Vol 6, No 12 (2015)

Available online throug

ISSN 2229 – 5046

RELATIONS ON GENERALIZED FUZZY SOFT SETS

MANASH JYOTI BORAH*

Department of Mathematics, Bahona College, Jorhat, Assam-785101, India.

(Received On: 05-12-15; Revised & Accepted On: 30-12-15)

ABSTRACT

In this paper, some properties of generalized fuzzy soft equivalence relation have been discussed and finally

applications of generalized fuzzy soft relation in decision making problem have been shown. Also we present the definition of union and intersection between two generalized fuzzy soft relations.

Keywords: Fuzzy soft sets, Generalized fuzzy soft set, union, intersection, generalized fuzzy soft relations.

2010 Mathematics Subject Classification: 03E72; 03B52.

1. INTRODUCTION

In real world, the complexity generally arises from uncertainty in the form of ambiguity. Uncertainty may arise due to partial information about the problem, or due to information which is not fully reliable, or due to receipt of information from more than one source. Fuzzy set theory, Rough set theory, Vague set, theory of probability are a mathematical tool to handle the uncertainty arising due to vagueness. In 1965, Zadeh [9] introduced the notion of fuzzy set theory. Later in 1999, Moldstov [7] initiated the novel concept of soft set theory. Maji, Biswas and Roy [5] studied the theory of soft sets initiated by Moldstov [7] in 2003 and put forward definitions of equality of two soft sets , subset and super set of a soft set, complement of a soft set. In 2009 Majumder and Samanta [6] initiated another important part, known as Generalized Fuzzy Soft Set Theory. While generalizing the concept of Fuzzy Soft Sets, Majumder and Samanta [6] considered the same set of parameter. But in 2012, Borah et.al [3] considering generalized fuzzy soft sets different sets of parameters. In 2014, Borah [4] introduced generalized fuzzy soft sets in new directions.

In this paper we give the proof of some propositions of soft relation and support them with examples. We also discussed irreflexive, equivalence relation, complement, union, intersection, Identity and some algebraic properties of fuzzy soft relation.

2.PRELIMINARIES

Definition 2.1[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 F

( )

ε

,

ε

∈

E, from this family may be considered as the set of

ε

-approximate elements of the soft set.

Definition 2.2[5]: A pair (F, A) is called a fuzzy soft set over U where F: A→P̃(U) is a mapping from A into P̃(U) .

Definition 2.3 [8]: A binary operation ∗: [0, 1] × [0, 1] → [0, 1] is continuous t - norm if * satisfies the following conditions.

(i) * is commutative and associative (ii) * is continuous

(iii)a * 1= a ∀𝑎𝑎 ∈ [0, 1]

(iv)a * b≤ 𝑐𝑐 ∗ 𝑑𝑑 whenever 𝑎𝑎 ≤ 𝑐𝑐, 𝑏𝑏 ≤ 𝑑𝑑 and 𝑎𝑎, 𝑏𝑏, 𝑐𝑐, 𝑑𝑑 ∈ [0, 1]

An example of continuous t - norm is a * b = ab.

Corresponding Author: Manash Jyoti Borah*

Definition 2.4 [8]: A binary operation ⟡: [0, 1] × [0, 1] → [0, 1] is continuous t-conorm if ⟡satisfies the following conditions:

(i) ⟡ is commutative and associative (ii) ⟡ is continuous

(iii)a⟡ 0 = a ∀𝑎𝑎 ∈ [0, 1]

(iv)a ⟡ b≤ 𝑐𝑐 ⟡ 𝑑𝑑 whenever 𝑎𝑎 ≤ 𝑐𝑐, 𝑏𝑏 ≤ 𝑑𝑑 and 𝑎𝑎, 𝑏𝑏, 𝑐𝑐, 𝑑𝑑 ∈ [0, 1]

An example of continuous t - conorm is a * b = a + b – ab.

