On Union and Intersection of Fuzzy Soft Set - IJCTAijcta.com/documents/volumes/vol2issue5/ijcta2011020504.pdf · On Union and Intersection of Fuzzy Soft Set . Tridiv Jyoti Neog1,

On Union and Intersection of Fuzzy Soft Set

Tridiv Jyoti Neog1, Dusmanta Kumar Sut 2

1. Research Scholar, Department of Mathematics, CMJ University, Shillong, Meghalaya email : [email protected]

2. Assistant Professor, Department of Mathematics, Jorhat Institute of Science and Technology, Jorhat, Assam

email : [email protected]

Abstract

Molodtsov introduced the theory of soft sets, which can be seen as a new mathematical approach to vagueness. Maji et al. have further initiated several basic notions of soft set theory. They have also introduced the concept of fuzzy soft set, a more generalized concept, which is a combination of fuzzy set and soft set. They introduced some properties regarding fuzzy soft union, intersection, complement of a fuzzy soft set, DeMorgan Laws etc. These results were further revised and improved by Ahmad and Kharal. They defined arbitrary fuzzy soft union and intersection and proved DeMorgan Inclusions and DeMorgan Laws in Fuzzy Soft Set Theory. In this paper, we give some propositions on fuzzy soft union and intersection with proof and examples. Using the definition of arbitrary fuzzy soft union and intersection proposed by Ahmad and Kharal, we are giving two more propositions with proof and examples. We further give the proof of DeMorgan Laws for a family of fuzzy soft sets in a fuzzy soft class proposed by Ahmad and Kharal and verify these laws with examples. Key words: Soft Set, Fuzzy Soft Set, Fuzzy Soft Class. 1. Introduction. In order to deal with many complicated problems in the fields of engineering, social science, economics, medical science etc involving uncertainties, classical methods are found to be inadequate in recent times. Molodstov [7] pointed out that the important existing theories viz. Probability Theory, Fuzzy Set Theory, Intuitionistic Fuzzy Set Theory, Rough Set Theory etc. which can be considered as mathematical tools for dealing with uncertainties, have their own difficulties. He further pointed out that the reason for these difficulties is, possibly, the inadequacy of the parameterization tool of the theory. In 1999 he proposed a new mathematical tool for dealing with uncertainties which is free of the difficulties present in these theories. He introduced the novel concept of Soft Sets and established the fundamental results of the new theory. He also showed how Soft Set Theory is free from parameterization inadequacy syndrome of Fuzzy Set Theory, Rough Set Theory, Probability Theory etc. Many of the established paradigms appear as special cases of Soft Set Theory. In 2003, P.K.Maji, R.Biswas and A.R.Roy [6] studied the theory of soft sets initiated by Molodstov. They defined equality of two soft sets, subset and super set of a soft set, complement of a soft set, null soft set, and absolute soft set with examples. Soft binary operations like AND, OR and also the operations of union, intersection were also defined. In 2005, Pei and Miao [8] and Chen et al. [3] improved the work of Maji et al. [4, 6]. In 2008, M.Irfan Ali, Feng Feng, Xiaoyan Liu,Won Keun Min, M.Shabir [2] gave some new notions such as the restricted intersection, the restricted union, the restricted difference and the extended intersection of two soft sets along with a new notion of complement of a soft set. In recent times, researches have contributed a lot towards fuzzification of Soft Set Theory. Maji et al. [5] introduced some properties regarding fuzzy soft union, intersection, complement of a fuzzy soft set, DeMorgan Law etc. These results were further revised and improved by Ahmad and Kharal [1]. They defined arbitrary fuzzy soft union and intersection and proved DeMorgan Inclusions and DeMorgan Laws in Fuzzy Soft Set Theory.

Dusmanta Kumar Sut et al, Int. J. Comp. Tech. Appl., Vol 2 (5), 1160-1176

IJCTA | SPT-OCT 2011 Available [email protected]

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In this paper we give the proof of some propositions introduced by Ahmad and Kharal [1] and support them with examples. We further give some more propositions regarding fuzzy soft union and intersection and support these propositions with proof and examples. Definition 1. [7] A pair (F, E) is called a soft set (over U) if and only if F is a mapping of E into the set of all subsets of the set U. In other words, the soft set is a parameterized family of subsets of the set U. Every set EF ∈εε ),( , from this family may be considered as the set of ε - elements of the soft set (F, E), or as the set of ε - approximate elements of the soft set. Example 1. Let { }4321 ,,, ccccU = be the set of four cars under consideration and

{ }Luxurious)(5),nologyModernTech(4,Efficient) Fuel(3),Beautiful(2),costly(1 eeeeeE = be the set of parameters and A = { e1,e2,e3} ⊆ E . Then (F, A) = { F(e1) = { c1,c4},F(e2) = { c1,c2,c4},F(e3) = { c3}} is the soft set representing the ‘attractiveness of the car’ which Mr. X is going to buy. We can represent this soft set in a tabular form as shown below [4].This style of representation will be useful for storing a soft set in a computer memory. U 1e 2e 3e

1c 1 1 0

2c 0 1 0

3c 0 0 1

4c 1 1 0

Definition 2. [5] A pair (F, A) is called a fuzzy soft set over U where )(~: UPAF → is a mapping from A into )(~ UP . Example 2. Let { }4321 ,,, ccccU = be the set of four cars under consideration and

{ }Luxurious)(5),nologyModernTech(4,Efficient) Fuel(3),Beautiful(2),costly(1 eeeeeE = be the set of parameters and A = { e1,e2,e3}⊆ E . Then (F, A) = { F(e1) = { c1/0.7,c2/0.1,c3/0.2,c4/0.6}, F(e2) = { c1/0.8,c2/0.6,c3/0.1,c4/0.5}, F(e3) = { c1/0.1,c2/0.2,c3/0.7,c4/0.3}} is the fuzzy soft set representing the ‘attractiveness of the car’ which Mr. X is going to buy. Definition 3. [1] Let U be a universe and E a set of attributes. Then the pair (U, E) denotes the collection of all fuzzy soft sets on U with attributes from E and is called a fuzzy soft class. Definition 4. [5] For two fuzzy soft sets (F, A) and (G, B) in a fuzzy soft class (U, E), we say that (F, A) is a fuzzy soft subset of ( G, B), if (i) BA ⊆ , (ii) For all A∈ε , ( ) ( )εε GF ≤ and is written as (F , A) ⊆~ ( G, B).



