International Journal of Pure and Applied Mathematics Volume 118 No. 20 2018, 4721-4734 ISSN: 1314-3395 (on-line version) url: http://www.ijpam.eu Special Issue ijpam.eu
FS-COMPLEMENT OPERATOR –A SYMMETRIC DIFFERENCE OPERATION BETWEEN FS-SUBSETS
1 2 3 4 VaddiparthiYogeswara ,K.V.Umakameswari ,D.Raghu Ram , Ch. Ramasaynasi Rao 1Associate Professor Dept. of Mathematics, GIT, GITAM University, Visakhapatnam-530045,A.P,India 2Research Scholar: Dept. of Mathematics, GIS, GITAM University, Visakhapatnam 530045, A.P, India 3Research Scholar: Dept. of Mathematics, GIS, GITAM University, Visakhapatnam 530045, A.P, India 4Research Scholar: Dept. of Mathematics, GIS, GITAM University, Visakhapatnam 530045, A.P , India [email protected] [email protected] [email protected] [email protected]
Abstract :In this paper ,we define a symmetric difference operation between two given Fs-subsets of a given Fs-set and prove collection of all Fs-subsets of a given Fs-set with this symmetric operation is a commutative group with some conditions Keywords:Fs-set, Fs-subset, Fs-union, Fs-intersection , Fs-Complement, symmetric difference operation . . I. INTRODUCTION
Ever since Zadeh [8] introduced the notion of fuzzy sets in his pioneering work, several mathematicians studied numerous aspects of fuzzy sets.
Recently many researchers put their efforts in order to prove collection of all fuzzy subsets of a given fuzzy set is Boolean algebra under suitable operations [21].VaddiparthiYogeswara ,G.Srinivas and BiswajitRath[11] introduced the concept of Fs-set and developed the theory of Fs-sets in order to prove collection of all Fs-subsets of given Fs-set is a complete Boolean algebra under Fs-unions, Fs- intersections and Fs-complements. The Fs-sets they introduced contain Boolean valued membership functions. In this paper, we define a binary operation namely ,symmetric differenceoperation between to Fs-subsets and prove collection of all Fs-sub sets of a given Fs-set under this operation is a commutative group along with some conditions. For smooth reading of the paper, the theory of Fs-sets is introduced briefly in the section-II. All other relevant results are proved in the next section. We denote the largest element of a complete Boolean algebra LA [1.1] by MA or 1or1A . For all lattice theoretic, set theoretic properties and Boolean algebraic properties one can refer Szasz [3], Garret Birkhoff[4],Steven Givant• Paul Halmos[2] and Thomas Jech[5].
II.THEORY OF FS-SETS
2.1 Fs-set: Let U be a universal set, A1 ⊆ U and let A⊆U be non-empty. A four tuple 𝒜 =
A1 ,A, A μ1A1 ,μ2A , LA is said be an Fs-set if, and only if (1) A ⊆ A1 (2) LA is a complete Boolean Algebra
(3) μ1A1 : A1 ⟶ LA ,μ2A : A ⟶ LA ,are functions such that μ1A1 |A ≥ μ2A c (4) A:A⟶ LA is defined by Ax = μ1A1 x ⋀ μ2Ax ,for each x ∈ A
2.2 Fs-subset: Let 𝒜= A1, A, A μ1A1,μ2A , LA and ℬ= B1, B, B μ1B1,μ2B , LB be a pair of Fs-sets. ℬ is said to be an Fs-subset of 𝒜, denoted by ℬ⊆𝒜, if, and only if
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(1) B1 ⊆ A1, A ⊆ B (2) LB is a complete subalgebra of LA or LB ≤ LA
(3) μ1B1 ≤ μ1A1 |B1, and μ2B|A ≥ μ2A
2.3Fs-empty set For some LX , such that LX ≤ LA a four tuple 풳 = X1, X, X μ1X1 , μ2X , LX is not an Fs-set if, and only if
(a)X ⊈ X1or
(b) μ1X1 x ≱ μ2X x , for some x ∈ X ∩ X1
Here onwards, any object of this type is called an Fs-empty set of first kind and we accept that it is an Fs-subset of 퓑 for any 퓑 ⊆ 퓐.