Definition 2.5 [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) A

⊆

B

(ii) For all

ε

∈

A, F(

ε

) ⊆ 𝐺𝐺(

ε

) and is written as (F, A) ⊆� (G, B)

𝐃𝐃𝐃𝐃𝐃𝐃𝐃𝐃𝐃𝐃𝐃𝐃𝐃𝐃𝐃𝐃𝐃𝐃𝐃𝐃𝐃𝐃. 𝟔𝟔[𝟓𝟓]: Union of two fuzzy soft sets (F, A) and (G, B) in a fuzzy soft class (U, E) is a fuzzy soft set (H, C) where

C

=

A

∪

B

and

∀

ε

∈

C

,











∩

∈

∪

−

∈

−

∈

=

B

A

G

F

A

B

G

B

A

F

H

ε

ε

ε

ε

ε

ε

ε

ε

if

),

(

)

(

if

),

(

if

),

(

)

(

and is written as

(

F

,

A

) (

∪

~

G

,

B

) (

=

H

,

C

)

.

Definition 2.7[5]: The complement of a fuzzy soft set (F, A) is denoted by (𝐹𝐹, 𝐴𝐴)𝑐𝑐and is defined by

(𝐹𝐹, 𝐴𝐴)𝑐𝑐_{= (𝐹𝐹}𝑐𝑐_{,¬𝐴𝐴 ) where 𝐹𝐹}𝑐𝑐_{:¬𝐴𝐴 →P̃(U) is a mapping given by 𝐹𝐹}𝑐𝑐_{(𝜎𝜎) = (𝐹𝐹(−𝜎𝜎))}𝑐𝑐 _{for all 𝜎𝜎 ∈ ¬𝐴𝐴}

Definition 2.8 [6]: Let F_µ be a GFSS over (U, E). Then the complement of F_µ, denoted by F_µc and is defined by

δ

µ G

F c = , where

δ

(e)=

µ

c(e)andG(e)=Fc(e),∀e∈E.

Definition 2.9 [3]: Let X ={x1,x2,x3,...,xn} be the universal set of elements and E={e1,e2,e3,...,em}be the

set of parameters. Let A⊆Eand F:A→IUand λbe a fuzzy subset of A i.e. λ:A→I=[0,1], where IUis the collection of all fuzzy subsets of U. Let F:A→IU×Ibe a function defined as follows:

(

( ), ( )

)

)

(e F e e

F_λA = λ , where U

I e

F( )∈ .Then F_λAis called a generalized fuzzy soft set (GFSS) over (U, E).

Here for each parameter , ( i)

A i F e

e λ indicates not only degree of belongingness of the elements of U in F(ei)but also

degree of preference of such belongingness which is represented byλ(ei).

Definition 2.10[4]: The union of two GFSS F_λAand G_µBover (U, E) is denoted by F_λA ∪~G_µBand defined by GFSS

I I B A

H_δA∪B: ∪ → U× such that for each e∈A∪B andA,B⊆E

Where











∩

∈

∗

◊

−

∈

−

∈

=

∪

B

A

e

e

e

e

G

e

F

A

B

e

e

e

G

B

A

e

e

e

F

e

H

A B

if

)),

(

)

(

),

(

)

(

(

if

)),

(

),

(

(

if

)),

(

),

(

(

)

(

µ

λ

µ

λ

δ

Definition 2.11[4]: The intersection of two GFSS F_λAand G_µBover (U, E) is denoted by F_λA∩~G_µBand defined by GFSS K_δA∩B:A∩B→IU ×Isuch that for each e∈A∩B andA,B⊆E K_δA∩B(e)=

(

K(e),δ(e)

)

,

Wh e

re

K e

( )

=

F e

( ) * ( ), ( )

G e

δ

e

=

λ

( )

e

◊

µ

( )

e

. In order to avoid degenerate case, we assume here thatA∩B≠ϕ.

3. MAIN RESULTS

Definition 3.1: Let F_λA,G_µBbe a GFSS over (U, E). Then generalized fuzzy soft relation (in short GFSR) R from

A

F_λ to G_µB is a function R:A×B→IU×I defined by

. ) , ( ), ( ~ ) ( ) ,

(e e F e G e e e A B

Definition 3.2: If R is a GFSR from F_λA to G_µB then R−1is defined as R−1(e_a,e_b)=R(e_b,e_a),∀(e_a,e_b)∈A×B

Definition 3.3: Let R and S be two generalized fuzzy soft relations from FλA to GµB andGµB to HδC respectively.

