∈ B(x) , μR (x, y) if y ∈ / B(x) .

is called the fuzzy equivalence class determined by x .

 = [x] :x∈X (b) [X] is called the set of all fuzzy equivalence classes.  be a Definition 2.6 (Weakly Empty Fuzzy Subset, see [4]) Let X be a nonempty set and A  weakly empty fuzzy subset of X if μ (x) < 0.5 ∀ x ∈ X. fuzzy subset of X. Then we call A A  be a fuzzy equivalence relation on a nonempty set X and Theorem 2.1 (see [2]) Let R  be the set of all fuzzy equivalence classes. Then [X]  ∩ [y]  < 0.5. (a) If x = y, then [x] (b)

 x∈X

 = X. [x]

 be a fuzzy equivalence relation on a nonempty set X and Theorem 2.2 (see [1]) Let R  be the set of all fuzzy equivalence classes. Then μ   (y) ≤ μ (x) ∀ x, y, z ∈ X. [X] [x]∩[z] [z]  Remark 2.1 For simplicity, we can use the notation A(x) instead of μA (x) to denote the    degree of x in the fuzzy set A on X. By A ≤ B, we mean μA (x) ≤ μB (x) ∀ x ∈ X.

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S-H Fuzzy Partition

Now, we will introduce a new definition of a fuzzy partition which contains an additional condition. This additional condition gives us the ability to generate a fuzzy equivalence relation. For the importance of this condition, we will call it the S-H condition and we will call this new partition the S-H fuzzy partition. Definition 3.1 (S-H Fuzzy Collection) Let F be a collection of fuzzy subsets of a nonempty ∩A  ≤ B(a)  B  ∈ F such set X. F is called S-H collection if and only if B whenever A,  that A(a) = 1.   Example 3.1 Let T := {Pγ : γ ∈ Γ} form an ordinary partition. Then T := χPγ : γ ∈ Γ forms an S-H fuzzy collection. Proof Let γ, η ∈ Γ and a ∈ X such that χPγ is a non weakly empty fuzzy subset and χPη (a) = 1. Now if Pγ = Pη , then we have χPγ (a) = 1. This implies that χPγ ∩ χPη ≤ χPγ (a) and we are done. Also if Pγ = Pη , then we have χPγ ∩ χPη = 0, which implies that χPγ ∩ χPη ≤ χPγ (a) and we are done. By above discussion T forms an S-H fuzzy collection. 2

1 , A 2 , A 3 , A 4 , A 5 be a collecExample 3.2 Let X = {x0 , x1 , x2 , x3 , x4 } and let F = A tion of fuzzy subsets of X, where 1 ={(x0 , 1.0), (x1 , 0.9), (x2 , 0.8), (x3, 0.7), (x4 , 0.1)} A 2 ={(x0 , 0.9), (x1 , 1.0), (x2 , 0.8), (x3, 0.7), (x4 , 0.1)} A 3 ={(x0 , 0.8), (x1 , 0.8), (x2 , 1.0), (x3, 0.7), (x4 , 0.1)} A 4 ={(x0 , 0.7), (x1 , 0.7), (x2 , 0.7), (x3, 1.0), (x4 , 0.1)} A 5 ={(x0 , 0.1), (x1 , 0.1), (x2 , 0.1), (x3, 0.1), (x4 , 0.1)} A i ∩ A j ≤ A j (x), where A i (x) = 1 and i ∈ {1, 2, 3, 4}. Therefore F is S-H Note that A 5 is a weakly empty fuzzy set and collection. We can see also that A  i = {(x0 , 1), (x1 , 1), (x2 , 1), (x3 , 1), (x4 , 0.1)} = X. A Also note that μA1 ∩A2 (x0 ) = 0.9 > 0.5. Definition 3.2 (The S-H Fuzzy Partition) Let X be a nonempty set. By an S-H fuzzy partition T of X, we mean a set of non weakly empty fuzzy subsets of X such that B  ∈ T and A  = B,  then (A  ∩ B)(x)  (a) If A, < 0.5, i.e., any two non identical fuzzy subsets are Weak-Separated fuzzy subsets.  (b) w∈T w  = X. (c) T is S-H collection.

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Remark 3.1 The last condition in Definition 3.2 is independent of the other conditions. In Example 3.2, we can see that Condition (c) does not imply Condition (a) or Condition (b). Also Condition (a) and Condition (b) do not imply Condition (c). To see that we need the following example.

