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Original Article
Fuzzy soft connected sets in fuzzy soft topological spaces II
A. Kandila, O.A. El-Tantawyb, S.A. El-Sheikhc, Sawsan S.S. El-Sayedc'*
a Mathematics Department, Faculty of Science, Helwan University, Helwan, Egypt b Mathematics Department, Faculty of Science, Zagazig University, Zagazig, Egypt c Mathematics Department, Faculty of Education, Ain Shams University, Cairo, Egypt
A R T I C L E I N F 0
Article history: Received 18 October 2016 Revised 27 December 2016 Accepted 8 January 2017 Available online xxx
Keywords:
Fuzzy soft sets
Fuzzy soft topological space
Fuzzy soft separated sets
Fuzzy soft connected sets
Fuzzy soft connected components
A B S T R A C T
In this paper, we introduce some different types of fuzzy soft connected components related to the different types of fuzzy soft connectedness and based on an equivalence relation defined on the set of fuzzy soft points of X. We have investigated some very interesting properties for fuzzy soft connected components. We show that the fuzzy soft C.-connected component may be not exists and if it exists, it may not be fuzzy soft closed set. Also, we introduced some very interesting properties for fuzzy soft connected components in discrete fuzzy soft topological spaces which is a departure from the general topology.
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1. Introduction
The concept of a fuzzy set was introduced by Zadeh [15] in his classical paper of 1965. In 1968, Chang [2] gave the definition of fuzzy topology. Since Chang applied fuzzy set theory into topology many topological notions were investigated in a fuzzy setting.
In 1999, the Russian researcher Molodtsov [9] introduced the soft set theory which is a completely new approach for modeling uncertainty. He established the fundamental results of this new theory and successfully applied the soft set theory into several directions. Maji et al. [8] defined and studied several basic notions of soft set theory in 2003. Shabir and Naz [12] introduced the concept of soft topological space.
Maji et al. [7] initiated the study involving both fuzzy sets and soft sets. In this paper, Maji et al. combined fuzzy sets and soft sets and introduced the concept of fuzzy soft sets. In 2011, Tanay Kandemir [14] gave the topological structure of fuzzy soft sets.
The notions of fuzzy soft connected sets and fuzzy soft connected components are very important in fuzzy soft topological spaces which in turn reflect the intrinsic nature of it that is in fact its peculiarity. In fuzzy soft setting, connectedness has been introduced by Mahanta and Das [6] and Karatas et al. [5]. Recently, Kandil et al. [4] introduced some types of separated sets and some types of connected sets. They studied the relationship between these types.
* Corresponding author. E-mail addresses: sawsan_809@yahoo.com, s.elsayed@mu.edu.sa (S.S.S. El-Sayed).
In this paper, we extend the notion of connected components of fuzzy topological space to fuzzy soft topological space. In Section 3, we introduce and investigate some very interesting properties for fuzzy soft connected components. We define an equivalence relation on the set of fuzzy soft points. The union of equivalence classes turns out to be a maximal fuzzy soft connected set which is called a fuzzy soft connected component. There are many types of connected components deduced from the many types of connected sets due to Kandil et al. [4]. Furthermore, we show that some of these connected components may be not exists and the some if exists, it may not be fuzzy soft closed set. Moreover, we introduced some very interesting properties for fuzzy soft connected components in discrete fuzzy soft topological spaces which is a departure from the general topology.
2. Preliminaries
Throughout this paper X denotes initial universe, E denotes the set of all possible parameters which are attributes, characteristic or properties of the objects in X. In this section, we present the basic definitions and results of fuzzy soft set theory which will be needed in the sequel.
Definition 2.1. [2] A fuzzy set A of a non-empty set X is characterized by a membership function ¡iA : X —> [0 . 1] = I whose value ¡iA(x ) represents the "degree of membership" of x in A for x e X. Let ¡X denotes the family of all fuzzy sets on X.
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Definition 2.2. [9] Let A be a non-empty subset of E. A pair (F, A) denoted by FA is called a soft set over X , where F is a mapping given by F: A — P(X). In other words, a soft set over X is a parametrized family of subsets of the universe X . For a particular e e A , F(e) may be considered the set of e-approximate elements of the soft set (F, A) and if e gA. then F(e) = <p i.e. F = {F (e) : e e A c E, F : A — P(X)}.
Aktas and Qagman [1] showed that every fuzzy set may be considered as a soft set. That is, fuzzy sets are a special class of soft sets.
Definition 2.3. [7] Let AcE. A pair (f A), denoted by fA, is called fuzzy soft set over X , where f is a mapping given by f : A —> Ix defined by fA (e) = if ; where if. = 0 if e eA. and ie. = 0 if e e
A. where 0 (x) = 0V x e X . The family of all these fuzzy soft sets over X denoted by FSS(X)E .
Definition 2.4. [3,7,10,11,13,14] The complement of a fuzzy soft set (. A) , denoted by (. A)c , and defined by (. A)c = (fc,A) , ff : A —> IX is a mapping given by ifc = 1 - iej Ve e A. Clearly,
(fA )c = fA . A f
Definition 2.5. [7,10,11,13,14] A fuzzy soft set fE over X is said to be a null-fuzzy soft set, denoted by 0E , if for all e e E. fE (e) = 0.