Then the composition



of R and S is defined by )

, ( ~ ) , ( ) , )(

(RS ea ec =Rea eb S eb ec

Definition 3.4: A generalized fuzzy soft relation R on F_λA is said to be generalized fuzzy soft reflexive relation if

)

,

(

)

,

(

e

_a

e

_b

R

e

_a

e

_a

R

⊆

and

R

(

e

_b

,

e

_a

)

⊆

R

(

e

_a

,

e

_a

),

∀

e

_a

,

e

_b

∈

A

Definition 3.5: A generalized fuzzy soft relation R on F_λA is said to be generalized fuzzy soft symmetric relation if

A

e

e

e

e

R

e

e

R

(

_a

,

_b

)

=

(

_b

,

_a

),

∀

_a

,

_b

∈

Definition 3.6: A generalized fuzzy soft relation R on F_λA is said to be generalized fuzzy soft transitive relation if

R

R

R



⊆

Definition 3.7: A generalized fuzzy soft relation R on F_λA is said to be generalized fuzzy soft irraflexive relation if

)

,

(

)

,

(

e

_a

e

_b

R

e

_a

e

_a

R

⊄

and

R

(

e

_b

,

e

_a

)

⊄

R

(

e

_a

,

e

_a

),

∀

e

_a

,

e

_b

∈

A

Definition 3.8: A generalized fuzzy soft relation R on F_λA is said to be generalized fuzzy soft equivalence relation if it is reflexive, symmetric and transitive.

Proposition 3.1: Ifa generalized fuzzy soft relation R is reflexive then

R

−1 is also reflexive.

Proof:

R

−1

(

e

_a

,

e

_b

)

=

R

(

e

_b

,

e

_a

)

⊆

R

(

e

_a

,

e

_a

)

=

R

−1

(

e

_a

,

e

_a

)

And

R

−1

(

e

_b

,

e

_a

)

=

R

(

e

_a

,

e

_b

)

⊆

R

(

e

_a

,

e

_a

)

=

R

−1

(

e

_a

,

e

_a

)

Hence

R

−1 is reflexive.

Proposition 3.2: A generalized fuzzy soft relation R is Symmetric if and only if

R

=

R

−1

.

Proof: Let R is symmetric, then

)

,

(

)

,

(

)

,

(

1

b a a

b b

a

e

R

e

e

R

e

e

e

R

−

=

=

1

,

So

R

=

R

−

Conversely, let

R

=

R

−1 then

)

,

(

)

,

(

)

,

(

e

_a

e

_b

R

1

e

_a

e

_b

R

e

_b

e

_a

R

=

−

=

So R is Symmetric.

Proposition 3.3: A generalized fuzzy soft relation R is Symmetric if and only if the

R

−1is symmetric.

Proof: Let R is symmetric, then

)

,

(

)

,

(

)

,

(

)

,

(

1

1

a b b

a a

b b

a

e

R

e

e

R

e

e

R

e

e

e

R

−

=

=

=

−

So R is Symmetric.

Conversely, let

R

−1is symmetric. Then

)

,

(

)

,

(

)

,

(

)

,

(

)

(

)

,

(

1 1 1 1

1

b a b

a a

b b

a b

a

e

R

e

e

R

e

e

R

e

e

R

e

e

e

R

−

=

− −

=

−

=

−

=

Proposition: 3.4. If a generalized fuzzy soft relation R is transitive then

R

−1 is also transitive.

Proof:

1

1 1 1 1

( ,

)

( ,

)

(

)( ,

)

( , )

( ,

)

( ,

)

( , )

( , )

( ,

)

(

)( ,

)

a b b a b a b c c a c a b c

a c c b a b

R

e e

R e e

R R e e

R e e

R e e

R e e

R e e

R

e e

R

e e

R

R

e e

−

− − − −

=

⊇

=

∩

=

=

=









Therefore

R

−1



R

−1

⊆

R

−1.

Proposition 3.5: If a generalized fuzzy soft relation R is transitive then

R



R

is also transitive.

)

,

)(

(

)

,

)(

(

)

,

)(

(

)

,

(

)

,

(

)

,

)(

(

R



R

e

_a

e

_b

=

R

e

_a

e

_c



R

e

_c

e

_b

⊇

R



R

e

_a

e

_c

∩

R



R

e

_c

e

_b

=

R



R



R



R

e

_a

e

_c

)

,

)(

(

)

,

)(

(

)

,

)(

(

)

,

(

)

,

(

)

,

)(

(

R



R

e

_a

e

_b

=

R

e

_a

e

_c



R

e

_c

e

_b

⊇

R



R

e

_a

e

_c

∩

R



R

e

_c

e

_b

=

R



R



R



R

e

_a

e

_c

Therefore

R



R



R



R

⊆

R



R

.