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Example 3. Let { }4321 ,,, ccccU = be the set of four cars under consideration and

{ }Luxurious)(5),nologyModernTech(4,Efficient) Fuel(3),Beautiful(2),costly(1 eeeeeE = be the set of parameters , A = { e1,e2,e3} ⊆ E and B = { e1,e2,e3,e5} ⊆ E. Then (F, A) = { F(e1) = { c1/0.7,c2/0.1,c3/0.2,c4/0.6},

F(e2) = { c1/0.8,c2/0.6,c3/0.1,c4/0.5}, F (e3) = { c1/0.1,c2/0.2,c3/0.7,c4/0.3}} is the fuzzy soft set representing the ‘attractiveness of the car’ which Mr. X is going to buy and (G, B) = { G(e1) = { c1/0.7,c2/0.2,c3/0.2,c4/0.7},

G(e2) = { c1/0.9,c2/0.6,c3/0.5,c4/1}, G(e3) = { c1/0.3,c2/0.2,c3/0.8,c4/0.3},

G(e5) ={ c1/0.1,c2/0.2,c3/0.7,c4/0.3}} is the fuzzy soft set representing the ‘attractiveness of the car’ which Mr. Y is going to buy. Here BA ⊆ , and for all A∈ε , ( ) ( )εε GF ≤ . Thus (F, A) ⊆~ (G, B). Definition 5. [5]

The complement of a fuzzy soft set (F, A) is denoted by (F, A) c and is defined by ( )cAF , = (F c, A),

where F c : A → )(~ UP is a mapping given by ( ) ( )( )cc FF σσ ¬= for all ∈σ A. Example 4. Let { }4321 ,,, ccccU = be the set of four cars under consideration and

{ }Luxurious)(5),nologyModernTech(4,Efficient) Fuel(3),Beautiful(2),costly(1 eeeeeE = be the set of parameters and A = { e1,e2,e3}⊆ E . Then (F, A) = { F(e1) = { c1/0.7, c2/0.1, c3/0.2, c4/0.6},

F(e2) = { c1/0.8, c2/0.6, c3/0.1, c4/0.5}, F (e3) = {c1/0.1, c2/0.2, c3/0.7, c4/0.3}} is the fuzzy soft set representing the ‘attractiveness of the car’ which Mr. X is going to buy. Here

( )cAF , = { F c ( ¬ e1) = { c1/0.3,c2/0.9,c3/0.8,c4/0.4}, F c ( ¬ e2) = { c1/0.2,c2/0.4,c3/0.9,c4/0.5}, F c ( ¬ e3) = { c1/0.9,c2/0.8,c3/0.3,c4/0.7}}

Definition 6. [5]

Union of two fuzzy soft sets (F, A) and (G, B) in a soft class (U, E) is a fuzzy soft set (H, C) where BAC ∪= and C∈∀ε ,

∩∈∨−∈−∈

=BAxGFABxGBAxF

H if ),()( if ),( if ),(

)(εε

εε

ε

and is written as ( ) ( ) ( )CHBGAF ,,~, =∨ .



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{ }Luxurious)(5),nologyModernTech(4,Efficient) Fuel(3),Beautiful(2),costly(1 eeeeeE = be the set of parameters , A = { e1,e2,e3} ⊆ E and B = { e1,e2,e3,e5} ⊆ E. We consider the fuzzy soft sets (F, A) = { F(e1) = { c1/0.9,c2/0.1,c3/0.4,c4/0.6},

F(e2) = { c1/1,c2/0,c3/0.9,c4/0.5}, F (e3) = { c1/0.8,c2/0.2,c3/0.7,c4/0.6}} and (G, B) = { G(e1) = { c1/0.7,c2/0.2,c3/0.2,c4/0.7},

G(e2) = { c1/0.9,c2/0.6,3/0.5,c4/1}, G(e3) = { c1/0.3,c2/0.2,c3/0.8,c4/0.3},

G(e5) ={ c1/0.1,c2/0.2,c3/0.7,c4/0.3}}

Then ( ) ( ) ( )CHBGAF ,,~, =∨ , where C = A ∪ B = {e1,e2,e3,e5} and (H, C) = { H(e1) = { c1/0.9,c2/0.2,c3/0.4,c4/0.7},

H(e2) = { c1/1,c2/0.6,c3/0.9,c4/1}, H(e3) = { c1/0.8,c2/0.2,c3/0.8,c4/0.6},

H(e5) ={ c1/0.1,c2/0.2,c3/0.7,c4/0.3}} Definition 7. [5]

Intersection of two fuzzy soft sets (F, A) and (G, B) in a soft class (U, E) is a fuzzy soft set (H, C) where BAC ∩= and C∈∀ε , )(or )()( εεε GFH = (as both are same fuzzy set) and is written as ( ) ( ) ( )CHBGAF ,,~, =∧ . Ahmad and Kharal [1] pointed out that generally )(εF or )(εG may not be identical. Moreover in order to avoid the degenerate case, he proposed that BA∩ must be non-empty and thus revised the above definition as follows. Definition 8. [1]