Definition An Fs-subset 풴= Y1, Y, Y μ1Y1 , μ2Y , LY of 𝒜, is said to be an Fs-empty set of second kind if, and only if
(a') Y1 = Y = A
(b') LY ≤ LA
(c') Y = 0 i.e. μ1Y1 = μ2Y Unless otherwise there is a specific situation we treat Fs-empty set of first kind and Fs-empty set of second kind are same and denoted by Φ𝒜and observe that [11]. Φ𝒜is the least Fs-subset among all Fs- subsets of 𝒜
Fs- Union and Fs-Intersection
2.4Definition :Let ℬ = B1, B, B μ1B1,μ2B , LB and 풞= C1, C, C μ1C1 , μ2C , LC be a pair of Fs-subsets of𝒜. Then, the Fs-union of ℬ and 풞, denoted by ℬ∪풞 is defined as
ℬ∪풞=풟= D1, D, D μ1D1 , μ2D , LD ,where
(1) D1 = B1 ∪ C1 , D = B ∩ C
(2) LD = LB ∨ LC = The complete subalgebra generated by LB ∪ LC
(3) μ1D1 : D1 LD is defined by
μ1D1 x = (μ1B1 ∨ μ1C1 )x
μ2D ∶ D LDis defined by
μ2Dx = μ2B x ∧ μ2Cxand
D : D LD is defined by c Dx = μ1D1 x ∧ μ2Dx
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2.5 Definition:Let ℬ = B1, B, B μ1B1,μ2B , LB and 풞= C1, C, C μ1C1 , μ2C , LC be a pair of Fs-subsets of𝒜. Then, the Fs-intersection of ℬ and 풞, denoted byℬ ∩풞 is defined as
ℬ ∩풞=풟= D1, D, D μ1D1 , μ2D , LD ,where
(1) LD = LB ∧ LC = The complete subalgebra generated by LB ∩ LC
(2) μ1D1 : D1 LD is defined by
μ1D1 x = μ1B1 x ∧ μ1C1 x
μ2D ∶ D LDis defined by
μ2Dx = μ2B ∨ μ2C x and
D : D LD is defined by c Dx = μ1D1 x ∧ μ2Dx
III. Fs- COMPLEMENT ANDSYMMETRIC DIFFERENCE OPERATION BETWEEN TWO FS-SUBSETS
3.0 Definition of Fs-complement of an Fs-subset:
Consider a particular Fs-set𝒜 = A1 ,A, A μ1A1 ,μ2A , LA ,A ≠ Φ,where
[1] A ⊆ A1 [2] LA = 0, MA , MA =∨ A A = a∈A A a [3] μ1A1 = MA,μ2A = 0 , c Ax = μ1A1 x ⋀ μ2Ax = MA , for each x ∈ A
C𝒜 Given ℬ = B1, B, B μ1B1,μ2B , LB ⊆ 𝒜We define Fs-complement of ℬ, denoted by ℬ for B=A and LB = LA as follows:
C𝒜 ℬ = 풟= D1, D, D μ1D1 , μ2D , LD , where