Proposition 3.6: If R and S are on symmetric generalized fuzzy soft relation F_λAthen

R



S

is symmetric on F_λAand then if and only if

R



S

=

S



R

.

Proof: Since R and S are symmetric. Therefore

R

−1

=

R

,

S

−1

=

S

.

Now

1 1 1

)

(

R



S

−

=

S

−



R

− . So

R



S

is symmetric.

Conversely,

S

R

R

S

R

S

S

R



)

−1

=

−1



−1

=



=



(

.

Hence

R



S

is symmetric.

Proposition 3.7: If R is symmetric and transitive generalized fuzzy soft relation then R is reflexive.

Proof:

),

,

)(

(

)

,

(

e

a

e

a

R

R

e

a

e

a

R

⊇



Since R is transitive.

=

R

(

e

a

,

e

b

)



R

(

e

b

,

e

a

)

=

R

(

e

a

,

e

b

)

∩

R

(

e

a

,

e

b

)

=

R

(

e

a

,

e

b

)

Similarly,

R

(

e

a

,

e

a

)

⊇

R

(

e

b

,

e

a

)

Hence the proof.

Proposition: 3.8: If S be a generalized fuzzy soft equivalence relation onF_λAthen

S



S

is also equivalence relation.

Proof: Since S is a generalized fuzzy soft equivalence relation therefore S is reflexive, symmetric and transitive.

Now

)

,

(

)

,

)(

(

S



S

e

_a

e

_b

⊆

S

e

_a

e

_a ∴ S



S is reflexive.

Again

(

S S e e



)( ,

_{a b}

)

=

S e e

( ,

_{a c}

)





S e e

( ,

_{c b}

)

=

S e e

( ,

_{c b}

)

∩



S e e

( ,

_{a c}

)

since S is symmetric

( ,

_{b c}

)

( ,

_{c a}

)

(

)( ,

_{b a}

)

S e e

S e e

S S e e

=



=



∴S



S is symmetric.

(

S S e e



)( ,

_{a b}

)

=

S e e

( ,

_{a c}

)





S e e

( ,

_{c b}

)

⊇

(

S S e e



)( ,

_{a c}

)

∩



(

S S e e



)( ,

_{c b}

)

since S is symmetric

(

S S S S e e

)( ,

a b

)

=

_{  }

∴S



S ∘ 𝑆𝑆



𝑆𝑆 ⊆S



S

∴S



S is transitive.

Proposition: 3.9: If S be a generalized fuzzy soft equivalence relation onF_λAthen

S

−1is also equivalence relation. Proof:

),

,

(

)

,

(

)

,

(

)

,

(

1 1 a a a a a b b

a

e

S

e

e

S

e

e

S

e

e

e

S

−

=

⊆

=

− since S is reflexive

),

,

(

)

,

(

)

,

(

)

,

(

1 1 a a a a b a a

b

e

S

e

e

S

e

e

S

e

e

e

S

−

=

⊆

=

−

𝑆𝑆¯1 is reflexive

),

,

(

)

,

(

)

,

(

)

,

(

1 1 a b b a a b b

a

e

S

e

e

S

e

e

S

e

e

e

S

−

=

=

=

− since S is symmetric

𝑆𝑆¯1 is symmetric. 1

( ,

_{a b}

)

( ,

_{b a}

)

(

)( ,

_{b a}

)

( ,

_{b c}

)

( ,

_{c a}

)

S

−

e e

=

S e e

⊇

S S e e



=

S e e

∩



S e e

1 1

( ,

_{c b}

)

( ,

_{a c}

)

( ,

_{b c}

)

( ,

_{c a}

)

S e e

S e e

S

−

e e

S

−

e e

=





=

∩



1 1

(

S

S

)( ,

e e

a b

)

− −

=

_

𝑆𝑆¯1



𝑆𝑆¯1⊆𝑆𝑆¯1

∴𝑆𝑆¯1 is transitive

Thus 𝑆𝑆¯1 is equivalence relation.