 B,  C  be a collection of fuzzy subsets Example 3.3 Let X = {x, y, z} and let F = A, of X, where  ={(x, 1.0), (y, 0.1), (z, 0.4)} A  ={(x, 0.1), (y, 1.0), (z, 0.4)} B  ={(x, 0.4), (y, 0.4), (z, 1.0)} C  ∩ B)(z)   Note that (A = 0.4 > B(x) = 0.1. Therefore, Condition (c) does not hold,  B  and C  are non weakly empty i.e., F is not an S-H collection. We can see also that A,      ∩C  < 0.5. fuzzy sets and Ai = X. Note also that A ∩ B < 0.5, A ∩ C < 0.5 and B The next lemma contains some hidden properties of the fuzzy partition introduced in [4] and [2]. Part (a) of this lemma will be used in proving that the fuzzy equivalence relation introduced in Definition 6.1 below is a well defined relation. It is also used in proving the next theorems. Lemma 3.1 Let T be S-H fuzzy partition of a nonempty set X.  ∈ T such that μ (x) = 1. (a) If x ∈ X, then there exist a unique A A  ∈ T such that μ (x) ≥ 0.5, then μ (x) = 1. (b) If x ∈ X, A A A Proof (a)



 ∈ T. Then max μ (x) : A  ∈ T = 1. • (Existence) Suppose that μA (x) = 1 ∀ A A  This implies that μ A (x) = 1 which contradicts part(b) of Definition 3.2. A∈T

B  ∈ T such that A  = B  and μ (x) = μ (x) = • (Uniqueness) Suppose that ∃A, A B  1. Then min μA (x), μB (x) = 1 ≥ 0.5. This implies that μA∩B (x) ≥ 0.5 which contradicts part (a) of Definition 3.2. (b) Suppose on contrary that μA (x) = 1. But by assumption we have μA (x) ≥ 0.5.     Then there  B ∈ T such that μB (x) = 1 and B = A which implies  exist a unique that min μA (x), μB (x) ≥ 0.5. Thus μA∩B (x) ≥ 0.5 which contradicts part (a) of 2 Definition 3.2 and so μA (x) = 1. Now, the remaining sections give us a feeling of the suitability of the definition of the S-H fuzzy partition.

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S-H Fuzzy Partition Covers The Ordinary Case

  Theorem 4.1 Let T := {Pγ : γ ∈ Γ} forms an ordinary partition. Then T := χPγ : γ ∈ Γ forms an S-H fuzzy partition. Proof Firstly, by Example 3.1, T forms an S-H fuzzy collection. If T forms an ordinary partition, then Pγ = φ. Therefore, ∃ x ∈ X such that x ∈ Pγ . This implies that ∃ x ∈ X such that χPγ (x) = 1, i.e., χPγ is not a weakly empty  fuzzy set. Also ∀ y ∈  X, ∃ γ ∈ Γ such that χPγ (y) = 1, which implies that i∈Γ χPi (y) = χPγ (y) = 1, i.e., i∈Γ χPi = X. Now if Pγ = Pη , then we claim that Pγ ∩ Pη is a weakly empty fuzzy set, i.e., Pγ ∩ Pη < 0.5 (Weak-Separated fuzzy subsets). To prove the claim we have two cases. If y ∈ / Pγ ,  then χPγ (y) = 0. And so min χPγ  (y), χPη (y) ≤ 0.5 ∀ y ∈ X. If y ∈ P , then y ∈ / Pη . γ  2 Therefore, χPη (y) = 0. And so min χPγ (y), χPη (y) ≤ 0.5 ∀ y ∈ X. The next theorem gives a wealthy collection of S-H fuzzy partitions.

5

A Way of Generating S-H Fuzzy Partition

 introduced in [1], The authors noticed that the set of all fuzzy equivalence classes [X] forms an S-H fuzzy partition of X. So we are catching a practical way to construct an S-H fuzzy partition. To construct an S-H fuzzy partition, it is just to search for any fuzzy  equivalence relation and to compute the set of all fuzzy equivalence classes [X].  is  be a fuzzy equivalence relation on a nonempty set X, then [X] Theorem 5.1 Let R S-H fuzzy partition of X.  is a non weakly empty since μ  (x) = 1. Thus [X]  is Proof Firstly, it is known that [x] [x] a collection of non weakly empty fuzzy subsets. (a) and (b) were proved in [2]. (c) follows directly from Theorem 2.2. 2  be the fuzzy relation in X × X given in Example 5.1 Let X = {x0 , x1 , x2 , x3 } and let R  is a fuzzy equivalence relation. Table 1. We can show that R

x0 x1 x2 x3

x0 x1 x2 x3 1 0.7 0.3 0.1 0.7 1 0.3 0.1 0.3 0.3 1 0.1 0.1 0.1 0.1 1 Table 1.