Definition 2.6. [7,10,11,13,14] A fuzzy soft set fE over X is said to be an absolute fuzzy soft set, denoted by 1E , if fE (e) = 1 Ve e E. Clearly, we have (0E ). = 1E and (1E )c = 0E.
Definition 2.7. [3,7,10,11,13,14] Let f A, gB e FSS(X)E. Then fA is fuzzy
soft subset of gB , denoted by fA ç
if AçB and ae, (x) <
ig. (x.Vx e X, Ve e E. Also, gB is called fuzzy soft superset of fA denoted by gB 5 fA. If fA is not fuzzy soft subset of gB, we written as
fA 5 gB .
Definition 2.13. [10,11] Let (X, t, E) be a fuzzy soft topological space and fA e FSS(X)E. The fuzzy soft closure of fA, denoted by Fcl(fA ) is the intersection of all fuzzy soft closed supersets of fA , i.e. Fcl(fA) = n{hC;hC e tc and fA c hC}. Clearly, Fcl(fA) is the smallest fuzzy soft closed set over X which contains fA, and Fcl(fA ) is fuzzy soft closed set.
Definition 2.14. [11,13] The fuzzy soft set fA e FSS(X)E is called fuzzy soft point if there exist x e X and e e E such that if (x) = a; (0 < a < 1) and u,er (y) = 0 Vy e X - {x} and this fuzzy soft point
is denoted by xe or fe . The class of all fuzzy soft points of X, de-
noted by FSP{X)e .
Definition 2.15. [6] The fuzzy soft point x% is said to be belonging to the fuzzy soft set fA, denoted by xea e fA, if for the element e e A. a < ief (x) . If x% is not belong to fA, we write x% e fA and implies that a > if. (x).
Definition 2.16. [11,13] A fuzzy soft point xea is said to be a quasi-coincident with a fuzzy soft set fA , denoted by x% q fA, if a + if (x) > 1. Otherwise, xea is non-quasi-coincident with fA and de-
Definition 2.17. [11,13] A fuzzy soft set fA is said to be quasi-coincident with gB, denoted by fA q gB, if there exists x e X such that if. (x) + ig. (x) > 1. for some e e A n B. If this is true we can say that fA and gB are quasi-coincident at x. Otherwise, fA and gB are not quasi-coincident and denoted by fA q gB.
Proposition 2.1. [11, 13] Let fA and gB be two fuzzy soft sets. Then, Ja C gB if and only if Ja q (gB T. In particular, x% e fA if and only if
xea q (Ja )..
Definition 2.8. [3,7,10,11,13,14] Two fuzzy soft sets fA and gB on X are called equal if fA c gB and gB c fA.
Definition 2.9. [7,10,11,13,14] The union of two fuzzy soft sets fA and gB over the common universe X. denoted by fA ugB , is also a fuzzy soft set hC , where C = A U B and for all e e C, hC (e) = ih =
f v ßeg/Ve
Definition 2.10. [7,10,11,13,14] The intersection of two fuzzy soft sets fA and gB over the common universe X. denoted by fA ngB , is also a fuzzy soft set hC , where C = A n B and for all e e C, hC (e ) =
ih. = if.A ig.Ve e E.
Definition 2.11. [14] Let FSS(X)E be a collection of fuzzy soft sets over a universe X with a fixed set of parameters E. Then t cFSS(X)E is called fuzzy soft topology on X if
Definition 2.18. [10] Let FSS(X)E and FSS(Y)K be families of fuzzy soft sets over X and Y. respectively. Let u : X —> Y and p : E —> K be mappings. Then the map fpu is called fuzzy soft mapping from FSS(X)E to FSS(Y)K, denoted by fpu: FSS(X)E FSS(Y)K. such that:
1. If gB e FSS(X)E, then the image of gB under the fuzzy soft mapping fpu is a fuzzy soft set over Y defined by fpu( gB ) where Vk e p(E), Vy e Y.
fpu (gB)(k/(y/ = v [ v (gB (e))](x ) if x e u
u (x) =y p(e) =k
2. If hC e FSS(Y)K, then the pre-image of hC under the fuzzy soft mapping fpu, fpu (hC) is a fuzzy soft set over X defined by Ve e
p-1 (K). Vx e X.
f(he)(e)(x) = he(p(e))(u(x))
for p(e/ e C,0
1. 0E , 1E e t, where 0E (e) = 0 and 1E (e) = 1Ve e E,
2. The union of any members of t belongs to t.
3. The intersection of any two members of t belongs to t.
The triplet (X, t, E) is called fuzzy soft topological space over X. Also, each member of t is called fuzzy soft open set in (X, t, E).
Definition 2.12. [14] Let (X, t, E) be a fuzzy soft topological space. A fuzzy soft set fA over X is said to be fuzzy soft closed set in X. if its relative complement fAc is fuzzy soft open set.