Definition 3.9: Let R and S be two generalized fuzzy soft relations from A to B. Then

R

≤

S

is defined as follows

))

,

(

)

,

(

),

,

(

)

,

(

(

)

,

(

)

,

(

e

a

e

b

S

e

a

e

b

F

R

e

a

e

b

G

S

e

a

e

b R

e

a

e

b S

e

a

e

b

R

≤

⇔

≤

λ

≥

µ

,

∀

(

e

a

,

e

b

)

∈

A

×

B

Definition 3.10: Let R be a generalized fuzzy soft relation from A to B. Then complement of R is denoted by

R

Cand it is defined as follows

))

,

(

),

,

(

(

)

,

(

a b

C R b a C R b a C

e

e

e

e

F

e

e

R

=

λ

,

∀

(

e

a

,

e

b

)

∈

A

×

B

Definition 3.11: Let R and S be two generalized fuzzy soft relations from A to B. Then RS,RS are defined as follows

))

,

(

)

,

(

),

,

(

)

,

(

(

)

,

)(

~

(

R



S

e

_a

e

_b

=

F

_R

e

_a

e

_b

◊

G

_S

e

_a

e

_b

λ

_R

e

_a

e

_b

∗

µ

_S

e

_a

e

_b

))

,

(

)

,

(

),

,

(

)

,

(

(

)

,

)(

~

(

R

∩

S

e

_a

e

_b

=

F

_R

e

_a

e

_b

∗

G

_S

e

_a

e

_b

λ

_R

e

_a

e

_b

◊

µ

_S

e

_a

e

_b

,

∀

(

e

_a

,

e

_b

)

∈

A

×

B

Proposition 3.10: Let R and S be two elements of GFSR

(

A

×

B

)

(i)

R

≤

R





S P

,

≤

R





S

(ii)

R

≥

R





S P

,

≥

R





S

1 1

(iii)

R

≤ ⇒

S

R

−

≤

S

−

1 1 1

(iv) (

R



S

)

−

=

R

−



S

−

1 1 1

(v) (

R



S

)

−

=

R

−



S

−

Proof:

(i) We have

))

,

(

)

,

(

),

,

(

)

,

(

(

)

,

)(

~

(

R



S

e

_a

e

_b

=

F

_R

e

_a

e

_b

◊

G

_S

e

_a

e

_b

λ

_R

e

_a

e

_b

∗

µ

_S

e

_a

e

_b

By definition of t-norm and t-co norm

)

,

(

)

,

(

)

,

(

_{a b S a b R a b}

R

e

e

G

e

e

F

e

e

F

◊

≥

)

,

(

)

,

(

)

,

(

_{a b S a b R a b}

R

e

e

µ

e

e

λ

e

e

λ

∗

≤

Therefore, by definition of

≤

)

,

(

)

,

)(

~

(

R



S

e

_a

e

_b

≥

R

e

_a

e

_b

Hence

S

R

(ii) similar to (i)

(iii)

R

(

e

_a

,

e

_b

)

≤

S

(

e

_a

,

e

_b

)

⇒

(

F

_R

(

e

_a

,

e

_b

)

≤

G

_S

(

e

_a

,

e

_b

),

λ

_R

(

e

_a

,

e

_b

)

≥

µ

_S

(

e

_a

,

e

_b

))

))

,

(

)

,

(

),

,

(

)

,

(

(

F

_R−1

e

_b

e

_a

≤

G

_S−1

e

_b

e

_{a R}−1

e

_b

e

_a

≥

_S−1

e

_b

e

_a

⇒

λ

µ

1 1 1 1

)

,

(

)

,

(

− − − −

_≤

_⇒

_≤

⇒

R

e

_b

e

_a

S

e

_b

e

_a

R

S

HENCE

R

≤

S

⇒

R

−1

≤

S

−1

(iv)

(

R

∩

~

S

)

−1

(

e

_a

,

e

_b

)

=

(

R



~

S

)(

e

_b

,

e

_a

)

=

(

F

_R

(

e

_b

,

e

_a

)

∗

G

_S

(

e

_b

,

e

_a

),

λ

_R

(

e

_b

,

e

_a

)

◊

µ

_S

(

e

_b

,

e

_a

))

)

,

)(

~

(

))

,

(

)

,

(

),

,

(

)

,

(

(

1 1 1 1 1 1 b a b a S b a R b a S b a R

e

e

S

R

e

e

e

e

e

e

G

e

e

F

− −

=

◊

∗

=

− − − −



µ

λ

Hence 1 1 1 )

(RS − =R− S− _.