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Now to construct an S-H fuzzy partition, we are to compute the set of all fuzzy equiv as follows alence classes [X] [x 0 ] ={(x0 , 1.0), (x1 , 1.0), (x2 , 0.3), (x3 , 0.1)} [x1 ] ={(x0 , 1.0), (x1, 1.0), (x2 , 0.3), (x3 , 0.1)} [x 2 ] ={(x0 , 0.3), (x1 , 0.3), (x2 , 1.0), (x3 , 0.1)} [x3 ] ={(x0 , 0.1), (x1, 0.1), (x2 , 0.1), (x3 , 1.0)}  we have an S-H fuzzy partition. Finally, by setting T = [X],

6

How to Generate Fuzzy Equivalence Relation Using an S-H Fuzzy Partition?

The next definition demonstrates a way of generating a fuzzy equivalence relation from the S-H fuzzy partition. We will prove that this relation is an equivalence relation and then when we start with any S-H fuzzy partition, we can get a fuzzy equivalence relation. Definition 6.1 Let T be an S-H fuzzy partition of a nonempty set X. We define a fuzzy relation X/T on X by the following,  ∈ T such that μ (x) = 1. μX/T (x, y) = μA (y) where A A Remark 6.1 We can see that the above definition is well defined by using Part(a) of Lemma 3.1. Theorem 6.1 Let T be an S-H fuzzy partition of a nonempty set X. Then the fuzzy relation X/T is a fuzzy equivalence relation on X. Proof  ∈ T and μ (x) = 1 by Definition 6.1 (a) (X/T is reflexive) μX/T (x, x) = μA (x) where A A  which implies that μ (x, x) = 1, and so X/T is reflexive on X. X/T

 E  ∈ T such that μ (d) = (b) (X/T is symmetric) Let d, e ∈ X, then by Lemma 3.1, ∃ D, D     μE (e) = 1. Now using part(c) of Definition 3.2 and taking A  = D and  B = E, we have μD∩E (e) ≤ μE (d). But μD∩E (e) = min μD (e), μE (e) = min μD (e), 1 = =E  and B  = D,  μD (e). And so μD (e) ≤ μE (d). In the same way and by taking A we have μE (d) ≤ μD (e) which implies that μE (d) = μD (e). But by Definition 6.1, we have μE (d) = μX/T (e, d) and μD (e) = μX/T (d, e) and so μX/T (e, d) = μX/T (d, e). Thus X/T is symmetric on X.  E,  F ∈ T such that (c) (X/T is transitive) Let d, e, f ∈ X, then by Lemma 3.1 ∃ D, =D  μD (d) = μE (e) = μF (f ) = 1. Now using part(c) of Definition 3.2 and taking A    = E,  we have μ and C D∩E (f ) ≤ μE (d). Thus min μD (f ), μE (f ) ≤ μE (d). But

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  by part(b) of this theorem, μE (f ) = μF (e). And so min μD (f ), μF (e) ≤ μE (d). Now using Definition 6.1, we have μD (f ) = μX/T (d, f ), μF (e) = μX/T (f, e) and

μE (d) = μX/T (e, d). And so min μX/T (d, f ), μX/T (f, e) ≤ μX/T (e, d). Thus X/T is transitive on X. Now from (a),(b) and (c), we conclude that X/T is a fuzzy equivalence relation on X. 2