c (a') D1 = CA B1 = B1 ∪ A, D = B = A (b') LD = LA
(c') μ1D1 : D1 ⟶ LA, is defined by μ1D1 x = MA c μ2D: A ⟶ LA , is defined by μ2Dx = Bx = μ1B1 x⋀ μ2Bx
c c c 3.1Remark:ClearlyD ∶ A ⟶ LA ,is given byDx = μ1D1 x⋀ μ2Dx = MA ∧ Bx = Bx .
3.2SYMMETRIC DIFFERENCE OPERATION BETWEENTWO FS-SUBSETS
A family 픊 of allFs-subsetsℬ⊆𝒜 and 𝒜 = A1 ,A, A μ1A1 ,μ2A , LA with
ℬ = B1, B, B μ1B1,μ2B , LB with B=A and LB = LA is a commutative group along with (1.3a) and (1.3b) with the operation ∆ as defined below
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퓑 ∆ 퓒= ℬ ∩ 풞퐶𝒜 ∪ ℬC𝒜 ∩ 풞 where ℬ &퐶 areFs- subsets of𝒜
3.3Proposition: We can easily observe that for any Fs-subsets ℬ &퐶, the following Resultsaretrue a)ℬ ∆ 풞 = 풞 ∆ ℬ
b) ℬ ∆Φ𝒜 = ℬ
c)ℬ ∆ℬ = Φ𝒜
3.4 Proposition : For any Fs-subsetsℬ, 풞&풟퓑 ∆ (퓒 ∆ 퓓) = (퓑 ∆ 퓒) ∆ 퓓
is true provided
a)μ1D1 x = μ1B1 x,
퐶 퐶 b)(휇2퐷푥) ∨ 휇2퐵푥 = (휇2퐵푥) ∨ 휇2퐷푥
c)휇2퐷푥 ∨ 퐵 푥 = 휇2퐵푥 ∨ 퐷 푥
for each x ∈ A
Proof : Say ℬ △ 풞 △ 풟 = 풬, 풞 △ 풟 = 풨 ,ℬ푐𝒜 = ℱ,풞푐𝒜 = 𝒢 ,풟푐𝒜 = ℋ, 풞 ∩ 풟퐶𝒜 = 풥 , 풞C𝒜 ∩ 풟 =풦 , 풥 ∪ 풦 = 풨, 풨푐𝒜 = 풩, ℬ ∩ 풨퐶𝒜 = 풫 ℬC𝒜 ∩ 풨 =풧 ,풫 ∪ 풧 = 풬,(ℬ △ 풞) △ 풟 = 풵 , ℬ △ 풞 = 풯 , ℬ ∩ 풞퐶𝒜 = 풭, ℬC𝒜 ∩ 풞 =풮 , 풭 ∪ 풮 = 풯, 풯푐𝒜 = 풰, 풯 ∩ 풟퐶𝒜 = 풫 풯C𝒜 ∩ 풟 =풲 , 풱 ∪ 풲 = 풵
푐𝒜 Thenℱ = ℬ = 퐹1, 퐹, 퐹 휇1퐹1,휇2퐹 , 퐿퐹 Where 퐶 퐶 (i) 퐹1 = 퐶퐴퐵1 = 퐵1 ∪ 퐴, F=B=A where퐵1 = 퐴1 − 퐵1
(ii) 퐿퐹=퐿퐴
(iii) 휇1퐹1 : 퐹1 퐿퐴is given by휇1퐹1 푥=푀퐴
푐 휇2퐹: 퐴 퐿퐴is given by휇2퐹 = 퐵푥 = 휇1퐵1 푥 ∧ 휇2퐵푥
푐 퐹: 퐴 퐿퐴is given by퐹푥 = 휇1퐹1 푥 ∧ 휇2퐵푥
푐 푐 = 푀퐴 ∧ 퐵 푥 = 퐵 푥
푐𝒜 𝒢 = 풞 = 퐺1, 퐺, 퐺 휇1퐺1,휇2퐺 , 퐿퐺 푤here
퐶 푖 퐺1 = 퐶퐴퐶1 = 퐶1 ∪ 퐴 , G=C=A
(ii)퐿퐺=퐿퐶=퐿퐴
(iii) 휇1퐺1 : 퐺1 퐿퐴is given by휇1퐺1 푥=푀퐴
휇2퐺 : 퐴 퐿퐴is given by휇2퐺 푥 = 퐶 푥
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푐 푐 푐 퐺 : 퐴 퐿퐴is given by퐺푥 = 휇1퐺1 푥 ∧ 휇2퐺 푥 = 푀퐴 ∧ 퐶푥 = 퐶푥
푐𝒜 ℋ = 풟 = 퐻1, 퐻, 퐻 휇1퐻1,휇2퐻 , 퐿퐻 Where
퐶 (푖) 퐻1 = 퐶퐴퐷1 = 퐷1 ∪ 퐴 , H=D=A
(푖푖) 퐿퐻=퐿퐷=퐿퐴
(iii)휇1퐻1 : 퐻1 퐿퐴is given by휇1퐻1 푥=푀퐴
휇2퐻: 퐻 퐿퐴is given by 휇2퐻푥 = 퐷 푥