(v) Similar to (iv).

Proposition 4.11: Let P, R and S be three elements of GFSR

(

A

×

B

)

(i) If

R

≤

S

then

R



P

≤

S



P

(ii) If

R

≤

S

then

R



R

≤

S



S

Proof:

(i)

R

(

e

a

,

e

b

)

≤

S

(

e

a

,

e

b

)

⇒

(

F

R

(

e

a

,

e

b

)

≤

G

S

(

e

a

,

e

b

),

λ

R

(

e

a

,

e

b

)

≥

µ

S

(

e

a

,

e

b

))

))

,

(

)

,

(

)

,

(

)

,

(

),

,

(

)

,

(

)

,

(

)

,

(

(

F

R

e

a

e

b

∗

H

P

e

b

e

c

≤

G

S

e

a

e

b

∗

H

P

e

b

e

c

λ

R

e

a

e

b

◊

δ

P

e

b

e

c

≥

µ

S

e

a

e

b

◊

δ

P

e

a

e

b

⇒

)

,

(

~

)

,

(

)

,

(

~

)

,

(

e

_a

e

_b

P

e

_b

e

_c

S

e

_a

e

_b

P

e

_b

e

_c

R



≤



⇒

P

S

P

R

e

e

P

S

e

e

P

R



_{a c}

≤



_{a c}

⇒



≤



⇒

(

)(

,

)

(

)(

,

)

HENCE

R

≤

S

then

R



P

≤

S



P

(ii) Similar to (i).

Definition 3.12: A generalized fuzzy soft relation is said to be a generalized Identity fuzzy soft relation, denoted by

∆

_I, if

(

( ), ( )

)

)

(e F e I e

I =

∆ Where

F

(

e

)

=

1

,

I

(

e

)

=

0

,

∀

e

∈

A

⊆

E

It is clear from our definition that the generalized fuzzy soft absolute set is also not unique in our way, it would depend upon the set of parameters under consideration.

Proposition 3.12:

(i)

∆

_I

=

∆

_I−1

(ii)

R



∆

_I

=

∆

_I



R

=

R

Proof:

(i) It is obvious. (ii)

(

R



∆

I

)(

e

a

,

e

c

)

)

,

(

~

)

,

(

e

_a

e

_{b I}

e

_b

e

_c

R

∆

=



)

0

)

,

(

,

1

)

,

(

(

∗

◊

=

F

_R

e

_a

e

_b

λ

_R

e

_a

e

_b

))

,

(

),

,

(

(

F

R

e

a

e

b

λ

R

e

a

e

b

=

)

,

(

e

a

e

c

R

=

R

R



∆

_I

=

Similarly we prove that

4. AN APPLICATION OF GENERALIZED FUZZY SOFT RELATION IN A DECISION MAKING PROBLEM

In this section, we present an application of generalized fuzzy soft relation in a candidate’s selection of an interview.

4.1: Algorithm:

Step-1: Construct the generalized fuzzy soft sets for each set of parameters.

Step-2: Construct the generalized fuzzy soft relation.

Step-3: Compute

F

(

e

)

*

λ

(

e

)

Step-4: Compute the total Score of each Candidate.

Step-5: Find the lowest Score.

Suppose U={c1,c2,c3,c4,c5}be the five candidates appearing in an interview and

E

=

{ (enterprising), (confident),

e

1

e

2

3

(wiling to take risk)}

e

be the set of parameters. Suppose two experts X and Y take interview of the five candidates and they consider different set of parameters and the following generalized fuzzy soft sets are constructed accordingly,

{

/

0

.

3

,

/

0

.

7

,

/

0

.

5

,

/

0

.

2

,

/

0

.

1

}

,

0

.

6

)

(

)

(

e

₁

C

₁

C

₂

C

₃

C

₄

C

₅

F

_λA

=

,

{

/

0

.

8

,

/

0

.

3

,

/

0

.

1

,

/

0

.

9

,

/

0

.

6

}

,

0

.

2

)

(

)

(

e

₂

C

₁

C

₂

C

₃

C

₄

C

₅

F

_λA

=

And

{

/

0

.