7

Example on Generating Fuzzy Equivalence Relation



1 , A 2 , A 3 , A 4 , A 5 Example 7.1 Let X = {x0 , x1 , x2 , x3 , x4 , x5 , x6 , x7 , x8 , x9 } and let T = A be a collection of fuzzy subsets of X, where 1 = A {(x0 , 1.0), (x1 , 1.0), (x2 , 1.0), (x3, 1.0), (x4 , 1.0), (x5 , 1.0), (x6 , 0.4), (x7, 0.3), (x8 , 0.2), (x9 , 0.1)} 2 = A {(x0 , 0.4), (x1 , 0.4), (x2 , 0.4), (x3, 0.4), (x4 , 0.4), (x5 , 0.4), (x6 , 1.0), (x7, 0.3), (x8 , 0.2), (x9 , 0.1)} 3 = A {(x0 , 0.3), (x1 , 0.3), (x2 , 0.3), (x3, 0.3), (x4 , 0.3), (x5 , 0.3), (x6 , 0.3), (x7, 1.0), (x8 , 0.2), (x9 , 0.1)} 4 = A {(x0 , 0.2), (x1 , 0.2), (x2 , 0.2), (x3, 0.2), (x4 , 0.2), (x5 , 0.2), (x6 , 0.2), (x7, 0.2), (x8 , 1.0), (x9 , 0.1)} 5 = A {(x0 , 0.1), (x1 , 0.1), (x2 , 0.1), (x3, 0.1), (x4 , 0.1), (x5 , 0.1), (x6 , 0.1), (x7, 0.1), (x8 , 0.1), (x9 , 1.0)} We recognize that μAi ∩Aj (x) < 0.5, where i, j ∈ {1, 2, 3, 4, 5} and i = j and that  i is not weakly empty fuzzy subset ∀ i ∈ {1, 2, 3, 4, 5}. Ai = X. We can also see that A i ∩ A j ≤ A j (x), where A i (x) = 1. Thus T is S-H collection. By the above Note also that A discussion, T forms an S-H partition. Now we are ready to generate a fuzzy equivalence relation by using Definition 6.1 to get the fuzzy equivalence relation X/T. i ∈ T such that A i (x0 ) = 1. But For example, to compute μR (x0 , x5 ), we search for A 1 (x5 ) = 1. In the same way we can compute 1 , thus μ (x0 , x5 ) = A this is true for A R μR (xi , xj ) ∀ i, j ∈ {1, 2, 3, 4, 5} and hence, we can get the fuzzy equivalence relation X/T as in Table 2.

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x0 x1 x2 x3 x4 x5 x6 x7 x8 x9

x0 1 1 1 1 1 1 0.4 0.3 0.2 0.1

x1 1 1 1 1 1 1 0.4 0.3 0.2 0.1

x2 1 1 1 1 1 1 0.4 0.3 0.2 0.1

x3 1 1 1 1 1 1 0.4 0.3 0.2 0.1

x4 1 1 1 1 1 1 0.4 0.3 0.2 0.1

x5 1 1 1 1 1 1 0.4 0.3 0.2 0.1

x6 0.4 0.4 0.4 0.4 0.4 0.4 1 0.3 0.2 0.1

x7 0.3 0.3 0.3 0.3 0.3 0.3 0.3 1 0.2 0.1

x8 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 1 0.1

x9 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 1

Table 2.

8

General Example on S-H Fuzzy Partition

The next theorem demonstrates a general way of generating S-H fuzzy partitions from ordinary partitions and decreasing sequences whose terms belong to the interval [0, 0.5). This theorem also can be used in proving that the above example is indeed forms an S-H partition. Theorem 8.1 Let X be a nonempty set, {Ai : i ∈ N} be an ordinary partition of

X and ∞ {αi }i=0 be a decreasing sequence such that αi ∈ [0, 0.5). If T := Tk : k ∈ N is a collection of fuzzy sets, given by ⎧  ⎨ αk if k > s , 1 if k = s , 1 if k = s , = Tk (x) = min {αk , αs } if k = s . ⎩ αs if k < s . where s ∈ N such that x ∈ As , then T is S-H fuzzy partition. Proof First, note that Tk (x) = 1 ⇔ x ∈ Ak .

(1)

T is a collection of nonempty fuzzy subsets since Tk (x) = 1 ∀ x ∈ Ak . Moreover T is a collection of non weakly empty fuzzy subsets. We are to show that whenever i, j ∈ N such that i = j, we have ⎧ ⎨ αs if i < j ≤ s or j < i ≤ s ,   αj if i < j and j > s, (Ti ∩ Tj )(x) = ⎩ αi if i > j and i > s. where s ∈ N such that x ∈ As .