푐 푐 푐 퐻: 퐴 퐿퐴 is given by퐻푥 = 휇1퐻1 푥 ∧ 휇2퐻푥 = 푀퐴 ∧ 퐷푥 = 퐷푥
풥 = 퐽1, 퐽, 퐽 휇1퐽1, 휇2퐽 , 퐿퐽 where
퐶 (1) 퐽1 = 퐶1 ∩ 퐻1 = (퐶1 ∩ 퐷1 ) ∪ 퐴, J=C ∪ H=A
2 퐿퐽 = 퐿퐶 ∧ 퐿퐻= 퐿퐴
(3) 휇1퐽1 : 퐽1 퐿퐽 is defined by휇1퐽1 푥 = 휇1퐶1 푥 ∧ 휇1퐻1 푥
=휇1퐶1 푥 ∧ 푀퐴=휇1퐶1 푥
휇2퐽 ∶ 퐽 퐿푗 is defined by휇2퐽 푥 = 휇2퐶푥 ∨ 휇2퐻푥 = 휇2퐶푥 ∨ 퐷 푥
푥 푐 퐽: 퐽 퐿 퐽 is definedby퐽푥 = 휇1퐽1 푥 ∧ 휇2퐽
푐 = 휇1퐽1 푥 ∧ ( 휇2푐 ∨ 휇2퐻 푥)
푥 푐 푥 푐 = 휇1퐽1 푥 ∧ (휇2푐 ) ∧ (휇2퐻 )
푥 푐 = 퐶 푥 ∧ 휇2퐻
=퐶 푥 ∧ 퐷 푥 푐
풦 = 퐾1, 퐾, 퐾 휇1퐾1,휇2퐾 , 퐿퐾 where
퐶 퐶 퐶 (4) 퐾1 = 퐺1 ∩ 퐷1 = (퐶1 ∪ 퐴) ∩ 퐷1=(퐶1 ∪ 퐷1) ∪ (퐴 ∩ 퐷1) = (퐶1 ∩ 퐷1) ∪ 퐴 , K=G∪ 퐷=A
(5)퐿퐾= 퐿퐺 ∧ 퐿퐷=퐿퐴
(6) 휇1퐾1 : 퐾1 퐿퐾is defined by휇1퐾1 푥 = 휇1퐺1 푥 ∧ 휇1퐷1 푥
= 푀퐴 ∧ 휇1퐷1 푥
=휇1퐷1 푥
푥 휇2퐾 ∶ 퐾 퐿퐾 isdefined by 휇2퐾 = 휇2퐺 ∨ 휇2퐷 푥 = 휇2퐺 푥 ∨ 휇2퐷푥
= 퐶푥 ∨ 휇2퐷푥
푐 퐾:퐾 퐿퐾 is definedby 퐾푥 = 휇1퐾1 푥 ∧ 휇2퐾푥
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푐 = 휇1퐷1 푥 ∧ ( 휇2퐺 ∨ 휇2퐷 푥)
푐 푐 = 휇1퐷1 푥 ∧ (휇2퐺 푥) ∧ (휇2퐷푥)
푐 푐 = 휇1퐷1 푥 ∧ 휇2퐷푥 ∧ 휇2퐺 푥
= 퐷 푥 ∧ 퐶 푥 푐
풨 = 푀1, 푀, 푀 휇1푀1,휇2푀 , 퐿푀 Where
퐶 퐶 (7) 푀1 = 퐽1 ∪ 퐾1 = [(퐶1 ∩ 퐷1 ) ∪ 퐴] ∪ [(퐶1 ∩ 퐷1) ∪ 퐴] = (퐶1∆ 퐷1) ∪ 퐴 ,
푀 = 퐽 ∩ 퐾 = 퐴 ∩ 퐴=퐴
(8) 퐿푀=퐿퐽 ∨ 퐿퐾=퐿퐴
(9) 휇1푀1 : 푀1 퐿퐴isdefined by휇1푀1 푥 = 휇1퐽1 ∨ 휇1퐾1 푥
Case (i) x∈A ⟹ 휇1푀1 푥 =휇1퐶1 푥 ∨ 휇1퐷1 푥
Case (ii) x∉ 퐴, x∈C ⟹ 휇1푀1 푥 = 휇1퐽1 푥=휇1퐶1 푥
Case (iii) x∉ 퐴, x∈D ⟹ 휇1푀1 푥 = 휇1퐾1 푥 =휇1퐷1 푥
휇2푀 ∶ 푀 퐿퐴isdefined by휇2푀푥 = 휇2퐽 푥 ∧ 휇2퐾푥
= (휇2퐶푥 ∨ 퐷 푥)∧ (퐶 푥 ∨ 휇2퐷푥)
푥 = (퐶 푥 ∧ 퐷 푥) ∨ (휇2퐶푥 ∧ 휇2퐷 )
푐 푀: M 퐿퐴 is defined by푀푥 = 휇1푀1 푥 ∧ 휇2푀푥
푐 = 휇1퐽1 ∨ 휇1퐾1 푥 ∧ 휇2퐽 푥 ∧ 휇2퐾푥 c =(휇1퐶1 푥 ∨ 휇1퐷1 푥) ∧ [휇2퐶푥 ∨ 퐷푥) ∧ (퐶푥 ∨ 휇2퐷푥) ] C c =(휇1퐶1 푥 ∨ 휇1퐷1 푥) ∧[(퐶푥 ∧ 퐷푥) ∨ (휇2퐶푥 ∧ 휇2퐷푥) ]
풩 = N1, N, N μ1N1 , μ2N , LN , where
푐 (a') N1 = CA 푀1 = 푀1 ∪ A, N = M = A (b') LD = LA
(c') μ1N1 : N1 ⟶ LA, is defined byμ1푁1 x = MA μ2N : N ⟶ LA, is defined byμ2Nx = 푀 x c N: N ⟶ LA ,isdefined byDx = μ1N1 x⋀ μ2N x
c c = MA ∧ M x = M x C 푥 푥 c c = (휇1퐶1 푥 ∨ 휇1퐷1 푥) ∧[(퐶푥 ∧ 퐷푥) ∨ (휇2푐 ∧ 휇2퐷 ) ]}