Definition 2.19. [10] The fuzzy soft mapping fpu is called surjective (resp. injective) if p and u are surjective (resp. injective), also f pu is said to be constant if p and u are constant.
Definition 2.20. [10] Let (X, t j , E) and (Y, t2 , K) be two fuzzy soft topological spaces and fpu : FSS(X)E —> FSS(Y)K be a fuzzy soft mapping. Then fpu is called:
1. Fuzzy soft continuous if J- (hC) e t1 V hC e t2 .
2. Fuzzy soft open if fpu (gB ) e t 2 V gB e t j .
noted by xe q }a
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Definition 2.21. [5] Two non-null fuzzy soft sets fE and gE are said to be fuzzy soft Q-separated in a fuzzy soft topological space (X, t, E) if Fclf )n gE = fE n Fcl(gE) = 0e.
Definition 2.22. [5] Let (X, t, E) be a fuzzy soft topological space and fE e FSS(X)e. Then, fE is called:
FSC1 -connected: if does not exist two non-null fuzzy soft open sets hE and sE such that fE c hEusE, hEnsE c fc , fE n hE = 0E , and fE n Se = 0e .
FSC2-connected: if does not exist two non-null fuzzy soft open sets hE and sE such that fE c hEusE , fE n hE n sE = 0E, fE n hE = 0E , and fE n se = 0e .
FSC3-connected: if does not exist two non-null fuzzy soft open sets hE and sE such that fE c hEusE, hEnsE c fji, hE ( fE, and sE (
fE ■
FSC4-connected: if does not exist two non-null fuzzy soft open sets hE and sE such that fE c hEusE , fE n hE n sE = 0E, hE ( fji, and
Se C fc ■
Otherwise, f. is called FSC.-disconnected set for i = 1. 2 . 3 . 4 . In the above definition, if we take 1. instead of f., then the fuzzy soft topological space (X, t, E) is called FSC. -connected space (i = 1. 2 . 3 . 4).
Remark 2.1. [5] The relationship between FSC.-connectedness (i = 1 , 2, 3, 4) can be described by the following diagram:
FSC1 FSC 3
F SC 2
Definition 2.23. [4] Two non-null fuzzy soft sets fE and gE are said to be:
1. Weakly separated sets in a fuzzy soft topological space (X, t, E) if FcI(Ee) q gE and fE q Fclg).
2. Separated sets in a fuzzy soft topological space (X, t, E) if there exist non-null fuzzy soft open sets hE and sE such that fE c hE,
gE c sE and fE n sE = gE n hE = c0E .
Definition 2.24. [4] Let fE e FSS(X)E. The support of fE(e), denoted by S(Ee(e)), is the set, S( fE(e)) = {x e XMe)(x) > 0}.
Definition 2.25. [4] Two fuzzy soft sets fE and gE are said to be quasi-coincident with respect to fE if if (x) + /g. (x) > 1 for every x e S(/e( e ) ). E
Definition 2.26. [4] Two non-null fuzzy soft sets fE and gE are said to be fuzzy soft strongly separated in a fuzzy soft topological space (X, t, E) if there exist hE and sE e t such that fE c hE, gE c sE, fE n sE = gE n hE = 0E, fE, hE are fuzzy soft quasi-coincident with respect to fE , and gE , sE are fuzzy soft quasi-coincident with respect to gE .
Remark 2.2. [4] In fuzzy soft topological space (X, t, E) the relationship between different notions of fuzzy soft separated sets can be described by the following diagram.
fuzzy soft strongly separated
fuzzy soft separated fuzzy soft weakly separated
fuzzy soft Q-separated
1. FSCM-disconnected set if there exist two non-null fuzzy soft Q-separated sets hE, sE in X such that fE = hE u sE. Otherwise, fE is called FSCM-connected set.
2. FSCS-disconnected set if there exist two non-null fuzzy soft weakly-separated sets hE , sE in X such that fE = hE u sE . Otherwise, fE is called FSCS-connected set.
3. FSO-disconnected (respectively, FSOq-disconnected) set if there exist two non-null fuzzy soft separated (respectively, strongly separated) sets hE, sE in X such that fE = hE u sE . Otherwise, fE is called FSO-connected (respectively, FSOq-connected) set.
4. FSC5-connected set in X if there does not exist any non-null proper fuzzy soft clopen set in (.E , tf., E). Note that, this kind of fuzzy soft connectedness was studied by Mahanta and Das [6], Shabir and Naz [12].
In the above definitions, if we take c1 E instead of fE , then the fuzzy soft topological space (X, t, E) is called FSCM-connected (respectively, FSCS-connected, FSO-connected, FSOq-connected, FSC5 -connected) space.
Remark 2.3. [4] In a fuzzy soft topological space (X, t, E). The classes of FSO-connected, FSOq-connected, and FSCt-connected sets for i = 1, 2 . 3 . 4 . S, M can be described by the following diagram.