(2)

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Case 1 (i < j ≤ s or j < i ≤ s) Here we have also two cases. Case (a) (j = s or i = s) = 1. But i < j, Suppose that j = s. Now by the definition of Tk , we have Tj (x) so i < s. This implies that Ti (x) = αs . Also (Ti ∩ Tj )(x) = min Ti (x), Tj (x) . Thus (Ti ∩ Tj )(x) = αs . Similarly when i = s. Case (b) (j < s or i < s)

     Using the definition

of Tk , we have Ti (x) = Tj (x) = αs . But (Ti ∩ Tj )(x) = min Ti (x), Tj (x) . And (Ti ∩ Tj )(x) = αs .

Case 2 (j > i and j > s) Using the definition of Tk , we have Tj (x) = αj . Also we have Ti (x) = 1, αs or αi . By ∞ sequence, we have αi ≥ αj and αs ≥ αj . using the fact that {αi } i=0 is a decreasing

But (Ti ∩ Tj )(x) = min Ti (x), Tj (x) , so (Ti ∩ Tj )(x) = αj . Case 3 (i > j and i > s) Similarly to Case 2. Now if Tn (y) = 1, then by ⎧ ⎨ αm 1 Tm (y) = ⎩ αn

(1) we have y ∈ An . So for any m ∈ N we have if n < m , if m = n , if n > m .

(3)

Now we want to prove that Tn ∩ Tm ≤ Tm (y). Case 1 (n = m) We have Tn ∩ Tm = Tn , Tn ≤ 1 and Tn (y) = Tm (y). But Tn (y) = 1, and so Tn ∩ Tm ≤ Tm (y). Case 2 (n > m) Using (2), we have Tn ∩ Tm = αn or αs , where m < n ≤ s. But by (3), Tm (y) = αn .    Now using the fact that {αi }∞ i=0 is a decreasing sequence, we have Tn ∩ Tm ≤ Tm (y). Case 3 (n < m) Using (2), we have Tn ∩ Tm = αm or αs , where n < m ≤ s. But by (3), Tm (y) = αm .    Now using the fact that {αi }∞ i=0 is a decreasing sequence, we have Tn ∩ Tm ≤ Tm (y).

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So T is an S-H fuzzy collection. Now from (2) and that αi ∈ [0, 0.5), we can conclude that i ∩ Tj < 0.5. Note also that (∞ Tk )(x) = whenever i, j ∈ N such that i =  j, we have T k=1

∞    Tk = X. So T is an supk Tk (x) = Ts (x) = 1 where s ∈ N such that x ∈ As , i.e, k=1

S-H fuzzy partition.

2

Remark 8.1 One can generate the fuzzy equivalence relation, X/T, which is associated with the above S-H fuzzy partition, T, using the Definition 6.1 as follows: k (y), where x ∈ Ak Tk (y), where  Tk ∈ T and  Tk (x) = 1 ⇐⇒ μX/T (x, y) = T μX/T (x, y) =  ⎧ ⎨ αk if s < k , 1 if s = k , ⇐⇒ μX/T (x, y) = ⎩ αs if s > k . where y ∈ As and x ∈ Ak  1 if s = k , ⇐⇒ μX/T (x, y) = min {αk , αs } if s = k . where y ∈ As and x ∈ Ak So we can guess that when we start with any function f : X −→ [0, 1], the following fuzzy  becomes a fuzzy equivalence relation. relation, R,  1 if f (x) = f (y) , μR (x, y) = min {f (x), f (y)} if f (x) = f (y) . Example 8.1 Now we are ready to make an S-H fuzzy partition on the unit circle given by S = {(x, y) ∈ R2 : x2 + y 2 ≤ 1}. This can be done by using the ordinary partition of S

 2 2 2 given by {Ai : i ∈ N}, where Ai = (x, y) ∈ R : 1/(i + 1) < x + y ≤ 1/i , and also i+1 using the sequence given by {αi }∞ . Note that this is a decreasing i=0 , where αi = 1/2 sequence whose terms belong to the interval [0, 0.5). So by the above theorem, T := {Tk : k ∈ N} given by ⎧  ⎨ 1/2i+1 if i > s , 1 if i = s , 1 if i = s , = Ti (x) = i+1 s+1 min {1/2 , 1/2 } if k = s . ⎩ 1/2s+1 if i < s .

where s ∈ N such that x ∈ As , forms an S-H fuzzy partition.

Acknowledgments. The authors are grateful to Miss Saja Hayajneh for carefully reading an earlier draft and making several corrections and suggestions.

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