풫 = 푃, 푃, 푃 휇1푃1,휇2푃 , 퐿푃 where
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퐶 퐶 (10) 푃1 = 퐵1⋂푁1=퐵1 ∩ 퐶1 △ 퐷1 ∪ 퐴 = 퐵1 ∩ (퐶1 △ 퐷1) ∪ 퐵1 ∩ 퐴
퐶 = 퐵1 ∩ (퐶1 △ 퐷1) ∪ A ,
P=B ∪ 푁 = 퐵 ∪ (퐽 ∪ K)=B∪ (퐴 ∪ 퐴)=B ∪ 퐴=A
(11) 퐿푃 = 퐿퐵⋀퐿푁퐿퐴
(12) 휇1푃1 : 푃1 퐿퐴is defined by 휇1푝1 푥 = 휇1퐵1 푥⋀ 휇1푁1 푥
=휇1퐵1 푥 ∧ 푀퐴
=휇1퐵1 푥
휇2푃: 푃 퐿퐴 is defined by휇2푝 푥 = 휇2퐵 ∨ 휇2푁 x
= 휇2퐵푥 ∨ 휇2푁푥
=휇2퐵푥 푀 x
퐶 =휇2퐵푥 ∨ 휇2퐽 푥 ∨ 휇2퐾푥
퐶 =휇2퐵푥 ∨ 휇1퐶1 푥 ∨ 휇1퐷1 푥 ∧ [(휇2퐶푥 ∨ 퐷푥) ∧ (퐶푥 ∨ 휇2퐷푥)]
푥 푐 푃: 푃 퐿퐴 is defined by푃푥 = 휇1푃1 ∧ 휇2푃푥
퐶 =휇1퐵1 푥 ∧[휇2퐵푥 ∨ 휇1퐶1 푥 ∨ 휇1퐷1 푥 ∧ [(휇2퐶푥 ∨ 퐷푥) ∧ (퐶푥 ∨ 휇2퐷푥)]
풧 = 퐿1, 퐿, 퐿 휇1퐿1,휇2퐿 , 퐿퐿 where
퐶 퐶 (13) 퐿1 = 퐹1⋂푀1= 퐵1 ∪ 퐴 ⋂ 퐽1 ∪ 퐾1 = 퐷1 ∪ 퐴) ∪ 퐶1 △ 퐷1 ∪ 퐴
퐶 = 퐵1 ∩ 퐶1 △ 퐷1 ∪ A
(14)퐿퐿 =퐿퐹 ∩ 퐿푀 =퐿퐴
(15)휇1퐿1 : 퐿1 퐿퐴is defind by 휇1퐿1 푥 = 휇1퐹1 푥 = MA ∩ 휇1퐽1 ∨ 휇1퐾1 푥
= 휇1퐽1 ∨ 휇1퐾1 푥
= 휇1퐶1 ∨ 휇1퐷1 푥
휇2퐿 : 퐿 퐿퐴 is defined by휇2퐿푥 = 휇2퐹 ∨ 휇2푀 x
= 퐵 푥 ∨ 휇2푀푥
= 퐵 푥 ∨ [(휇2퐶푥 ∨ 퐷 푥)∧ (퐶 푥 ∨ 휇2퐷푥)]
= [(휇2퐶푥 ∨ 퐷 푥 ∨ 퐵 푥)∧ (퐶 푥 ∨ 휇2퐷푥 ∨ 퐵 푥)]
푥 푐 퐿: L 퐿퐴 is defined by퐿푥 = 휇1퐿1 ∧ 휇2퐿푥
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푥 푥 퐶 =휇1퐵1 푥 ∧[휇2퐵푥 ∨ (휇1퐶1 ∨ 휇1퐷1 ) ∧ [(휇2퐶푥 ∨ 퐷푥) ∧ (퐶푥 ∨ 휇2퐷푥)]
풬 = 푄1, 푄, 푄 휇1푄1,휇2푄 , 퐿푄 where
퐶 퐶 16 푄1 = 푃1 ∪ 푀1= 퐵1 ⋂ (퐶1∆퐷1 ) ∪ 퐴 ∪ [(퐵1 ∩ 퐶1 △ 퐷1 ) ∪ A]
=[퐵1 △ 퐶1 △ 퐷1 ] ∪ A , Q=P ∩L =A
(17)퐿푄 =퐿푃 ∩ 퐿퐿 =퐿퐴
(18)휇1푄1 : 푄1 퐿퐴is defind by 휇1푄1 푥 = (휇1푃1 ∨ 휇1퐿1 ) x
Case (i) x∈A ⟹ 휇1푄1 푥 = 휇1푃1 푥 ∨ 휇1퐿1 푥 = 휇1퐵1 푥 ∨ 휇1퐶1 푥 ∨ 휇1퐷1 푥
Case (ii) x∉ 퐴, x∈B ⟹ 휇1푄1 푥 = 휇1푃1 푥 = 휇1퐵1 푥
Case (iii) x∉ 퐴, x∈C ⟹ 휇1푄1 푥 = 휇1퐿1 푥 =휇1퐶1 푥
Case (iv) x∉ 퐴, x∈D ⟹ 휇1푄1 푥 = 휇1퐿1 푥 =휇1퐷1 푥
휇2푄 ∶ 푄 퐿퐴 is defined by 휇2푄푥 = 휇2푃푥 ∧ 휇2퐿푥
퐶 퐶 퐶 퐶 = 휇2퐵푥 ∨ [(휇1퐶1 푥 ∨ 휇1퐷1 푥) ∧ ((퐶푥) ∨ (퐷푥) )) ∧ ((휇2퐶푥) ∨ 휇2퐷푥) ]
∧ [(휇2퐶푥 ∨ 퐷 푥 ∨ 퐵 푥)∧ (퐶 푥 ∨ 휇2퐷푥 ∨ 퐵 푥)]
퐶 퐶 퐶 퐶 = (휇1퐶1 푥 ∨ 휇1퐷1 푥 ∨ 휇2퐵푥) ∧ ((휇2퐶푥) ∨ (휇2퐷푥) ∨ 휇2퐵푥) ∧ ( 퐶푥) ∨ (퐷푥) ∨ 휇2퐵푥) ∧ (휇2퐶푥 ∨ 퐷 푥 ∨ 퐵 푥) ∧ (퐶 푥 ∨ 휇2퐷푥 ∨ 퐵 푥)
푐 푄:푄 퐿퐴 is defined by 풬푥 = 휇1푄1 푥 ∧ 휇2푄푥