F SC M
F SC 1 ; F SC S
; FSOq
F SC 2
Definition 2.27. [4] A fuzzy soft set fE in a fuzzy soft topological space (X, t, E) is called:
3. Equivalence relations and components
In disconnected fuzzy soft topological space (X, t, E), the universe fuzzy soft set c1 E can be decomposed into several pieces of fuzzy soft sets, each of which is connected. As in general topologi-cal space, the whole space is decomposed into components.
In fuzzy soft setting, this decomposition is obtained in form of unions of equivalence classes of a certain equivalence relation, defined on the set of fuzzy soft points in X. The union of equivalence classes turns out to be a maximal fuzzy soft connected set. Accordingly, we have many types of notions of components in fuzzy soft setting.
Proposition 3.1. For fuzzy soft points xOE and yeE in X define a relation Ej as follows:
Ej = { (xeof, ye.). x^E , ye/E e FSP(X)e and there exists a FSC. -connected set fA such that x^E s fA and y^ e fA for i = 1 , 2, S, M, O, Oq }
Then, Ej is an equivalence relation on FSP(X)E.
Proof. As a sample we will prove the case of i = 1. Reflexivity follows from the fact that for each fuzzy soft point xea in X. there exists a fuzzy soft point xet in X. which is a FSC^ -connected and obviously contains xea . Symmetry is obvious. To show transitivity, let x°E, ye. and zY. be fuzzy soft points in X such that (x°E, y6.) e E1
and (ye., J*) e E1 . Then, there exist FSC^ -connected sets fA and gB in X such that xa e.A , y. e fA and y. e gB , zeyE s gB. Therefore, . < if (y) and . < /gjj (y). Hence, fA n gB = 0e . So by Theorem 4.10 in [9], fAugB is a FSC^ -connected. Also, we have xO. s fAugB and zO3 e fA ugB. Therefore, E^ is an equivalence relation. Similarly, E . is an equivalence relation for i = 2, S, M, O, Oq .
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Let xsa be a fuzzy soft point in X and Ej be the equivalence relation on FSP(X)E , described as above. Then the equivalence class determined by x% , is denoted by E. (xea) for i = 1. 2, S, M, O, Oq. □
Definition 3.1. The union u E. (xea) of all fuzzy soft points contained in the equivalence class Ei (xe ) is called a Ci -component of the universe fuzzy soft set 1E , determined by x%. We denoted it by C. (xea) for i = 1. 2, S, M, O, Oq.
Theorem 3.1. For each fuzzy soft point x<sa s FSP(X)E, the component C. (xea) is the maximal FSC.-connected (respectively, FSO-connected, FSOq-connected) set in X containing x% for i = 1. 2, S, M.
Proof. As a sample we will prove the case of C1 (xe ). Let xe ce FSP(X)E and let {(fA ) i ; i e I} be the family of FSCj -connected sets in X. containing xea. We claim Q (xea) = u (fA) . .
Firstly, we show that u (fA). c Q (x<sa) . Let y e X, e e E.
ie(.) (y) = fti for each i e I and sup fti = ft. Then, ie (f ,(y) =
(W. ie. UJ(Ja h
sup Pi = ft.
Now, if ft = 0 . we have nothing to prove. Suppose ft = 0. Then for every real number e > 0, there exists i e I such that if). (y) = fti > ft - e. Therefore, for each fuzzy soft point yft-e where 0 < e < ft, there exists a fuzzy soft set (JA). such that yft-e e (fA). . Since (JA ). is a FSCj -connected set, containing x% , it follows that (xa, yft-e) e Ej and hence yft-e e Ej(xf) for every 0 < e < ft.
Now, let {(yft).; j e J } be the family of all fuzzy soft points in X with support y which are Ej -related to x%. Then {yft-e }0<e<ft C{ (yft).; j e J } c Ej (xea). Therefore, 0<u<ftyft -e c u. (yft). e Ej (xea).
But 0<u<ftyft-e = yft. Hence, yft e uEj (x% ) = Cj (x% ) and so u (Ja)i
c Cj(xa).
Conversely, we show that Q (x% ) c u (fA) .. Let y e X, e e E
and {(yft)j; j e J } be the family of all fuzzy soft points in X with
support y such that (yft). ee Ej (xea). Suppose, supftj = ft . Then,
ft jeJ
ieE (x.) (y ) = ieu (yft). (y ) = ft.
Now, since (yft)j e Ej (xea) , there exists for every j e J a FSCj -connected set (fA). such that xa s (fA). and (yft). s (fA )j . Hence, the family of fuzzy soft sets {(JA ),■ ; j e J }c {(JA ). ; i e
I}. Therefore, sup[iej/) / (y)} = i
j((fA )/ (У ) i ie/((f/)/
(У / . But, ß =
i /1 '
supßj < sup{ie,f. (y)j. Therefore, ß < ieu (f.. (y). Hence, Ci (Xa) c
N j j^f (fA)f iJjA)f
H ( fA )f
That C1 (x'a) is the maximal FSC1 -connected set containing xf., now follows from the fact that, if gB is any FSC1 -connected set in X containing xa, then gB e {(fA ; i e 1} and hence gB c u (fA). =
Theorem 3.2. In a fuzzy soft topological space (X, t, E), the universe fuzzy soft set 1E is the disjoint union of its C. -components for i = 1. 2, S, M, O, O q .