Let (F, A) and (G, B) be two fuzzy soft sets in a soft class (U, E) with φ≠∩ BA .Then Intersection of two fuzzy soft sets (F, A) and (G, B) in a soft class (U, E) is a fuzzy soft set (H,C) where BAC ∩= and

C∈∀ε , )()()( εεε GFH ∧= . We write ( ) ( ) ( )CHBGAF ,,~, =∧ . Example 6. For the two fuzzy soft sets (F, A) and (G, B) given in Example 5, ( ) ( )BGAF ,~, ∧ = (H, C), where C = A ∩ B = { e1,e2,e3} and (H, C) = { H(e1) = { c1/0.7,c2/0.1,c3/0.2,c4/0.6},

H(e2) = { c1/0.9,c2/0,c3/0.5,c4/0.5}, H (e3) = { c1/0.3,c2/0.2,c3/0.7,c4/0.3}} 2. Some Propositions on Fuzzy Soft Union and Intersection.

Ahmad and Kharal [1] gave some propositions on fuzzy soft union and intersection. Here we give the proof of those propositions along with some additional propositions.

Let (F1, A1), (F2, A2) and (F3, A3) be three fuzzy soft sets in a soft class (U, E).



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(i) Commutative Property. (i) ( ) ( ) ( ) ( )11222211 ,~,,~, AFAFAFAF ∧=∧ (ii) ( ) ( ) ( ) ( )11222211 ,~,,~, AFAFAFAF ∨=∨

Proof. (i) Let ( ) ( ) ( )112211 ,,~, CHAFAF =∧ Where 211 AAC ∩= and ,1C∈∀ε )()()( 211 εεε FFH ∧=

Again, let ( ) ( ) ( )221122 ,,~, CHAFAF =∧ Where 122 AAC ∩= and ,2C∈∀ε )()()( 122 εεε FFH ∧=

Clearly 21 CC = and 21 HH = .

Thus ( ) ( ) ( ) ( )11222211 ,~,,~, AFAFAFAF ∧=∧ . (ii) Let ( ) ( ) ( )112211 ,,~, CHAFAF =∨ Where 211 AAC ∪= and

∩∈∧−∈−∈

=∈∀

2121

122

211

11 if )()(

if )( if )(

)(,AAFF

AAFAAF

HCεεε

εεεε

εε

Again let ( ) ( ) ( )221122 ,,~, CHAFAF =∨

Where 122 AAC ∪= and

∩∈∧−∈

−∈=∈∀

1212

211

122

22 if )()(

if )( if )(

)(,AAFF

AAFAAF

HCεεε

εεεε

εε .

Thus 21 CC = and 21 HH =

Hence ( ) ( ) ( ) ( )11222211 ,~,,~, AFAFAFAF ∨=∨ . (ii) Associative Property. (i) ( ) ( ) ( )( ) ( ) ( )( ) ( )3,3

~2,2

~1,13,3

~2,2

~1,1 AFAFAFAFAFAF ∧∧=∧∧

(ii) ( ) ( ) ( )( ) ( ) ( )( ) ( )3,3~

2,2~

1,13,3~

2,2~

1,1 AFAFAFAFAFAF ∨∨=∨∨ Proof. (i) Let ( ) ( ) ( )113322 ,,~, CHAFAF =∧ and ( ) ( ) ( )221111 ,,~, CHCHAF =∧ Where 321 AAC ∩= and )()()(, 3211 εεεε FFHC ∧=∈∀

And ( )321112 AAACAC ∩∩=∩= ,

)(1)(1)(2,2 εεεε HFHC ∧=∈∀

( ))()()( 321 εεε FFF ∧∧=

Again, let ( ) ( ) ( )332211 ,,~, CHAFAF =∧ and ( ) ( ) ( )443333 ,,~, CHAFCH =∧

Where 213 AAC ∩= and )()()(, 2133 εεεε FFHC ∧=∈∀

And ( ) 321334 AAAACC ∩∩=∩= , )()()(, 3344 εεεε FHHC ∧=∈∀

( ) )()()( 321 εεε FFF ∧∧=



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It is clear that 42 CC = and 42 HH =

Thus ( ) ( ) ( )( ) ( ) ( )( ) ( )332211332211 ,~,~,,~,~, AFAFAFAFAFAF ∧∧=∧∧ (ii) Let ( ) ( ) ( )113322 ,,~, CHAFAF =∨ and ( ) ( ) ( )221111 ,,~, CHCHAF =∨ Where 321 AAC ∪= and ( )321112 AAACAC ∪∪=∪= ……………..(1) Case I.

When 11 CA −∈ε = ( )321 AAA ∪− , )()( 12 εε FH = …..(2) Case – II

When 11 AC −∈ε = ( ) 132 AAA −∪ ,

)()( 12 εε HH = =

∩∈∨−∈−∈

3232

233

322

if )()( if )( if )(

AAFFAAFAAF

εεεεεεε

……..(3)

Case – III When 11 CA ∩∈ε = ( )321 AAA ∪∩ ,

)()()( 112 εεε HFH ∨= Now, 11 CA ∩ = ( )321 AAA ∪∩

= ( ) ( ) ( )( )3223321 AAAAAAA ∩∪−∪−∩

= ( )( ) ( )( ) ( )( )321231321 AAAAAAAAA ∩∩∪−∩∪−∩

When ( )321 AAA −∩∈ε , )()( 21 εε FH = , by (3) Thus )()()( 112 εεε HFH ∨= )()( 21 εε FF ∨= ……………..(4) When ( )231 AAA −∩∈ε , )()( 31 εε FH = , by (3) Thus )()()( 112 εεε HFH ∨= )()( 31 εε FF ∨= ……………..(5) When ( )321 AAA ∩∩∈ε , )()()( 321 εεε FFH ∨= , by (3) Thus )()()( 112 εεε HFH ∨= ( ))()()( 321 εεε FFF ∨∨= ……………..(6) Again, Let ( ) ( ) ( )332211 ,,~, CHAFAF =∨

and ( ) ( ) ( )443333 ,,~, CHAFCH =∨ Where 213 AAC ∪= and ( ) 321334 AAAACC ∪∪=∪= ……………..(7) Case – I When 33 AC −∈ε = ( ) 321 AAA −∪ ,