퐶 퐶 = 휇1퐵1 푥 ∨ 휇1퐶1 푥 ∨ 휇1퐷1 푥 ∧ [(휇1퐶1 푥 ∨ 휇1퐷1 푥 ∨ 휇2퐵푥) ∧ ((휇2퐶푥) ∨ (휇2퐷푥) ∨ 휇2퐵푥) ∧ 퐶 퐶 ( 퐶 푥) ∨ (퐷푥) ∨ 휇2퐵푥) ∧ (휇2퐶푥 ∨ 퐷 푥 ∨ 퐵 푥)
C ∧ (퐶 푥 ∨ 휇2퐷푥 ∨ 퐵 푥)]
풭 = 푅1, 푅, 푅 휇1푅1,휇2푅 , 퐿푅 Where
퐶 퐶 (19) 푅1 = 퐵1⋂퐺1=퐵1 ∩ 퐶1 ∪ 퐴 = 퐵1 ∩ 퐶1 ∪ 퐴
R=B∪ 퐺 =B∪ 퐴=A
(20) 퐿푅 = 퐿퐵⋀ 퐿퐺 = 퐿푅
(21) 휇1푅1 ∶ 푅1 퐿퐴is defind by 휇1푅1 푥 = 휇1퐵1 푥⋀ 휇1퐺1 푥
=휇1퐵1 푥 ∧ 푀퐴
=휇1퐵1 푥
휇2푅 : 푅 퐿퐴 is defined by 휇2푅푥 = 휇2퐵 ∨ 휇2퐺 x
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= 휇2퐵푥 ∨ 휇2퐺 푥
=휇2퐵푥 퐶 x
푐 푅:푅 퐿퐴 is defined by 푅푥 = 휇1푅1 푥 ∧ 휇2푅푥
푐 = 휇1푅1 푥 ∧ 휇2푅푥 퐶푥)
=퐵 푥 ∧ 퐶 푥 푐
풮 = 푆1, 푆, 푆 휇1푆1,휇2푆 , 퐿푆 where
퐶 퐶 22 푆1 = 퐹1⋂퐶1= 퐵1 ∪ 퐴 ∩ 퐶1 = 퐵1 ∪ 퐶1 ∪ 퐴, S=F∪ 퐶 =A
23 퐿푆 = 퐿퐹⋀ 퐿퐶 = 퐿퐴
(24)휇1푆1 ∶ 푆1 퐿퐴is defined by휇1푆1 푥 = 휇1퐹1 푥⋀ 휇1퐶1 푥
=휇1퐶1 푥 ∧ 푀퐴
=휇1퐶1 푥
휇2푆: 푆 퐿퐴 is defined by휇2푆푥 = 휇2퐹 ∨ 휇2퐶 x
= 휇2퐹푥 ∨ 휇2퐶푥
=휇2퐶푥 퐵 x
푐 푐 푆:푆 퐿퐴 is defined by 푆푥 = 휇1푆1 푥 ∧ 휇2푆푥 = 휇1퐶1 푥 ∧ 휇2퐶푥 퐵푥)
C 푥 퐶 = 휇1퐶1 푥 ∧ 퐵푥 ∧ 휇2퐶
= 퐶 푥 ∧ 퐵 푥 푐
풯 = 푇1, 푇, 푇 휇1푇1,휇2푇 , 퐿푇 Where
퐶 퐶 25 푇1 = 푅1⋂푆1= 퐵1 ∩ 퐶1 ∪ 퐴 = 퐵1 ∩ 퐶1 ∪ 퐴 = 퐵1∆퐶1 ∪ 퐴,
T=R∩S=A
26 퐿푇 = 퐿푅⋀ 퐿푆 = 퐿퐴
(27)휇1푇1 : 푇1 퐿퐴is defind by 휇1푇1 푥 = 휇1푅1 ∨ 휇1푆1 푥
= 휇1퐵1 ∨ 휇1퐶1 푥
Case (i) x∈A 휇1푇1 푥 = 휇1퐵1 푥 ∨ 휇1퐶1 푥
= 휇1퐵1 푥 ∨ 휇1퐶1 푥
Case (ii)x ∉ A, x ∈ B 휇1푇1 푥 = 휇1퐵1 푥
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Case (iii) x ∉ A , x ∈ C 휇1푇1 푥 = 휇1퐶1 푥
휇2푇: 푇 퐿퐴 is defined by휇2푇푥 = 휇2푅 ∨ 휇2푆 x
= 휇2퐵푥 ∨ 퐶 푥 ∧ 퐵 푥 ∨ 휇2퐶푥
= 퐶 푥 ∧ 퐵 푥 ∨ 휇2퐵푥 ∧ 휇2퐶푥
푐 푇:푇 퐿퐴 is defined by 푇푥 = 휇1푇1 푥 ∧ 휇2푇푥
푐 = 휇1푇1 푥 ∧ 퐶푥 ∧ 퐵푥 ∨ 휇2퐵 ∧ 휇2퐶푥
퐶 퐶 = 휇1푅1 푥 ∨ 휇1푆1 푥 ∧ ( 퐶푥 ∧ 퐵푥) ∨ 휇2퐵 ∧ 휇2퐶푥
퐶 푋 퐶 = 휇1퐵1 푥 ∨ 휇1퐶1 푥 ∧ 퐶푥 ∧ 퐵푥) ∧ 휇2퐵 ∧ 휇2퐶푥
풰= U1, U, U μ1U1 , μ2U , LU , where
c 퐶 (28)U1 = CA T1 = T1 ∪ A = [(퐵1 △ 퐶1) ∪ A] U = T = A
29 LU = LT = LA
30 μ1U1 : T1 ⟶ LA, is defined by μ1U1 x = MA c μ2U : U ⟶ LA, is defined by μ2U x = Tx = μ1T1 x⋀ μ2Tx c c c U: U ⟶ LA ,is defined byUx = μ1U1 x⋀ μ2U x = MA ∧ Tx = Tx
