International Journal of Applied Mathematics

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The fuzzy set theory was introduced by Zedeh (1965) in [6] in order to provide a scheme for handling a variety of problems in which a fundamental role is.
International Journal of Applied Mathematics ————————————————————– Volume 22 No. 4 2009, 581-587

SOME RESULTS ON FUZZY EQUIVALENCE CLASSES M.A. Shakhatreh1 § , M.A. Hayajneh2 Department of Mathematics Yarmouk University Irbid, JORDAN 1 e-mail: [email protected] 2 e-mail: [email protected] Abstract: In this paper we make a useful simplification in Definition 2.5 of fuzzy equivalence class that introduced in [1]. We give further characterizations of the equality of any two fuzzy equivalence classes and summarize them. We also give some useful properties of the set of all fuzzy equivalence classes. AMS Subject Classification: 03E72, 46S40, 26E50 Key Words: fuzzy sets, fuzzy equivalence class, fuzzy equivalence relation, fuzzy partition

1. Introduction The fuzzy set theory was introduced by Zedeh (1965) in [6] in order to provide a scheme for handling a variety of problems in which a fundamental role is played by an indefiniteness arising from a sort of intrinsic ambiguity. The fuzzy equivalence classes and fuzzy partitions play a major role in many topics in fuzzy theory, such as fuzzy measure, fuzzy integration and fuzzy algebra, see e.g. [2] and [3]. In [1], 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 [2]. In this article, we have made a useful simplification in Definition 2.5 so that this definition has been rewritten in more effective way giving us the ability to introduce some new properties of the set of all fuzzy equivalence classes which is given in Theorem 3.1. Received:

February 25, 2009

§ Correspondence

author

c 2009 Academic Publications 

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M.A. Shakhatreh, M.A. Hayajneh 2. Definition and Preliminaries

Now, we will state some definitions and theorems which are given in [2], [4], [5], and [1]. Definition 2.1. (see [4]) Let F be a collection of fuzzy subsets of a B  ∈ F with A  = B.  nonempty set X and A,   If μA∩  B  (x) < 0.5, ∀x ∈ X, then A and B are called Weak-Separated fuzzy subsets.  be a fuzzy relation on a nonempty set X. Definition 2.2. (see [5]) Let R  is reflexive on X iff μ  (x, x) = 1, ∀(x, x) ∈ X × X. (a) R R  (b) R is symmetric on X iff μR (x, y) = μR (y, x), ∀(x, y), (y, x) ∈ X × X.     is transitive on X iff μ  (x, z) ≥ maxy min μ  (x, y), μ  (y, z) , (c) R R R R ∀(x, z), (x, y), (y, z) ∈ X × X.

Definition 2.3. (see [2]) Let X be a nonempty set. By a fuzzy partition  T of X is a set of nonempty fuzzy subsets of X such that B  ∈ T and A  = B,  then μ   (x) < 0.5. (a) If A, A∩B  w  = X. (b) w∈  T  be a fuzzy relation on a nonempty set X Definition 2.4. (see [1]) Let R and x ∈ X. Then   B(x) = y ∈ X : μR (y, x) ≥ 0.5 is called the set of all elements which has strong bound with x.  be a fuzzy equivalence relation on a Definition 2.5. (see [1]) Let R nonempty set X and x ∈ X. Then    = (y, μ (y)) : y ∈ X , where (a) [x]  [x] 1   if y ∈ B(x) , μ[x]  (y) = / B(x) . min μR (z, y) : z ∈ B(x) if y ∈ 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 fuzzy subset Definition 2.6. (see [2]) Let X be a nonempty set and A  of X, then we call A weakly empty fuzzy subset of X if μA (x) < 0.5 ∀ x ∈ X.

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 be a fuzzy equivalence relation on a Theorem 2.1. (see [1]) Let R nonempty set X and x, y ∈ X. Then  has an element with degree 1, more over μ (x) = 1. (a) Each [x]  [x]

 = [y]  ⇐⇒ B(x) = B(y). (b) μR (x, y) ≥ 0.5 ⇐⇒ [x] (c) μR (x, y) < 0.5 ⇐⇒ μ[x]∩  [y]  (w) < 0.5 ∀w ∈ X ⇐⇒ B(x) ∩ B(y) = φ.