)()( 34 εε HH = =

∩∈∨−∈

−∈

2121

122

211

if )()( if )( if )(

AAFFAAF

AAF

εεεεεεε

……………..(8)

Case – II When 33 CA −∈ε = ( )213 AAA ∪− = ( ) ( )2313 AAAA −∩− )()( 34 εε FH = ……………..(9)



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Case – III When 33 AC ∩∈ε = ( ) 321 AAA ∩∪

= ( ) ( ) ( )( ) 3211221 AAAAAAA ∩∩∪−∪−

= ( )( ) ( )( ) ( )( )321312321 AAAAAAAAA ∩∩∪∩−∪∩−

)()()( 334 εεε FHH ∨=

When ∈ε ( ) 321 AAA ∩− )()( 13 εε FH = , by (8) Thus )(4 εH )()( 33 εε FH ∨=

i.e. )(4 εH )()( 31 εε FF ∨= ……………..(10)

When ∈ε ( ) 312 AAA ∩−

)()( 23 εε FH = , by (8) Thus )(4 εH )()( 33 εε FH ∨=

i.e. )(4 εH )()( 32 εε FF ∨= ……………..(11) When ∈ε ( ) 321 AAA ∩∩

)()()( 213 εεε FFH ∨= , by (8) Thus )(4 εH )()( 33 εε FH ∨=

i.e. )(4 εH ( ) )()()( 321 εεε FFF ∨∨= ……………..(12) Now, when ∈ε ( )321 AAA ∪− = ( ) ( )3121 AAAA −∩−

)()( 12 εε FH = , by (2) And )()( 14 εε FH = , by (8)

When ∈ε ( ) 132 AAA −∪ = ( ) ( ) ( )( ) 1322332 AAAAAAA −∩∪−∪−

= ( )( ) ( )( ) ( )( )132123132 AAAAAAAAA −∩∪−−∪−− When ( ) 132 AAA −−∈ε

)()( 22 εε FH = , by (3) )()( 24 εε FH = , by (8)

{ ( ) 132 AAA −− = ( ) cc AAA 132 ∩∩

= ( )cc AAA 132 ∩∩

= ( )cAAA 132 `∪∩

= ( )132 AAA ∪−

= ( ) ( )1232 AAAA −∩− } When ( ) 123 AAA −−∈ε

)()( 32 εε FH = , by (3) )()( 34 εε FH = , by (9)



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{ ( ) 123 AAA −− = ( ) cc AAA 123 ∩∩

= ( )cc AAA 213 ∩∩

= ( )cAAA 213 ∪∩

= ( )213 AAA ∪−

= 33 CA − }

When ( ) 132 AAA −∩∈ε

{ ( ) 132 AAA −∩ = ( ) cAAA 132 ∩∩

= ( ) 312 AAA c ∩∩ = ( ) 312 AAA ∩− }

)()()( 322 εεε FFH ∨= , by (3) )()()( 324 εεε FFH ∨= , by (11)

When ( )321 AAA ∪∩∈ε ( ) ( ) ( )( )3223321 AAAAAAA ∩∪−∪−∩=

( )( ) ( )( ) ( )( )321231321 AAAAAAAAA ∩∩∪−∩∪−∩=

When ( )321 AAA −∩∈ε ( )cAAA 321 ∩∩=

( ) cAAA 321 ∩∩=

( ) 321 AAA −∩=

)(2 εH )()( 21 εε FF ∨= , by (4) and )(4 εH )()( 21 εε FF ∨= , by (8)

When ( )231 AAA −∩∈ε ( )cAAA 231 ∩∩=

( )cAAA 213 ∩∩=

( )cAAA 213 ∩∩=

( )213 AAA −∩=

)()()( 314 εεε FFH ∨= , by (5) )()()( 312 εεε FFH ∨= , by (10)

When ( )321 AAA ∩∩∈ε

)(2 εH ( ))()()( 321 εεε FFF ∨∨= , by (6) And )(4 εH ( ) )()()( 321 εεε FFF ∨∨= , by (12) Thus ( ) ( ) ( )( ) ( ) ( )( ) ( )332211332211 ,~,~,,~,~, AFAFAFAFAFAF ∨∨=∨∨ (iii) Idempotent Property. (i) ( ) ( ) ( )111111 ,,~, AFAFAF =∧ (ii) ( ) ( ) ( )111111 ,,~, AFAFAF =∨



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Proof. (i) Let ( ) ( ) ( )CHAFAF ,,~, 1111 =∧

Where 111 AAAC =∩= and )()()()(, 1111 εεεεε FFFHAC =∧==∈∀

Thus ( ) ( ) ( )111111 ,,~, AFAFAF =∧ (ii) Let ( ) ( ) ( )CHAFAF ,,~, 1111 =∨

Where 111 AAAC =∪= and )()()()(, 1111 εεεεε FFFHAC =∨==∈∀ Thus ( ) ( ) ( )111111 ,,~, AFAFAF =∧ (iv) Absorption Property. (i) ( ) ( ) ( )( ) ( )11221111 ,,~,~, AFAFAFAF =∨∧ (ii) ( ) ( ) ( )( ) ( )11221111 ,,~,~, AFAFAFAF =∧∨ Proof. (i) Let ( ) ( ) ( )112211 ,,~, CHAFAF =∨ and