C c C = {[μ1B1 푥 ∨ μ1C1 푥) ∧[(퐶푥 ∧ 퐵푥) ∨ (μ2B 푥 ∧ μ2C푥) ]}
풱 = 푉1, 푉, 푉 휇1푉1,휇2푉 , 퐿푉 where
퐶 (31) 푉1 = 푇1⋂퐺1= (퐶1 △ 퐵1) ∩ 퐷1 ∪ 퐴 , V=T∪ 퐺=A
(32)퐿푉 = 퐿푇 ∨ 퐿퐺 =퐿퐴
(33) 휇1푉1 : 푉1 퐿퐴is defined by 휇1푉1 푥 = 휇1푇1 푥⋀ 휇1퐺1 푥
= (휇1퐵1 ∨ 휇1퐶1 )x ∧ 푀퐴
= (휇1퐵1 ∨ 휇1퐶1 )x
휇2푉: 푉 퐿퐴is defined by휇2푉푥 = 휇2푇 ∨ 휇2퐺 x
= 휇2푇푥 ∨ 휇2퐺 푥
= 휇2푇푥 퐷 푥
= [(휇2퐶푥 ∨ 퐵 푥) ∧ (퐶 푥 ∨ 휇2퐵푥)] 퐷 푥
= [(휇2퐶푥 ∨ 퐵 푥 퐷 x) ∧ (퐶 푥 ∨ 휇2퐵푥 퐷 푥)]
푐 푉:푉 퐿퐴 is defined by푉푥 = 휇1푉1 푥 ∧ 휇2푉푥
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C = (휇1퐵1 ∨ 휇1퐶1 )x ∧ [(휇2퐶푥 ∨ 퐵푥 퐷푥) ∧ (퐶푥 ∨ 휇2퐵푥 퐷푥)]
풲 = 푊1, 푊, 푊 휇1푊1,휇2푊 , 퐿푊 Where
퐶 퐶 34 푊1 = 푈1⋂퐷1= 푇1 ∩ 퐷1 = (퐵1∆퐶1) ∩ 퐷1 ∪ 퐴, W=U∪ 퐷 =A
35 퐿푊 = 퐿푈⋀ 퐿퐶 = 퐿퐴
36 휇1푊1 : 푊1 퐿퐴 is defined by 휇1푊1 푥 = 휇1푈1 푥⋀ 휇1퐷1 푥
= 휇1퐷1 푥 ∧ 푀퐴 = 휇1퐷1 푥
휇2푊 ∶ 푊 퐿퐴is defined by휇2푊푥 = 휇2푈 ∨ 휇2퐷 x
= 휇2푈푥 ∨ 휇2퐷푥
퐶 퐶 = 휇1퐵1 푥⋀ 휇1퐶1 푥 ⋀(퐶푥 ∧ 퐵푥) ⋀ 휇2퐵푥⋀ 휇2퐶푥 ∨ 휇2퐷푥
푐 푊:푊 퐿퐴 is defined by 푊푥 = 휇1푊1 푥 ∧ 휇2푊푥
푐 퐶 퐶 = 휇1퐷1 푥 ∧ 휇1퐵1 푥⋀ 휇1퐶1 푥 ⋀(퐶푥 ∧ 퐵푥) ⋀ 휇2퐵푥⋀ 휇2퐶푥
퐶 퐶
= 휇1퐷1 푥 ∧ 휇1퐵1 푥 ⋀ 휇1퐶1 푥 ∨ 퐶푥 ∧ 퐵푥 ∨ 휇2퐵푥⋀ 휇2퐶푥
풵 = 푍1, 푍, 푍 휇1푍1,휇2푍 , 퐿푍 where
퐶 퐶 37 푍1 = 푉1 ∪ 푊1 = (퐵1∆퐶1) ∩ 퐷1 ) ∪ 퐴 ∪ [(퐵1 △ 퐶1) ∩ 퐷1) ∪ A]
= [퐵1 △ 퐶1 △ 퐷1 ] ∪ A Z=V ∩W =A
(38)퐿푍 =퐿푉 ∩ 퐿푊 =퐿퐴
(39)휇1푍1 : 푍1 퐿퐴is defind by 휇1푍1 푥 = (휇1푉1 ∨ 휇1푊1 ) x
푥 Case (i) x∈A ⟹ 휇1푍1 = 휇1푉1 푥 ∨ 휇1푊1 푥 = 휇1퐵1 푥 ∨ 휇1퐶1 푥 ∨ 휇1퐷1 푥
Case (ii) x∉ 퐴, x∈B ⟹ 휇1푍1 푥 = 휇1푉1 푥 = 휇1퐵1 푥
Case (iii) x∉ 퐴, x∈C ⟹ 휇1푍1 푥 = 휇1푉1 푥 = 휇1퐶1 푥
Case (iii) x∉ 퐴, x∈D ⟹ 휇1푍1 푥 = 휇1푊1 푥 = 휇1퐷1 푥
휇2푍 ∶ 푍 퐿퐴 is defined by 휇2푍푥 = 휇2푉푥 ∧ 휇2푊푥
= [(휇2퐵푥 ∨ 퐷푥 ∨ 퐶푥) ∧ (휇2푍푥 ∨ 퐷푥 ∨ 퐵푥)] ∧ [ 휇1퐵1 푥 ∨ 휇1퐶1 푥 ∨ 휇2퐷푥
퐶 퐶 퐶 퐶 ∧ 퐶 푥) ∨ (퐵 푥) ∨ 휇2퐷푥 ∧((휇2퐶푥) ∨ (휇2퐵푥) ∨ 휇2퐷푥)]
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퐶 퐶 퐶 = 휇1퐵1 푥 ∨ 휇1퐶1 푥 ∨ 휇2퐷푥 ∧((휇2퐶푥) ∨ 휇2퐵푥) ∨ 휇2퐷푥 ∧ 퐶푥) ∨ 퐶 (퐵 푥) ∨ 휇2퐷푥 ∧ (휇2퐵푥 ∨ 퐷 푥 ∨ 퐶 푥) ∧ (휇2푍푥 ∨ 퐷 푥 ∨ 퐵 푥)
푐 푍:푍 퐿퐴 is defined by푍푥 = 휇1푍1 푥 ∧ 휇2푍푥
푥 푥 푐 = 휇1푉1 ∨ 휇1푊1 푥 ∧ 휇2푉 ∧ 휇2푊