3. Main Results In this section we will state and prove a useful lemma that will be used frequently through this paper. Specifically, using this lemma, we have made a useful simplification in Definition 2.5 so that this definition has been rewritten in more effective way giving us the ability to introduce some new properties of the set of all fuzzy equivalence classes defined in [1] which is given in Theorem 3.1.  be a fuzzy equivalence relation on a nonempty set X Lemma 3.1. Let R and x, y, z ∈ X. If B(x) = B(y) = B(z), then μR (z, x) = μR (z, y).  is a fuzzy equivalence relation, it follows that μ  (z, x) ≥ Proof. Since R R     min μR (z, y), μR (y, x) and μR (z, y) ≥ min μR (z, x), μR (x, y) . But B(x) = B(y) = B(z), so using part (b) of Theorem 2.1, we have μR (x, y) ≥ 0.5, μR (z, x) < 0.5 and μR (z, y) < 0.5 which implies that μR (z, x) ≥ μR (z, y) and  μR (z, y) ≥ μR (z, x) and hence μR (z, x) = μR (z, y).  be a fuzzy equivalence relation on a nonempty set Corollary 3.1. Let R   X and x, y ∈ X. If y ∈ / B(x) then, min μR (z, y) : z ∈ B(x) = μR (x, y). Proof. Suppose that y ∈ / B(x) and z ∈ B(x). Then: y∈ / B(x) and z ∈ B(x) =⇒ B(x) = B(z) = B(y) =⇒ μR (y, x) = μR (y, z)     =⇒ min μR (z, y) : z ∈ B(x) = min μR (y, x) : z ∈ B(x)   =⇒ min μR (z, y) : z ∈ B(x) = μR (y, x) .  Now, we are ready to set Definition 2.5 in more effective way.  be a fuzzy equivalence relation on a nonempty set Definition 3.1. Let R

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   = (y, μ (y)) : y ∈ X , where X and x ∈ X. Then [x]  [x] 1 if y ∈ B(x) , μ[x]  (y) = / B(x) . μR (x, y) if y ∈ is called the fuzzy equivalence class determined by x. Now, we will introduce some new properties of the set of all fuzzy equivalence classes defined in [1] where the first property gives us a new characterization of the equality of any two fuzzy equivalence classes and the second property, which we may call it the symmetric property in the sense that the degree of the element in a specific class equal the degree of the representative element of that class in the class which the first element represents it.  be a fuzzy equivalence relation on a nonempty set Theorem 3.1. Let R

X, x, y, z ∈ X and [X] be the set of all fuzzy equivalence classes. Then  = [y]  ⇐⇒ ∃w ∈ X such that μ (w) = μ (w) = 1. (a) [x]   [x] [y] (b) μ[x]  (y) = μ[y]  (x). (c) B(x) = B(y) ⇐⇒ μ[z]  (x) = μ[z]  (y) ∀z ∈ X. Proof. (a)  = [y]  ⇐⇒ μ (x, y) ≥ 0.5 [x]  R

by part (b) of Theorem 2.1

⇐⇒ ∃w ∈ X such that μ[x]∩  [y]  (w) ≥ 0.5

by part (c) of Theorem 2.1   (w), μ (w) ≥ 0.5 ⇐⇒ ∃w ∈ X such that min μ[x]   [y] ⇐⇒ ∃w ∈ X such that μ[x]  (w) ≥ 0.5 and μ[y]  (w) ≥ 0.5 ⇐⇒ ∃w ∈ X such that μ[x]  (w) = μ[y]  (w) = 1

by Definition 3.1.

(b) Case 1. B(x) = B(y). Since B(x) = B(y), it follows that y ∈ B(x) and x ∈ B(y). This implies that μ[y]  (x) = μ[x]  (y) = 1 by Definition 3.1. Case 2. B(x) = B(y) Since B(x) = B(y), it follows y ∈ / B(x) and x ∈ / B(y). This implies that μ[y]  (x, y) = μ[x]  (x) = μR  (y) by Definition 3.1. (c) (⇒) Case 1. B(x) = B(y) = B(z). Since B(x) = B(y) = B(z), it follows that x ∈ B(z) and y ∈ B(z) which implies that μ[z]  (x) = μ[z]  (y) = 1 by Definition 3.1.

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Case 2. B(x) = B(y) = B(z) Since B(x) = B(y) = B(z), it follows that x ∈ / B(z) and y ∈ / B(z) by (x) = μ (x, z) and μ[z] part(b) of Theorem 2.1. This implies that μ[z]    (y) = R μR (y, z) by Definition 3.1. But since B(x) = B(y) = B(z), it follows that μR (y, z) = μR (x, z) by Lemma 3.1 and so μ[z]  (x) = μ[z]  (y). (⇐) Suppose that μ[z]  (x) = μ[z]  (y) ∀z ∈ X. By taking z = x, we get μ[x]  (x, y) ≥ 0.5 by Definition  (y) = μ[x]  (x) = 1. But this holds if and only if μR 3.1. Thus B(x) = B(y).  The next theorem states that the degree of any element in the intersection of any two fuzzy equivalence classes is less than or equal the degree of the representative element of one of these two fuzzy equivalence classes in the other class.  be a fuzzy equivalence relation on a nonempty Theorem 3.2. Let R

set X and [X] be the set of all fuzzy equivalence classes. Then μ (y) ≤  [z]  [x]∩

μ[z]  (x) ∀ x, y, z ∈ X.  = [z].  Proof. Case 1. [x]