( ) ( ) ( )221111 ,,~, CHAHAF =∧

Where 211 AAC ∪= and

∩∈∨−∈−∈

=∈∀

2121

122

211

11 if )()( if )( if )(

)(,AAFFAAFAAF

HCεεεεεεε

εε

Also 1112 AAAC =∩= and )()()(, 1122 εεεε HFHC ∧=∈∀

Now if 21 AA −∈ε , )()()()( 1112 εεεε FFFH =∧=

And if 21 AA ∩∈ε , )()()( 112 εεε HFH ∧= = ( ))()()( 211 εεε FFF ∨∧ )(1 εF=

Thus ( ) ( ) ( )( ) ( )11221111 ,,~,~, AFAFAFAF =∨∧

(ii) Let ( ) ( ) ( )112211 ,,~, CHAFAF =∧ and ( ) ( ) ( )221111 ,,~, CHAHAF =∨

Where 211 AAC ∩= and )()()(, 2111 εεεε FFHC ∧=∈∀

Also 1112 AAAC =∪= and )()()(, 1122 εεεε HFHC ∨=∈∀

= ( ))()()( 211 εεε FFF ∧∨ )(1 εF=

Thus ( ) ( ) ( )( ) ( )11221111 ,,~,~, AFAFAFAF =∧∨ (v) Distributive Property. (i) ( ) ( ) ( )( ) ( ) ( )( ) ( ) ( )( )3,3

~1,1

~2,2

~1,13,3

~2,2

~1,1 AFAFAFAFAFAFAF ∧∨∧=∨∧

(ii) ( ) ( ) ( )( ) ( ) ( )( ) ( ) ( )( )3,3~

1,1~

2,2~

1,13,3~

2,2~

1,1 AFAFAFAFAFAFAF ∨∧∨=∧∨ Proof. (i) Let ( ) ( ) ( )112211 ,,~, CHAFAF =∧ and

( ) ( ) ( )223311 ,,~, CHAFAF =∧

Where 211 AAC ∩= , 312 AAC ∩=

Let ( ) ( ) ( )3,32,2~

1,1 CHCHCH =∨ where

213 CCC ∪= ( ) ( )3121 AAAA ∩∪∩=

( )321 AAA ∪∩=



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Case I. When 21 CC −∈ε ( ) ( )3121 AAAA ∩−∩= ( )321 AAA −∩= ,

)(3 εH )(1 εH= )()( 21 εε FF ∧= Case - II When 12 CC −∈ε ( ) ( )2131 AAAA ∩−∩= ( )231 AAA −∩= , )(3 εH )(2 εH=

)()( 31 εε FF ∧= Case - III When 21 CC ∩∈ε ( ) ( )3121 AAAA ∩∩∩= 321 AAA ∩∩= , )(3 εH )()( 21 εε HH ∨=

( ) ( ))()()()( 3121 εεεε FFFF ∧∨∧=

( ))()()( 321 εεε FFF ∨∧=

Let ( ) ( ) ( )443322 ,,~, CHAFAF =∨ and ( ) ( ) ( )554411 ,,~, CHCHAF =∧

Where ,324 AAC ∪=

415 CAC ∩= ( )321 AAA ∪∩=

( ) ( ) ( )( )3223321 AAAAAAA ∩∪−∪−∩=

( )( ) ( )( ) ( )( )321231321 AAAAAAAAA ∩∩∪−∩∪−∩=

3C=

When )(, 432 εε HAA −∈ )(2 εF=

)(, 423 εε HAA −∈ )(3 εF=

)(, 432 εε HAA ∩∈ )()( 32 εε FF ∨= Case I.

,5C∈∀ε When ( )321 AAA −∩∈ε , )(5 εH )()( 41 εε HF ∧=

)()( 21 εε FF ∧= (When )()(, 2432 εεε FHAA =−∈ ) = )(3 εH

Case II. ,5C∈∀ε When ( )231 AAA −∩∈ε ,

)(5 εH )()( 41 εε HF ∧=

)()( 31 εε FF ∧= (When )()(, 3423 εεε FHAA =−∈ ) = )(3 εH

Case III. ,5C∈∀ε When ( )321 AAA ∩∩∈ε ,

)(5 εH )()( 41 εε HF ∧=

( ))()()( 321 εεε FFF ∨∧= (When )()()(, 32432 εεεε FFHAA ∨=∩∈ ) = )(3 εH



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It is clear from above that ( ) ( ) ( )( ) ( ) ( )( ) ( ) ( )( )33112211332211 ,~,~,~,,~,~, AFAFAFAFAFAFAF ∧∨∧=∨∧ (ii) This can be proved in a similar way. (vi) (i) ( ) ( ) ( ) ( ) ( ) ( )222211112211 ,~,~,,,~,~, AFAFAFAFAFAF ⊆∧⊆∧

(ii) ( ) ( ) ( )221111 ,~,~, AFAFAF ∨⊆ , ( ) ( ) ( )221122 ,~,~, AFAFAF ∨⊆ Proof . (i) Let ( ) ( ) ( )CHAFAF ,,~, 2211 =∧ , where 21 AAC ∩=

and )()()(, 21 εεεε FFHC ∧=∈∀ Now, 121 AAAC ⊆∩= and )()()()(, 12121 εεεεε FFFHAAC ≤∧=∩=∈∀

Thus ( ) ( ) ( )112211 ,~,~, AFAFAF ⊆∧ . The other result can also be proved in a similar way. (ii) Let ( ) ( ) ( )CHAFAF ,,~, 2211 =∨ , where 21 AAC ∪=