Proof. As a sample we will prove the case of C1 -component. Let {Cj (xea). i e I} be the family of Q -components of 1E in X. Then u Cj (xea) c 1E . Since each fuzzy soft point xj c Ej (xea)
c CI (xea). then 1E C u C\ (xea). Moreover, if two Q -components
1 i e. 1
C1 (xea) and C1 (ytf) are intersecting, then C (xea) u C1 (yß ) is a FSC1 -connected set in X. Hence C1 (xea) and C1 (yß) are identical in view
of Theorem 3.1 □
In analogy with the general topological spaces, in an indiscrete fuzzy soft topological space, 1E is the only C[ -components (C2 -components). In a discrete general topological space, singletons are connected sets and hence components. This feature is too is retained in the fuzzy soft setting but with an interesting departure in the case of FSCj -connectedness, as reflected in the following results.
Theorem 3.3. In a discrete fuzzy soft topological space, the only FSC1 -connected sets are fuzzy soft points with value one.
Proof. Let xe1 be a fuzzy soft point in a discrete fuzzy soft topolog-ical space (X, t, E). Let hC and sD be fuzzy soft open sets in X such that xj s hC usD , h C nsD C (xj )c . Then we have either (ih (x) = 1 and ie$D (x) = 0) or (ih (x) = 0 and ie$D (x) = 1). Therefore, xj n
hC = 0E or xj n sD = 0E. Hence, xj is a FSCj -connected.
Next, to show that each fuzzy soft point xe , where 0 < < 1, has a FSC 1 -disconnection, what is required, is the construction of two fuzzy soft open sets uN and j. in X satisfying x% i uNujL, uNnjL c (xea). and xea n uN = 0E = xea n j.. Now, consider any fuzzy soft sets uN and jL in X such that iuN (x) = max{a, 1 - a} and i j (x) = min{a, 1 - a}. Then, uN and jL are FSCj -disconnection of xea.
Finally, we construct a FSC 1 -disconnection for any fuzzy soft set in X, which is not a fuzzy soft point. Let fA be any fuzzy soft set which takes non-zero values at least at two distinct points y and z in X. Suppose if (y) = a and ie. (z) = ft . Now, define fuzzy soft sets hC and sD , as follows:
ih. (y) = a, ih. (z) = 0 and ih. (x) = ieh (x) Vx e X - {y, z}
ies. (y) = 0 . il. (z) = ft and il. (x) = 1 - if. (x) Vx e X - {y, z} It is clear that, hC and sD form FSCj -disconnection of fA. □
Theorem 3.4. In a discrete fuzzy soft topological space, fuzzy soft points are only FSC2-connected sets.
logical space. Let hC and s D be fuzzy soft open sets in X such that xS c hCusD , xea n hC n sD = 0E. Then, we have either (ih (x) >
a, iS.(x) = 0 ) or ( ih.(x) = 0 . iSD (x) > a). Therefore, xsa n hC = 0E or xe n sD = c0E . Hence, xe is a FSC 2 -connected.
Next, we construct a FSC 2 -disconnection for any fuzzy soft in X, which is not a fuzzy soft point. Let fA be any fuzzy soft set which takes non-zero values at least at two distinct points y and z in X. Suppose is. (y) = ft and if (z) = y. Now, define fuzzy soft sets hC and s D , as follows:
ih. (y) = ft, ih.(z) = 0 and ih. (x) = if. (x) Vx e X - {y, z}
iS. (y) = 0 . il . (z) = Y and if. (x) = 0 Vx e X - {y, z}
It is clear that, hC and sD form FSC2-disconnection of fA. □
Corollary 3.1. In a discrete fuzzy soft topological space, fuzzy soft points with value 1 are the only C[ -components (C2 -components).
Let ft j be the set of all fuzzy soft points in a fuzzy soft topological
space (X, t, E), whose values are greater than j .
Proposition 3.2. For fuzzy soft points xS and yft in ft 1 define a reft 2
lation E* (E4 ) as follows:
E* (E4) = {(xsa, yft). there exists a FSC3-connected (FSC4 -connected) set fA in X such that xe c fA and ytft c fA } Then E3 (E4) is an equivalence relation on ft j .
Proof. Let xe be a fuzzy soft point in discrete fuzzy soft topo-
Ci (Ж* ). □
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Proof. Reflexivity follows from the fact that each fuzzy soft point
x% in . 1_ is a FSC3-connected set and obviously contains x%. Sym-
metry is obvious. To show transitivity, let x°E, ye. and z0. be fuzzy
soft points in . 1 such that (x° , y.) e E* and (ye., z°j) e E* .