4

,

/

0

.

2

,

/

0

.

3

,

/

0

.

7

,

/

0

.

3

}

,

0

.

4

)

(

)

(

e

1

C

1

C

2

C

3

C

4

C

5

G

_µB

=

,

{

/

0

.

6

,

/

0

.

4

,

/

0

.

3

,

/

0

.

4

,

/

0

.

6

}

,

0

.

7

)

(

)

(

e

3

C

1

C

2

C

3

C

4

C

5

G

_µB

=

Now, the following generalized fuzzy soft relations are constructing

{

}

(

)

1 1 1 2 3 4 5

( , )

/ 0.12,

/ 0.14,

/ 0.15,

/ 0.14,

/ 0.03 , 0.76

A B

R

_δ ∩

e e

=

C

C

C

C

C

{

}

(

)

1 3 1 2 3 4 5

( ,

)

/ 0.18,

/ 0.28,

/ 0.15,

/ 0.08,

/ 0.06 , 0.88

A B

R

_δ ∩

e e

=

C

C

C

C

C

.

{

}

(

)

2 1 1 2 3 4 5

( , )

/ 0.32,

/ 0.06,

/ 0.03,

/ 0.63,

/ 0.18 , 0.52

A B

R

_δ ∩

e e

=

C

C

C

C

C

{

}

(

)

2 3 1 2 3 4 5

( ,

)

/ 0.48,

/ 0.12,

/ 0.03,

/ 0.36,

/ 0.36 , 0.76

A B

R

_δ ∩

e e

=

C

C

C

C

C

1

c

c

₂

c

3

c

4

c

5

)

,

(

e

₁

e

₁ 0.0912 0.1064 0.114 0.1064 0.0228

)

,

(

e

₁

e

₃ 0.1584 0.2464 0.132 0.0704 0.0528

)

,

(

e

₂

e

₁ 0.1664 0.0312 0.0156 0.3276 0.0936

)

,

(

e

2

e

3 0.3648 0.0912 0.0228 0.2736 0.2736 Table-1

Candidates Total Score

1

c

0.7808

2

c

0.4752

3

c

0.2844

4

c

0.778

5

c

0.4428

Table-2

5.CONCLUSION

In this paper we discussed the fuzzy soft set and extend the concept of relation in fuzzy soft set theory. We also establish some properties of fuzzy soft relation such as symmetric, transitive, reflexive, irreflexive, equivalence etc. Also we present the definition of union and intersection between two fuzzy soft sets with a new approach.

6. ACKNOWLEDGEMENT

The author would like to thank the fund for University Grant Commission (UGC), India, for funding the research elaborated on in this paper. [Grant No. F.5-60/2013-14/MRP/NERO/17424]

7.REFERENCES

1. B. Ahmad and A.Kharal, “On Fuzzy Soft Sets”, Advances in Fuzzy systems, Volume 2009.

2. M.J.Borah, T.J.Neog, D.K.Sut, “Relation on Fuzzy Soft Set”, Journal of Mathematics Computer Science, Vol. 2(3), pp.515-534.

3. M.J.Borah, T.J.Neog and D.K.Sut, “Some Results on Generalized Fuzzy Soft Sets”, International Journal of Computer Technology and Application, Vol. 3(2), 583-591.

4. M.J.Borah, “Generalized Fuzzy soft Sets: A New Approach”, International Journal of Mathematical Archive, Vol. 5(7), pp.18-26.

5. P.K.Maji, R.Biswas and A.R.Roy, “Fuzzy soft Sets “, Journal of Fuzzy Mathematics, Vol9 , no.3, pp.-589-602, 2001.

6. P. Majumdar, S. K. Samanta, “Generalized Fuzzy Soft Sets”, Computers and Mathematics with Applications, 59(2010), pp.1425-1432.

7. D.A.Molodstov, “Soft Set theory-First Result”, Computer and Mathematics with Applications 37 (1999) 19-31.

8. B.Schweirer, A. Sklar, “Statistical metric space”, Pacific journal of mathematics 10(1960), 314-334. 9. L.A.Zadeh, “Fuzzy Sets”, Information and Control, 8(1965), pp 338-353.

Source of support: University Grant Commission (UGC), India. Conflict of interest: None Declared