And C∈∀ε ,

∩∈∨−∈

−∈=

2121

122

211

if )( )( if )( if )(

)(AAFF

AAFAAF

Hεεεεεεε

ε

Now, CA ⊆1 and

∩∈∨∈

=∈∀2121

2111 if )()(

if )()(,

AAεFF-AAεF

HAεε

εεε

Thus )()(, 11 εεε FHA ≤∈∀ Thus ( ) ( ) ( )112211 ,~,~, AFAFAF ⊆∧ . The other result can also be proved in a similar way. (vii) (i) ( ) ( ) ( ) ( ) ( )1122112211 ,,~,,~, AFAFAFAFAF =∧⇒⊆

(ii) ( ) ( ) ( ) ( ) ( )2222112211 ,,~,,~, AFAFAFAFAF =∨⇒⊆ Proof . (i) Let ( ) ( )2211 ,~, AFAF ⊆ . Then 21 AA ⊆ and )()(, 211 εεε FFA ≤∈∀

Now, let ( ) ( ) ( )CHAFAF ,,~, 2211 =∧ . Then 121 AAAC =∩= , as 21 AA ⊆ And 1AC =∈∀ε , )()()()( 121 εεεε FFFH =∧= as )()(, 211 εεε FFA ≤∈∀

Thus ( ) ( ) ( ) ( ) ( )1122112211 ,,~,,~, AFAFAFAFAF =∧⇒⊆ (ii) Let ( ) ( )2211 ,~, AFAF ⊆ . Then 21 AA ⊆ and )()(, 211 εεε FFA ≤∈∀

Now, let ( ) ( ) ( )CHAFAF ,,~, 2211 =∨ .

Then 221 AAAC =∪= , as 21 AA ⊆ And 2AC =∈∀ε , )()()()( 221 εεεε FFFH =∨= as )()(, 211 εεε FFA ≤∈∀

Thus ( ) ( ) ( ) ( ) ( )2222112211 ,,~,,~, AFAFAFAFAF =∨⇒⊆



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Definition 9. [1] Let ( ){ }IiAF ii ∈=ℑ |, be a family of fuzzy soft sets in a fuzzy soft class (U, E).Then the union of fuzzy soft sets in ℑ is a fuzzy soft set ( )CH , , where i

iAC ∪= and for all C∈ε ,

( )iii

AH ,)( εε ∆∨=

Where ( )

∉∈

=∆i

iiii A

AFA

εϕεε

ε if , if ),(

,


{ }Luxurious)(5),nologyModernTech(4,Efficient) Fuel(3),Beautiful(2),costly(1 eeeeeE = be the set of parameters and A1 = { e1,e2,e3} ⊆ E , A2 = { e1,e4} ⊆ E , A3 = { e1,e2,e3,e4} ⊆ E. We consider three fuzzy soft sets (F1, A1), (F2, A2) and (F3, A3) as follows. (F1, A1) = { F1(e1) = { c1/0.7,c2/0.1,c3/0.2,c4/0.6},

F1(e2) = { c1/0.8,c2/0.6,c3/0.1,c4/0.5}, F1(e3) = { c1/0.1,c2/0.2,c3/0.7,c4/0.3}} (F2, A2) = { F2(e1) = { c1/0.4,c2/0.6,c3/0.1,c4/0.6},

F2(e4) = { c1/0.8,c2/0.1,c3/0.4,c4/0.5}} (F3, A3) = { F3(e1) = { c1/0.2,c2/0.3,c3/0.5,c4/0},

F3(e2) = { c1/0.3,c2/0.9,c3/0.5,c4/0.5}, F3(e3) = { c1/0.1,c2/0.2,c3/0.6,c4/0.3}, F3(e4) = { c1/0.8,c2/0.3,c3/0.5,c4/0.1}}

Thus ( ) ( ) ( ) ( )CHAFAFAF ,,~,~, 332211 =∨∨ , where 321 AAAC ∪∪= = { e1,e2,e3,e4} and (H, C) = { H(e1) = { c1/0.7,c2/0.6,c3/0.5,c4/0.6},

H(e2) = { c1/0.8,c2/0.9,c3/0.5,c4/0.5}, H(e3) = { c1/0.1,c2/0.2,c3/0.7,c4/0.3}, H(e4) = { c1/0.8,c2/0.3,c3/0.5,c4/0.5}} Definition 10. [1] Let ( ){ }IiAF ii ∈=ℑ |, be a family of fuzzy soft sets in a fuzzy soft class (U, E),with ϕ≠∩ i

iA

.Then the intersection of fuzzy soft sets in ℑ is a fuzzy soft set ( )CH , ,where ii

AC ∩= and for all C∈ε ,

)()( εε ii

FH ∧=



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{ }Luxurious)(5),nologyModernTech(4,Efficient) Fuel(3),Beautiful(2),costly(1 eeeeeE = be the set of parameters and A1 = { e1,e2,e3} ⊆ E , A2 = { e1,e4} ⊆ E , A3 = { e1,e2,e3,e4} ⊆ E. We consider three fuzzy soft sets (F1, A1), (F2, A2) and (F3, A3) as follows. (F1, A1) = { F1(e1) = { c1/0.7,c2/0.1,c3/0.2,c4/0.6},

F1(e2) = { c1/0.8,c2/0.6,c3/0.1,c4/0.5}, F1(e3) = { c1/0.1,c2/0.2,c3/0.7,c4/0.3}} (F2, A2) = { F2(e1) = { c1/0.4,c2/0.6,c3/0.1,c4/0.6},

F2(e4) = { c1/0.8,c2/0.1,c3/0.4,c4/0.5}} (F3, A3) = { F3(e1) = { c1/0.2,c2/0.3,c3/0.5,c4/0},