Then, there exist FSC3-connected sets fA and gB in X such that x°E B , fy. e gB . Therefore, . < i0. (y) and
e fA, yeß s fA and yeß e
fi < figl (y)■ Hence, fA and gB are overlapping at y. So by Theorem 4.12 in [9], fAugB is a FSC3-connected. Also, we have xeJ e fAugB and zey3 e fA ugB. Therefore, E* is an equivalence relation. Similarly, E4 is an equivalence relation.
Here, the equivalence relation E* (£4 ) partitions the set of fuzzy
soft points fi 1 into equivalence classes. As usual, we shall de-
note an equivalence class containing the fuzzy soft point xea by defined in Proposition 3.2 Let E3 (xea)(E4(xea)) denoted the equiva-
there exist FSC3 -connected sets fA and gB in p such that xea I fA, yß I fA and yß I gB, 4 I gB• Now, two cases arise:
Case I. ß > 1 . Then, the fuzzy soft sets overlap at y. Hence, by Theorem 4.12 in [9], fA u gB is a FSC3 -connected set in p such that xea I fA u gB and zsy I fA ugB . So, x% E3ZY.
Case II. ß < 2 . Choose the C3 -quasicomponent C3 (y\) . which contains y1, hence also yß . Now, the fuzzy soft sets gB and C3 y ) overlap at y. so gB u C3 (yf. ) is a FSC3-connected set. By the same argument fA u gB u C3 (y\) is also FSC3 -connected set in p, containing both xea and zsy . Hence, E3 is an equivalence relation.
Now, we attain the desired objective of decomposing 1E into disjoint, maximal FSC3-connected (FSC4-connected) sets via the equivalence classes defected by the equivalence relation E3 ( E4 ), as
e*(xa)(e*(xa)) . □
lence class containing the fuzzy soft point xea
Definition 3.2. Let xea be a fuzzy soft point in . 1 and {(fA )e ; i e Definition 3.3. The union u E3(xf )(uE4(xf)) of all fuzzy soft
/} be the family of FSC3-connected (FSC4 -connected ) sets in X con- points contained in the equivalence dass E3(x.ea)(E4(x.ea)) is
taining x0a. Then, the union u (fA). is called a C3 -quasicomponent called C3 -component (C4 -component) of 1e , and is denoted by
iel ^ -.-..-.-..
(respectively, C4 -quasicomponent) of 1E containing xf and is denoted by C* (xf) (respectively, C4 (xf)).
C* (xea ) (C4 (xea )).
Theorem 3.5. For each fuzzy soft point xea in ß 1 , the quasicompo-
nent C3 (xea)(C|(xea)) is a FSC3-connected (FSC4-connected) set in X, containing the union u Ei (xea)(u E4 (xea)).
set in X. since xea I n (fA).. Now, E3 (xea) c Ci (xea) follows from
the fact that if yß E3 xea, there is a FSC3-connected set (fA). in X containing x% and yß . □
Theorem 3.6. In a fuzzy soft topological space (X, t, E), the universe fuzzy soft set 1E is the overlapping union of its C3 -quasicomponents (C4 -quasicomponents).
Proof. Let {C3*. (xea ). i e /} be the family of C3 -quasicomponents of 1E in X. Then u C3 ■ (xea ) c 1E . Since each fuzzy soft point x\ e
E* (xea ) c C*i(xea ). then 11E c u C*■ (xea ). Moreover, let C3* (xea ) be the C3 -quasicomponents of 1E containing xea, and yfi be a fuzzy soft point in fi 1 such that yfi e E* (xea ) . Now, if the quasicomponent C* (xea ) and C* (yfi ) are overlapping, then C* (xea ) u C* (yfi ) is a FSC3 -connected set in X, by Theorem 3.6 and Theorem 4.12 in [9]. Hence, yfi E* xea and so yfi e E* (xea ) which is a contradiction. □
Now, in order to introduce the concept of C* -components (respectively, C4 -components), we begin with the following notions. Let be the family of C* -quasicomponents (C4 -quasicomponents) of 1E and let ^ be the family of arbitrary unions of members of Then, we prove the following proposition.
Proposition 3.3. For any fuzzy soft points x% and yfi in FSP(X)E, define a relation E*( E4 ), as follows: x% E^fi (xea E4yfi) iff there exists a FSC* -connected (FSC4-connected) set fA in ^ such that x% e fA and yfi e fA. Then E*( E4 ) is an equivalence relation on FSP(X)E.
Proof. Let xe be a fuzzy soft point in X. Then there exists a C* -component, in particular C* (x^ ). which contains xea. Hence, the relation E* is reflexive. Symmetry is obvious. Next, let xea, yfi and zsy be fuzzy soft points in fi 1 such that x% E3yfi and yfi E* zsy . Then,
Theorem 3.7. For each fuzzy soft point xea I FSP(X)E, the C3 - component (C4-component) C3(xea)(C4 (xea)) is the maximal FSC3- connected (FSC4 -connected) set in X, containing xe .