 = [z]  ⇒ μ (x, z) ≥ 0.5 ⇒ x ∈ B(z) ⇒ μ[z] [x]   (x) = 1 R   ⇒ min μ[x]  (y), μ[z]  (y) ≤ 1 = μ[z]  (x) ⇒ μ[x]∩  [z]  (y) ≤ μ[z]  (x). =  Case 2. [x]  [z]  = [z]  ⇒ μ (x, z) < 0.5 [x]  R

by part (b) of Theorem 2.1

⇒x∈ / B(z)

by Definition 2.4 ⇒ μ[z]  (x, z) by Definition 3.1. (1)  (x) = μR

Case 1. y ∈ B(x) y ∈ B(x) ⇒ μR (x, y) ≥ 0.5

by Definition 2.4

⇒ μ[x]  (y) = 1 and B(x) = B(y)

by Definition 3.1 and part (b) of Theorem 2.1   (y), μ (y) = μ[z] ⇒ min μ[x]    (y) and μ[z]  (y) ≤ μ[z]  (x) [z] by part (c) of Theorem 3.1   (y), μ (y) ≤ μ[z] ⇒ min μ[x]    (x). ⇒ μ[x]∩  [z]  (y) ≤ μ[z]  (x). [z]

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M.A. Shakhatreh, M.A. Hayajneh Case 2. y ∈ B(z).

y ∈ B(z) ⇒ μR (z, y) ≥ 0.5

by Definition 2.4   ⇒ μ[z]  (y) = 1 and [y] = [z]

by Definition 3.1 and part (b) of Theorem 2.1  ⇒ min μ[x]  (y), μ[z]  (y) = μ[x]  (y) and μ[x]  (y) ≤ μ[y]  (x) = μ[z]  (x) 

by part (b) of Theorem 3.1   (y), μ (y) ≤ μ[z] ⇒ min μ[x]    (x). ⇒ μ[x]∩  [z]  (y) ≤ μ[z]  (x). [z] Case 3. y ∈ / B(z) and y ∈ / B(x) y∈ / B(z) and y ∈ / B(x) ⇒ μ[z]  (y, z) and μ[x]  (x, y)  (y) = μR  (y) = μR by Definition 3.1 .

 is a fuzzy equivalence relation, we have But since R     μR (x, z) ≥ min μR (x, y), μR (y, z) ⇐⇒ min μ[x] (y), μ (y) ≤ μ[z]    (x) [z] by (1) ⇐⇒ μ[x]∩  [z]  (y) ≤ μ[z]  (x). 

4. Summary  Remark 4.1. 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. Now by the above remark we can simplifiy Theorem 3.2 and Part(b) of Theorem 3.1 as follows.  be a fuzzy equivalence relation on a nonempty set Theorem 4.1. Let R

be the set of all fuzzy equivalence classes. Then X and [X]   (a) [x](y) = [y](x).  ∩ [y]  ≤ [x](y).  (b) [x] Now we are ready to summarize some characterizations of the equality of any two fuzzy equivalence classes that introduced in this paper.  be a fuzzy equivalence relation on a nonempty set Theorem 4.2. Let R

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X and x, y ∈ X. Then the following are equivalent:  = [y].  (a) [x] (b) μR (x, y) ≥ 0.5. (c) B(x) = B(y).

(d) ∃ w ∈ X such that w ∈ B(x) ∩ B(y).  ∩ [y](w)  (e) ∃ w ∈ X such that [x] ≥ 0.5.   (f) ∃ w ∈ X such that [x](w) = [y](w) = 1.   (g) [w](x) = [w](y) ∀ w ∈ X.   (h) [x](w) = [y](w) ∀ w ∈ X.

Acknowledgments The authors are grateful to Miss Saja Hayajneh for carefully reading an earlier draft and making several corrections and suggestions. References [1] M.A. Shakhatreh, M.A. Hayajneh, Fuzzy equivalence classes and fuzzy partitions, International Journal of Applied Mathematics, 21, No. 4 (2008), 607-613. [2] M.A. Shakhatreh, A.A. Fora, M.M. Jaradat, W.A. Shatanawi, Partition and fuzzy partition, International Journal of Applied Mathematics, 20, No.1 (2007), 57-66. [3] M.A. Shakhatreh, W.A. Shatanawi, M.M. Jaradat, A.A. Fora, On fuzzy partitions, International Journal of Pure and Applied Mathematics, 30, No. 4 (2006), 467-474. [4] M.A. Shakhatreh, W-separated fuzzy subsets, Jour. Inst. Math and Comp. Sci. (Math. Ser.), 7, No. 3 (1994), 213-216. [5] A. Kaufmann, Theory of Fuzzy Subsets, Volume 1, Academic Press (1975). [6] L.A. Zedeh, Fuzzy sets, Inform. and Control, 8 (1965), 338-353.

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