F3(e2) = { c1/0.3,c2/0.9,c3/0.5,c4/0.5}, F3(e3) = { c1/0.1,c2/0.2,c3/0.6,c4/0.3},

F3(e4) = { c1/0.8,c2/0.3,c3/0.5,c4/0.1}}

Thus ( ) ( ) ( ) ( )CHAFAFAF ,,~,~, 332211 =∧∧ , where 321 AAAC ∩∩= = { e1} and (H, C) = { H(e1) = { c1/0.2,c2/0.3,c3/0.1,c4/0}} Following the definitions 9 and 10, we now propose the following two propositions. Proposition 4. Let ( ){ }IiAF ii ∈=ℑ |, be a family of fuzzy soft sets in a fuzzy soft class (U, E).Then

( ) ( )iii

ii AFAFIi ,~~,, ∨⊆∈∀

Proof. Let ( ) ( )CHAF ii

i,,~ =∨ , where i

iAC ∪= and C∈∀α , Ii ∈

( )iii

AH ,)( αα ∆∨= , where

( )

∉∈

=∆i

iiii A

AFA

αϕαα

α if if )(

,

Clearly ii

i AA ∪⊆ i.e CAi ⊆ and iA∈∀α , Ii ∈ , ( )iii

i AF ,)( αα ∆∨≤ i.e. ( )αα HFi ≤)(

Thus ( ) ( )iii

ii AFAFIi ,~~,, ∨⊆∈∀

Example 9. For the three fuzzy soft sets (F1, A1), (F2, A2) and (F3, A3) given in Example 7, we see

that ( ) ( ) ( ) ( )CHAFAFAF ,,~,~, 332211 =∨∨ , where 321 AAAC ∪∪= = { e1,e2,e3,e4} and (H, C) = { H(e1) = { c1/0.7,c2/0.6,c3/0.5,c4/0.6},

H(e2) = { c1/0.8,c2/0.9,c3/0.5,c4/0.5}, H(e3) = { c1/0.1,c2/0.2,c3/0.7,c4/0.3},

H(e4) = { c1/0.8,c2/0.3,c3/0.5,c4/0.5}}

Thus CAi ⊆ and C∈∀ε , )()( εε HFi ≤ for 3,2,1=i



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Proposition 5. Let ( ){ }IiAF ii ∈=ℑ |, be a family of fuzzy soft sets in a fuzzy soft class (U, E).Then

( ) ( )iiiii

AFAFIi ,~,~, ⊆∧∈∀

Proof. Suppose that, ( ) ( )CHAF iii

,,~ =∧ , where ii

AC ∩= and ,C∈∀α Ii ∈

)()( αα ii

FH ∧=

Now, iii

AAC ⊆∩= and C∈∀α , Ii ∈ , )()()( ααα iii

FFH ≤∧=

Thus CFH i ∈∀≤ ααα )()( and hence the result follows. Example 10.

For the three fuzzy soft sets (F1, A1) , (F2, A2) and (F3, A3) given in Example 8, we see that

( ) ( ) ( ) ( )CHAFAFAF ,,~,~, 332211 =∧∧ , where 321 AAAC ∩∩= = { e1} and (H, C) = { H(e1) = { c1/0.2,c2/0.3,c3/0.1,c4/0}.

Thus iAC ⊆ and C∈∀ε , )()( εε iFH ≤ for 3,2,1=i . Ahmad and Kharal [1] proved DeMorgan Laws for soft sets (F, A) and (G, A) in a soft class (U, E). He further generalized DeMorgan Laws for a family of fuzzy soft sets in a fuzzy soft class (U, E) as follows- Theorem. [1] Let ( ){ }IiAFi ∈=ℑ |, be a family of fuzzy soft sets in a fuzzy soft class (U, E).Then one has the following -

( ) ( )c

ii

ci

iAFAF

∨=∧ ,~,~.1

( ) ( )cii

ci

iAFAF ,~,~.2 ∨=

∧

Here we give the proof of this theorem. Proof.

1.We have, ( )cii

AF ,~∧ = i∧~ ( c

iF , A)

= ( H , A), say

Where ∈∀¬α A, )( α¬H

)( α¬∧= ci

iF .........................(1)

Again suppose that ( )AFii

,~∨

( )AI ,= .Then

( )c

ii

AF

∨ ,~ = ( )cAI ,

= ( cI , A), where



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)( α¬cI

[ ]cI )(α= c

ii

F

∨= )(α

∈∀¬α A, we have, )( α¬cI = c

ii

F

∨ )(α

= )( α¬∧ ci

iF .........................(2)

From (1) and (2), we get the desired result.

2. We have ( )cii

AF ,~∨ =i∨~ ( c

iF , A)

= ( H , A), say

where ∈∀¬α A, )( α¬H

)( α¬∨= ci

iF .........................(1)

Again suppose that ( )AFii

,~∧ ( )AI ,= .Then

( )c

ii

AF

∧ ,~ = ( )cAI ,

= ( cI , A), where

)( α¬cI

[ ]cI )(α= c

ii

F

∧= )(α

∈∀¬α A, we have, )( α¬cI = c

ii

F

∧ )(α

= )( α¬∨ ci

iF .........................(2)

From (1) and (2), we get the desired result. Example 12. Let { }4321 ,,, ccccU = be the set of four cars under consideration and

{ }Luxurious)(5),nologyModernTech(4,Efficient) Fuel(3),Beautiful(2),costly(1 eeeeeE = be the set of parameters and A = { e1,e4}⊆ E , We consider three fuzzy soft sets (F1, A), (F2, A) and (F3, A) as follows. (F1, A) = { F1(e1) = { c1/0.7,c2/0.1,c3/0.2,c4/0.6},