Proof. In view of Corollary 4.2 in [9], C* (xea) is a FSC3-connected ieI
Proof. We claim that, for each fuzzy soft point xe , C3 (xe ) = u (fA )j, where {(fA ). ; i e /} is the family of those members of ^
which are FSC3-connected, and contains the fuzzy soft point x^ . The family {(jA ) i ; i e /} is non-empty since C3 -quasicomponent
C3(x0) e f.
Firstly, we show that u (fA)i c C3 (xf). Let y e X, t e E and
suppose it (E ) (y) = .. If . = 0 , we have nothing to prove. If .
.u.( fA ) . 1 el
= 0, suppose if). (y) = . for each i e I. Now, the fuzzy soft
point yß I (fA )j for each i e I . Therefore, yß E3 x1, since the ßi ßi 1 C3 -quasicomponent C3 (x1) is a FSC3-connected set such that xea
I C| (x1 ) I ip . Therefore, yß E3 x% for each i s I . Hence, yß I C3 (xea) = u E3 (xea). for each i 6 I and so ßi < /A (.) (y) for each i 6
C3 (xa )
I. Then ß = sup{ßi} < ßj-. (x.) (y) implies yß I C3 (xea). Hence u.(fA)
Conversely, we show that C3 (xea) c u (fA).. Let y i X, t i
E and suppose ßC (xe ^y ) = y. Again if y = 0 , we have noth-
C 3 (x )
ing to prove. Suppose y = 0, and {yY.; i s I1 } be the family of fuzzy soft points such that yly. E3 xea. Then, clearly, /ß. (xe) (y) = ßt t (y) = sup{ y.} = y. Since yy E3 xea for each i s I1 , there ex-
ists a FSC3-connected set (fA)y. s p such that xea I (fA)y. and yy. I (fA )yt . Now, for each i s I 1 , consider the fuzzy soft set (fA)., define as follows: (fA). = (fA)yt u C3 (x1 ). Then, fA)i s p is a FSC3-connected set, such that x1 I (fA). and yß. I (fA).. Therefore, u yß.
1 ßl ief Yl
I u (fA) t, but {(.A )f ; i e I1 } 1 {(fA )t ; i s I} and so, we have u yß.
iet t ief '
1 iu( (fA )f . Now, ß = ßlu yff (yf < ßtf (f A). (yf . Therefore, ßtf x.) (y) <
ßtf (fA)t (y) and so, C3 (xea) 11 .u (fA)..
In view of Corollary 4.2 in [9], the C3 -component C3 (xea) is a FSC3-connected set, since x\ I n (f A).. To show that C3 (xea) is a
maximal FSC3-connected set containing xea, let gB be any FSC3 -connected set containing xea, such that C3 (xea) c gB. Then, the fuzzy soft set, defined as gB u C3 (y\) for every y e S(gB( e)). Therefore, gB
Ç C* (xe )
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[m5G; February 10, 2017;22:55J
cu E3 (x1 ) C C3 (xi ) . But, C3 (x1 ) c C3 (xea) as C| (xi ) e f is a FSC3 -connected set, containing xi . Thus, we have gB = C3 (x%) . □
Theorem 3.8. In a fuzzy topological space (X, t, E), the universe fuzzy soft set 1E is the disjoint union of its C3-components (respectively, C4 -components).
Definition 3.5. For any fuzzy soft points xe and ytft in FSP(X)E , define a relation E5 , as follows: x% E5yft iff there exists a FSC5 -connected set fA such that x e c fA and yt ft c fA .
Example 3.1 shows that E5 may not be reflexive. E5 is obviously symmetric. By using Theorem 4.10 in [9], it can be readily verified that E5 is transitive.
Proof. As a sample, we prove the case of C3 -components. Let Theorem 3.11. Let (X, t, E) be a fuzzy soft topological space and xsa
{C3 (xa). i e I} be the family of C3 -components of 1E in X. Then,
it can be verified that u C3 (xsa) = 1E . Next, suppose the C3 -
components C3 (xf) and C3 (yft) intersect at a point z. Then
iC/ (xa ) C y. )(z) = u Hence, i'c/ (x/ )(z) = У1 and i'c3 (yt/ )(z) = У2
where Y1 = 0 and Y2 = 0. Now, consider the C3 -quasicomponent C3* (zs1 ), containing the fuzzy soft point zs1 , which overlaps with C3-component C3 (xsa) and C3 (yft) at z. Therefore, C3 (xsa) u C| (zj ) and C3 (ytft ) u C3* (zs1 ) are FSC3 -connected sets, containing the fuzzy soft points xe and ytft respectively, and also the fuzzy soft point zj . Since C3 z) is the maximal FSC3 -connected set containing the fuzzy soft point zj . Therefore, C3 (xsa) u C| (zj ) C C3 (z^) and C3 (yft) u C3* (zj ) c C3 (zj ) so that C3 (xsa) c C3 (zj ) and C3 (yft) C C3 (zs1 ). Now, as C3 (x e ) and C3 (ytft ) are maximal FSC3 -connected sets containing the fuzzy soft points xe and ytft respectively, C3 (xe ) and C3 (ytft ) are identical. □
Theorem 3.9. For each fuzzy soft point xe in X, the C3 -component (C4 -component) C3 (xe ) (C4 (xe )) is a fuzzy soft closed set in X.