퐶 퐶 퐶 = (휇1퐵1 푥 ∨ 휇1퐶1 푥 ∨ 휇1퐷1 푥) ∧ [ 휇1퐵1 푥 ∨ 휇1퐶1 푥 ∨ 휇2퐷푥 ∧((휇2퐶푥) ∨ 휇2퐵푥) ∨ 휇2퐷푥 ∧ 퐶푥) ∨ 퐶 C (퐵 푥) ∨ 휇2퐷푥 ∧ (휇2퐵푥 ∨ 퐷 푥 ∨ 퐶 푥) ∧ (휇2푍푥 ∨ 퐷 푥 ∨ 퐵 푥)]
As푄1 = 푍1, Q=Z,퐿푄 = 퐿푍 = 퐿퐴 ,휇1푄1 =휇1푍1 in 풬 푎푛푑 풵,
Sufficientto show that 휇2푄 = 휇2푍
But 휇2푄 = (1) ∧ (2) ∧ (3) ∧ (4) ∧ (5) and 휇2푍 = (1′) ∧ (2′) ∧ (3′) ∧ (4′) ∧ (5′) where
(1) 휇1퐶1 푥 ∨ 휇1퐷1 푥 ∨ 휇2퐵푥 (1′)휇1퐵1 푥 ∨ 휇1퐶1 푥 ∨ 휇2퐷푥 퐶 퐶 퐶 퐶 (2) (휇2퐶푥) ∨ (휇2퐷푥) ∨ 휇2퐵푥 (2′)(휇2퐶푥) ∨ (휇2퐵푥) ∨ 휇2퐷푥 퐶 퐶 퐶 퐶 (3) 퐶 푥) ∨ (퐷푥) ∨ 휇2퐵푥 (3′) 퐶 푥) ∨ (퐵 푥) ∨ 휇2퐷푥
(4) (휇2퐶푥 ∨ 퐷 푥 ∨ 퐵 푥) (4′) (휇2퐶푥 ∨ 퐷 푥 ∨ 퐵 푥)
(5) (퐶 푥 ∨ 휇2퐷푥 ∨ 퐵 푥) (5′) (휇2퐵푥 ∨ 퐷 푥 ∨ 퐶 푥)
And observe that using theconditionsgiven in the statement we can see (1) = (1′),
(2)=(2′) , (3)= (3′) , (4)=(4′) and 5 = 5′
so that휇2푄 = 휇2푍
ACKNOWLEDGEMENTS
The authorsacknowledgeGITAM(Deemed to be University) ,Visakhapatnam and V.E.S. MURTHY NISTLA , Professor and Head- Department of Mathematics, Andhra University, Visakhapatnam. REFERENCES [1] J.A.Goguen ,L-Fuzzy Sets, JMAA,Vol.18, P145-174,1967 [2] Steven Givant• Paul Halmos, Introduction to Boolean algebras, Springer [3] Szasz, G., An Introduction to Lattice Theory, Academic Press, New York. [4] Garret Birkhoff, Lattice Theory, American Mathematical Society Colloquium publications Volume-xxv [5] Thomas Jech ,Set Theory, The Third Millennium Edition revised and expanded, Springer [6] George J. Klir and Bo Yuan ,Fuzzy Sets, Fuzzy Logic, and Fuzzy Systems: Selected Papers by Lotfi A. Zadeh ,Advances in Fuzzy Systems-Applications and Theory Vol-6,World Scientific [7] James Dugundji, Topology, Universal Book Stall, Delhi. [8] L.Zadeh, Fuzzy Sets, Information and Control,Vol.8,P338-353,1965 [9] U.Höhle,S.E.Rodabaugh, Mathematics of fuzzy Sets Logic, Topology, and Measure Theory, Kluwer Academic Publishers, Boston
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