F1(e4) = { c1/0.8,c2/0.6,c3/0.1,c4/0.5}}

(F1, A) c = { F1 c ( ¬ e1) = { c1/0.3,c2/0.9,c3/0.8,c4/0.4}, F1 c ( ¬ e4) = { c1/0.2,c2/0.4,c3/0.9,c4/0.5}} (F2, A) = { F2 (e1) = { c1/0.4,c2/0.6,c3/0.1,c4/0.6},

F2(e4) = { c1/0.8,c2/0.1,c3/0.4,c4/0.5}}

(F2, A) c = { F2 c ( ¬ e1) = { c1/0.6,c2/0.4,c3/0.9,c4/0.4}, F2 c ( ¬ e4) = { c1/0.2,c2/0.9,c3/0.6,c4/0.5}}



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(F3, A) = { F3(e1) = { c1/0.2,c2/0.3,c3/0.5,c4/0},

F3(e4) = { c1/0.8,c2/0.3,c3/0.5,c4/0.1}}

(F3, A) c = { F3 c ( ¬ e1) = { c1/0.8,c2/0.7,c3/0.5,c4/1}, F3 c ( ¬ e4) = { c1/0.2,c2/0.7,c3/0.5,c4/0.9}}

Thus ( ) ( ) ( )AFAFAF ,~,~, 321 ∨∨ = ( )AH ,1 , where (H1, A) = { H1(e1) = { c1/0.7,c2/0.6,c3/0.5,c4/0.6},

H1(e4) = { c1/0.8,c2/0.6,c3/0.5,c4/0.5}}

(H1, A) c = { H1 c ( ¬ e1) = { c1/0.3,c2/0.4,c3/0.5,c4/0.4},

H1 c ( ¬ e4) = { c1/0.2,c2/0.4,c3/0.5,c4/0.5}}

and ( ) ( ) ( )AFAFAF ,~,~, 321 ∧∧ = ( )AH ,2 , where (H2, A) = { H2(e1) = { c1/0.2,c2/0.1,c3/0.1,c4/0},

H2(e4) = { c1/0.8,c2/0.1,c3/0.1,c4/0.1}}

(H2,A) c = { H2 c ( ¬ e1) = { c1/0.8,c2/0.9,c3/0.9,c4/1},

H2 c ( ¬ e4) = { c1/0.2,c2/0.9,c3/0.9,c4/0.9}}

Now, ( ) ( ) ( )ccc AFAFAF ,~,~, 321 ∨∨

= ( cF1 , A) ∨~ ( cF2 , A) ∨~ ( cF3 , A) = ( I , A) = { I( ¬ e1) = { c1/0.8,c2/0.9,c3/0.9,c4/1},

I( ¬ e4) = { c1/0.2,c2/0.9,c3/0.9,c4/0.9}}

And ( ) ( ) ( )ccc AFAFAF ,~,~, 321 ∧∧

= ( cF1 , A) ∧~ ( cF2 , A) ∧~ ( cF3 , A) = ( J , A) = {J( ¬ e1) = { c1/0.3,c2/0.4,c3/0.5,c4/0.4},

J( ¬ e4) = { c1/0.2,c2/0.4,c3/0.5,c4/0.5}} It is clear that

( ) ( ) ( )ccc AFAFAF ,~,~, 321 ∧∧ = ( ) ( ) ( )( )cAFAFAF ,~,~, 321 ∨∨ and

( ) ( ) ( )ccc AFAFAF ,~,~, 321 ∨∨ = ( ) ( ) ( )( )cAFAFAF ,~,~, 321 ∧∧ 3. Conclusion. The Soft Set Theory of Molodstov [7] offers a general mathematical tool for dealing with uncertain and vague objects. At present, work on the extension of soft set theory is progressing rapidly. Maji et al. [5] proposed the concept of fuzzy soft set and developed some properties of fuzzy soft sets and in recent years the researchers have contributed a lot towards the fuzzification of Soft Set Theory. This paper contributes some more properties regarding fuzzy soft union and intersection and support these propositions with proof and examples. We further give the proof of some propositions introduced by Ahmad and Kharal [1] and support them with examples. We hope that our findings will help enhancing this study on fuzzy soft sets.



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References 1. Ahmad B and Kharal Athar, On Fuzzy Soft Sets, Advances in Fuzzy Systems, Volume 2009.

2. Ali M.I, Feng F, Liu XY, Min WK, Shabir M (2009) “On some new operations in soft set theory”. Computers and Mathematics with Applications 57:1547–1553

3. Chen D, Tsang E C C, Yeung D S, and Wang X, “The parameterization reduction of soft sets and its applications,” Computers & Mathematics with Applications, vol. 49, no.5-6, pp. 757–763, 2005.

4. Maji P K and Roy A R, “An Application of Soft Sets in A Decision Making Problem”, Computers and Mathematics with Applications 44 (2002) 1077-1083

5. Maji P K, Biswas R and Roy A R, “Fuzzy Soft Sets”, Journal of Fuzzy Mathematics, Vol 9 , no.3,pp.-589-602,2001

6. Maji P K and Roy A R, “Soft Set Theory”, Computers and Mathematics with Applications 45 (2003) 555 – 562

7. Molodstov D A, Soft Set Theory - First Result, Computers and Mathematics with Applications 37 (1999) 19-31

8. Pei D and Miao D, “From soft sets to information systems,” in Proceedings of the IEEE International Conference on Granular Computing, vol. 2, pp. 617–621, 2005.



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On Union and Intersection of Fuzzy Soft Set - IJCTAijcta.com/documents/volumes/vol2issue5/ijcta2011020504.pdf · On Union and Intersection of Fuzzy Soft Set . Tridiv Jyoti Neog1,