Proof. In view of Theorem 4.15 in [9], Fcl(C3(xsa)) is a FSC3-connected set in X. Moreover, xsa is contained in Fcl(C3 (xsa)). as xe c C3 (xe ) cc F cl(C3 (xe )). Since C3 (xe ) is the maximal FSC3 -connected set containing xsa, it follows that C3 (xsa) and Fcl(C3(xsa)) are identical. □
Again, it is obvious that an indiscrete fuzzy soft topological space, 1E is the only C3 -component (C4 -component). Moreover, when the fuzzy soft topological space is discrete, we state the following result:
Theorem 3.10. In a discrete fuzzy soft topological space, fuzzy soft points are the only FSC3-connected (FSC3 -connected) sets.
c FSP(X)E. Then, xsa is not a FSC5 -connected iff there exists a 0 = ft < a, fA and gB e t such that if (x) = ft and ig. (x) = 1 - ft.
Proof. Let xsa be not FSC5-connected. Then, xsa contains a non-null proper fuzzy soft clopen set xeft (say). Therefore, there exist fuzzy soft sets fA e t, gB e tc such that fA n xsa = gB n xea = xft . Since gB
e tc. then gB e t and iS. (x) = 1 - ft.
Conversely, let there exist a 0 = ft < such that there exist Ja and gB e t satisfying f (x) = ft and ig. (x) = 1 - ft. Then, gBB e tc and ig. (x) = ft. Also, fA n xsa = gB n xsa = xft and so xft is nonnull proper fuzzy soft clopen set in xsa. Therefore, xa is not FSC5 -connected. □
Remark 3.2. E5 is an equivalence relation iff 1E is a FSC5 -connected set and then it is the only C5 -component of (X, t, E).
Remark 3.3. The C5 -component of a fuzzy soft set if it exists, may not be fuzzy soft closed as shown by the following example:
Example 3.2. Consider the fuzzy soft topological space (X, t, E)
defined in Example 3.1. fE = {(e^, {a 1 }). (e2 , {b2})} is a FSC5-
connected.
Solution. Let gE be any fuzzy soft subset of X containing fE . Then, gE is of the form fE c gE = {(ej, { aa, bft}), S, { aY, b.})}
where a, S > 2 and y, ft > 0. Then, {(ej , {a 1 , ba}). (e2, {aY, b 1 })}
2 2 ft 2
or {(ej , {a 1 , b 1 }). (e2 , {a2 , b 1 })} is a non-null proper fuzzy soft
2 2 2 2
clopen set in gE according as y, ft < 2 or y, ft > 2 . So, gE is not FSC5 -connected. Therefore, fE is the C5 -component of fE and it is not fuzzy soft closed.
4. Conclusion
Proof. Immediate. □
Therefore, in a discrete fuzzy soft topological space, the C3 -component (C4 -component) are only the fuzzy soft points with value 1.
Definition 3.4. Let fA be a fuzzy soft set in a fuzzy soft topological space (X, t, E). The maximal FSC5-connected set containing fA is called the C5 -component of fA .
Remark 3.1. The C5 -component of a fuzzy soft set may not exist as shown by the following example:
Example 3.1. Let X = {a, b}, E = {ej, e2} and t = {1E , 0E , {(ex, {a j } ). (e2 , {b 2} )} . {(ej , {b , } ) . S , {a j} )} , {(ej , {a 1 , b , } ) , S , {a j,
bj })}} be a fuzzy soft topology defined on X. Let fA = { S, {b07 })}.
Since {(ej, {b j})} is a non-null proper fuzzy soft clopen set in fA, then fA is not a FSC5 -connected set. Also, there does not exist any FSC5 -connected set containing fA . So, fA has no C5 -component.
In this paper, we define on the set of fuzzy soft points in X an equivalence relation. The union of equivalence classes turns out to be a maximal fuzzy soft connected set which is called a fuzzy soft connected component. According to Remark 2.3, we have many types of connected components in fuzzy soft setting. The universe fuzzy soft set c1 E is the disjoint union of its Ci -components for i = 1. 2, S, M, O, Oq. Furthermore, we introduced some very interesting properties for fuzzy soft connected components in discrete fuzzy soft topological spaces which is a departure from the general topology such that in a discrete fuzzy soft topological space, the C3 -component ( C4 -component) are only the fuzzy soft points with value 1. Also, we find that: for each fuzzy soft point xe in X, the C3 -component (C4 -component) C3 (xsa) (C4 (xsa)) is a fuzzy soft closed set in X. Moreover, we prove that the C5 -component of a fuzzy soft set may not exist and the C5 -component of a fuzzy soft set if it exists, may not be fuzzy soft closed set.
Acknowledgment
The author would like to thank the referees for their useful comments and valuable suggestions given to this paper.
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